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DDDD Basic Photography in 180 Days Book IX - Composition Editor: Ramon F. aeroramon.com Contents

1 Day 1 1 1.1 Composition (visual arts) ...... 1 1.1.1 Elements of design ...... 1 1.1.2 Principles of organization ...... 3 1.1.3 Compositional techniques ...... 4 1.1.4 Example ...... 8 1.1.5 See also ...... 9 1.1.6 References ...... 9 1.1.7 Further reading ...... 9 1.1.8 External links ...... 9 1.2 Elements of art ...... 9 1.2.1 Form ...... 10 1.2.2 Line ...... 10 1.2.3 Color ...... 10 1.2.4 Space ...... 10 1.2.5 Texture ...... 10 1.2.6 See also ...... 10 1.2.7 References ...... 10 1.2.8 External links ...... 11

2 Day 2 12 2.1 Visual design elements and principles ...... 12 2.1.1 Design elements ...... 12 2.1.2 Principles of design ...... 15 2.1.3 See also ...... 18 2.1.4 Notes ...... 18 2.1.5 References ...... 18 2.1.6 External links ...... 18

3 Day 3 20 3.1 Shape ...... 20 3.1.1 Classiﬁcation of simple shapes ...... 20 3.1.2 Shape in geometry ...... 21

i ii CONTENTS

3.1.3 Shape analysis ...... 23 3.1.4 Similarity classes ...... 23 3.1.5 See also ...... 23 3.1.6 References ...... 24 3.1.7 External links ...... 24

4 Day 4 25 4.1 Color ...... 25 4.1.1 Physics of color ...... 27 4.1.2 Perception ...... 29 4.1.3 Associations ...... 33 4.1.4 Spectral colors and color reproduction ...... 33 4.1.5 Additive coloring ...... 35 4.1.6 Subtractive coloring ...... 35 4.1.7 Structural color ...... 36 4.1.8 Mentions of color in social media ...... 37 4.1.9 Additional terms ...... 37 4.1.10 See also ...... 37 4.1.11 References ...... 38 4.1.12 External links and sources ...... 38

5 Day 5 39 5.1 Texture (visual arts) ...... 39 5.1.1 Three varieties of texture ...... 39 5.1.2 Hypertexture ...... 40 5.1.3 Examples of physical texture ...... 40 5.1.4 Examples of visual texture ...... 41 5.1.5 See also ...... 42 5.1.6 Notes ...... 42 5.1.7 References ...... 42

6 Day 6 43 6.1 Lightness ...... 43 6.1.1 Relationship between lightness, value, and relative luminance ...... 43 6.1.2 Other psychological eﬀects ...... 44 6.1.3 See also ...... 45 6.1.4 References ...... 45 6.1.5 External links ...... 45

7 Day 7 48 7.1 Space ...... 48 7.1.1 Philosophy of space ...... 48 7.1.2 Mathematics ...... 56 CONTENTS iii

7.1.3 Physics ...... 56 7.1.4 Spatial measurement ...... 58 7.1.5 Geographical space ...... 58 7.1.6 In psychology ...... 59 7.1.7 See also ...... 59 7.1.8 References ...... 59 7.1.9 External links ...... 60

8 Day 8 61 8.1 Rule of thirds ...... 61 8.1.1 Use ...... 61 8.1.2 History ...... 61 8.1.3 See also ...... 64 8.1.4 References ...... 65

9 Day 9 66 9.1 Golden ratio ...... 66 9.1.1 Calculation ...... 67 9.1.2 History ...... 69 9.1.3 Applications and observations ...... 71 9.1.4 Mathematics ...... 74 9.1.5 Pyramids ...... 80 9.1.6 Disputed observations ...... 81 9.1.7 See also ...... 81 9.1.8 References and footnotes ...... 82 9.1.9 Further reading ...... 85 9.1.10 External links ...... 86 9.2 Rabatment of the rectangle ...... 101 9.2.1 Theory ...... 101 9.2.2 Practice ...... 101 9.2.3 Examples ...... 102 9.2.4 References ...... 102 9.2.5 External links ...... 103 9.3 Headroom (photographic framing) ...... 103 9.3.1 Examples ...... 104 9.3.2 See also ...... 104 9.3.3 References ...... 104 9.3.4 Further reading ...... 105

10 Day 10 106 10.1 Perspective (graphical) ...... 106 10.1.1 Overview ...... 107 iv CONTENTS

10.1.2 Types of perspective ...... 113 10.1.3 Methods of construction ...... 116 10.1.4 Example ...... 116 10.1.5 Limitations ...... 117 10.1.6 See also ...... 117 10.1.7 Notes ...... 118 10.1.8 References ...... 118 10.1.9 Further reading ...... 119 10.1.10 External links ...... 119

11 Text and image sources, contributors, and licenses 132 11.1 Text ...... 132 11.2 Images ...... 139 11.3 Content license ...... 145 Chapter 1

Day 1

1.1 Composition (visual arts)

In the visual arts, composition is the placement or arrangement of visual elements or ingredients in a work of art, as distinct from the subject. It can also be thought of as the organization of the elements of art according to the principles of art. The composition of a picture is diﬀerent from its subject, which what is shown, whether a moment from a story, a person or a place. Many subjects, for example Saint George and the Dragon, are often shown in art, but using a great range of compositions, even though the two ﬁgures are typically the only ones shown. The term composition means 'putting together' and can apply to any work of art, from music to writing to photography, that is arranged using conscious thought. In the visual arts, composition is often used interchangeably with various terms such as design, form, visual ordering, or formal structure, depending on the context. In graphic design for press and desktop publishing, composition is commonly referred to as page layout.

1.1.1 Elements of design

Main article: Elements of art

The various visual elements, known as elements of design, formal elements, or elements of art, are the vocabulary with which the visual artist composes. These elements in the overall design usually relate to each other and to the whole art work. The elements of design are:

• Line — the visual path that enables the eye to move within the piece

• Shape — areas deﬁned by edges within the piece, whether geometric or organic

• Colour — hues with their various values and intensities

• Texture — surface qualities which translate into tactile illusions

• Tone — Shading used to emphasize form

• Form — 3-D length, width, or depth

• Space — the space taken up by (positive) or in between (negative) objects

• Depth — perceived distance from the observer, separated in foreground, background, and optionally middle ground

1 2 CHAPTER 1. DAY 1

The Art of Painting by Jan Vermeer, noted for his subtle compositions

Line and shape

Lines are optical phenomena that allow the artist to direct the eye of the viewer. The optical illusion of lines do exist in nature and visual arts elements can be arranged to create this illusion. The viewer unconsciously reads near continuous arrangement of diﬀerent elements and subjects at varying distances. Such elements can be of dramatic use in the composition of the image. These could be literal lines such as telephone and power cables or rigging on boats. Lines can derive also from the borders of areas of diﬀering color or contrast, or sequences of discrete elements. Movement is also a source of line, and blur can also create a reaction. Subject lines contribute to both mood and linear perspective, giving the viewer the illusion of depth. Oblique lines convey a sense of movement and angular lines generally convey a sense of dynamism and possibly tension. Lines can also direct attention towards the main subject of picture, or contribute to organization by dividing it into compartments. 1.1. COMPOSITION (VISUAL ARTS) 3

The artist may exaggerate or create lines perhaps as part of their message to the viewer. Many lines without a clear subject point suggest chaos in the image and may conflict with the mood the artist is trying to evoke. Straight left lines create different moods and add affection to visual arts. A line’s angle and its relationship to the size of the frame influence the mood of the image. Horizontal lines, commonly found in landscape photography, can give the impression of calm, tranquility, and space. An image filled with strong vertical lines tends to have the impression of height and grandeur. Tightly angled convergent lines give a dynamic, lively, and active effect to the image. Strongly angled, almost diagonal lines produce tension in the image. The viewpoint of visual art is very important because every different perspective views different angled lines. This change of perspective elicits a different response to the image. By changing the perspective only by some degrees or some centimetres lines in images can change tremendously and a totally different feeling can be transported. Straight lines are also strongly influenced by tone, color, and repetition in relation to the rest of the image. Compared to straight lines, curves provide a greater dynamic influence in a picture. They are also generally more aesthetically pleasing, as the viewer associates them with softness. In photography, curved lines can give graduated shadows when paired with soft-directional lighting, which usually results in a very harmonious line structure within the image.

Colour

Color is characterized by attributes such as hue, brightness, and saturation. Color symbolism assigns additional associations, dependent on culture. For example, white has long suggested purity, but it can also take slightly diﬀerent meanings such as peace, or innocence. However, in some places (for instance, Japan and China) it signiﬁes death.

1.1.2 Principles of organization

Main article: Principles of art

The artist determines what the center of interest (focus in photography) of the art work will be, and composes the elements accordingly. The gaze of the viewer will then tend to linger over these points of interest, elements are arranged with consideration of several factors (known variously as the principles of organization, principles of art, or principles of design) into a harmonious whole which works together to produce the desired statement – a phenomenon commonly referred to as unity. Such factors in composition should not be confused with the elements of art (or elements of design) themselves. For example, shape is an element; the usage of shape is characterized by various principles. Some principles of organization aﬀecting the composition of a picture are:

• Shape and proportion • Positioning/orientation/balance/harmony among the elements • The area within the ﬁeld of view used for the picture ("cropping") • The path or direction followed by the viewer’s eye when they observe the image. • Negative space • Color • Contrast: the value, or degree of lightness and darkness, used within the picture. • Arrangement: for example, use of the golden mean or the rule of thirds • Lines • Rhythm • Illumination or lighting • Repetition (sometimes building into pattern; rhythm also comes into play, as does geometry) • Perspective • Breaking the rules can create tension or unease, yet it can add interest to the picture if used carefully 4 CHAPTER 1. DAY 1

Viewpoint (leading with the eye)

The position of the viewer can strongly influence the aesthetics of an image, even if the subject is entirely imaginary and viewed “within the mind’s eye”. Not only does it influence the elements within the picture, but it also influences the viewer’s interpretation of the subject. For example, if a boy is photographed from above, perhaps from the eye level of an adult, he is diminished in stature. A photograph taken at the child’s level would treat him as an equal, and one taken from below could result in an impression of dominance. Therefore, the photographer is choosing the viewer’s positioning. A subject can be rendered more dramatic when it fills the frame. There exists a tendency to perceive things as larger than they actually are, and filling the frame full fills this psychological mechanism. This can be used to eliminate distractions from the background. In photography, altering the position of the camera can change the image so that the subject has fewer or more distractions with which to compete. This may be achieved by getting closer, moving laterally, tilting, panning, or moving the camera vertically.

1.1.3 Compositional techniques

There are numerous approaches or “compositional techniques” to achieve a sense of unity within an artwork, depending on the goals of the artist. For example, a work of art is said to be aesthetically pleasing to the eye if the elements within the work are arranged in a balanced compositional way.[1] However, there are artists such as Salvador Dalí who aim to disrupt traditional composition and challenge the viewer to rethink balance and design elements within art works. Conventional composition can be achieved with a number of techniques:

Rule of thirds

Main article: Rule of thirds

The rule of thirds is a composition guide that states that arranging the important features of an image on or near the horizontal and vertical lines that would divide the image into thirds horizontally and vertically is visually pleasing. The objective is to stop the subject(s) and areas of interest (such as the horizon) from bisecting the image, by placing them near one of the lines that would divide the image into three equal columns and rows, ideally near the intersection of those lines. The rule of thirds is thought to be a simpliﬁcation of the golden ratio. The golden ratio is thought to have been used by artists throughout history as a composition guide, but there is little evidence to support this claim.

Rule of odds

The “rule of odds” states that by framing the object of interest with an even number of surrounding objects, it becomes more comforting to the eye, thus creates a feeling of ease and pleasure. It is based on the assumption that humans tend to find visual images that reflect their own preferences/wishes in life more pleasing and attractive. The “rule of odds” suggests that an odd number of subjects in an image is more interesting than an even number. Thus if you have more than one subject in your picture, the suggestion is to choose an arrangement with at least three subjects. An even number of subjects produces symmetries in the image, which can appear less natural for a naturalistic, informal composition. An image of a person surrounded/framed by two other persons, for instance, where the person in the center is the object of interest in that image/artwork, is more likely to be perceived as friendly and comforting by the viewer, than an image of a single person with no significant surroundings. 1.1. COMPOSITION (VISUAL ARTS) 5

Rule of thirds: Note how the horizon falls close to the bottom grid line, and how the dark areas are in the left third, the overexposed in the right third.

Rule of space

Main article: Lead room

The rule of space applies to artwork (photography, advertising, illustration) picturing object(s) to which the artist wants to apply the illusion of movement, or which is supposed to create a contextual bubble in the viewer’s mind. This can be achieved, for instance, by leaving white space in the direction the eyes of a portrayed person are looking, or, when picturing a runner, adding white space in front of him rather than behind him to indicate movement.

Simpliﬁcation

Images with clutter can distract from the main elements within the picture and make it diﬃcult to identify the subject. By decreasing the extraneous content, the viewer is more likely to focus on the primary objects. Clutter can also be reduced through the use of lighting, as the brighter areas of the image tend to draw the eye, as do lines, squares and colour. In painting, the artist may use less detailed and deﬁned brushwork towards the edges of the picture.Removing the elements to the focus of the object, taking only the needed components.

Shallow Depth of Field In photography, and also (via software simulation of real lens limitations) in 3D graphics, one approach to achieving simplification is to use a wide aperture when shooting to limit the depth of field. When used properly in the right setting, this technique can place everything that is not the subject of the photograph out of focus. A similar approach, given the right equipment, is to take advantage of the Scheimpflug principle to change the plane of focus. 6 CHAPTER 1. DAY 1

A simple composition with cloud and rooftop that creates asymmetry.

Geometry and symmetry

Related to the rule of odds is the observation that triangles are an aesthetically pleasing implied shape within an image. In a canonically attractive face, the mouth and eyes fall within the corners of the area of an equilateral triangle. Paul Cézanne successfully used triangles in his compositions of still lifes. A triangular format creates a sense of stability and strength.

Creating movement

It is generally thought to be more pleasing to the viewer if the image encourages the eye to move around the image, rather than immediately fixating on a single place or no place in particular. Artists will often strive to avoid creating compositions that feel “static” or “flat” by incorporating movement into the image. In image A the 2 mountains are equally sized and positioned beside each other creating a very static and uninteresting image. In image B the mountains are differently sized and one is placed closer to the horizon, guiding the eye to move from one mountain to the other creating a more interesting and pleasing image. This also feels more natural because in nature objects are rarely the same size and evenly spaced.

Other techniques

• There should be a center of interest or focus in the work, to prevent it becoming a pattern in itself;

• The direction followed by the viewer’s eye should lead the viewer’s gaze around all elements in the work before leading out of the picture;

• The subject should not be facing out of the image;

• Exact bisections of the picture space should be avoided; 1.1. COMPOSITION (VISUAL ARTS) 7

ImageA

ImageB

• Small, high contrast, elements have as much impact as larger, duller elements;

• The prominent subject should be oﬀ-centre, unless a symmetrical or formal composition is desired, and can be balanced by smaller satellite elements

• the horizon line should not divide the art work in two equal parts but be positioned to emphasize either the sky or ground; showing more sky if painting is of clouds, sun rise/set, and more ground if a landscape 8 CHAPTER 1. DAY 1

These principles can be means of a good composition yet they cannot be applied separately but should act together to form a good composition.

• Also, in your work no spaces between the objects should be the same. They should vary in shape and size. That creates a much more interesting image.

1.1.4 Example

These paintings all show the same subject, the Raising of Lazarus, and essentially the same ﬁgures, but have very diﬀerent compositions:

• Duccio, 1310–11

• Geertgen tot Sint Jans, 1480s

• Guercino, c. 1619 1.2. ELEMENTS OF ART 9

• Rembrandt, c. 1630

1.1.5 See also

• Miksang (contemplative photography)

• New Epoch Notation Painting (a notation system for painting)

• Page layout (graphic design)

• CLACL (a computer language for composition)

1.1.6 References

[1] Dunstan, Bernard. (1979). Composing Your Paintings. London, Studio Vista.

1.1.7 Further reading

• Arnheim, Rudolf (1974). Art and Visual Perception: A Psychology of the Creative Eye. University of California Press. ISBN 978-0-520-02613-1.

• Downer, Marion (1947). Discovering Design. Lothrop Lee & Shepard. ISBN 0-688-41266-1.

• Graham, Peter (2004). An Introduction to Painting Still Life. Chartwell Books Inc. ISBN 0-7858-1750-6.

• Grill, Tom; Scanlon, Mark (1990). Photographic Composition. Watson-Guptill Publications. ISBN 0-8174- 5427-6.

• Peterson, Bryan (1988). Learning to See Creatively. Watson-Guptill Publications. ISBN 0-8174-4177-8.

• Langford, Michael (1982). The Master Guide to Photography. New York: Dorling Kindersley Limited. ISBN 0-394-50873-4.

1.1.8 External links

• Percy Principles of Art and Composition, Goshen College Art Department

1.2 Elements of art

A work of art can be analysed by considering a variety of aspects of it individually. These aspects are often called the elements of art. A commonly used list of the main elements include form, shape, line, color, value, space and texture. 10 CHAPTER 1. DAY 1

1.2.1 Form

The form of a work is its shape, including its volume or perceived volume. A three-dimensional artwork has depth as well as width and height. Three-dimensional form is the basis of sculpture.[1] However, two-dimensional artwork can achieve the illusion of form with the use of perspective and/or shading or modelling techniques.[2][3] Formalism is the analysis of works by their form or shapes in art history or archeology.

1.2.2 Line

Lines and curves are marks that span a distance between two points (or the path of a moving point). As an element of visual art, line is the use of various marks, outlines, and implied lines in artwork and design. A line has a width, direction, and length.[1] A line’s width is sometimes called its “thickness”. Lines are sometimes called “strokes”, especially when referring to lines in digital artwork.

1.2.3 Color

Color is the element of art that is produced when light, striking an object, is reflected back to the eye.[1] There are three properties to color. The first is hue, which simply means the name we give to a color (red, yellow, blue, green, etc.). The second property is intensity, which refers to the vividness of the color. A color’s intensity is sometimes referred to as its "colorfulness", its “saturation”, its “purity” or its “strength”.The third and final property of color is its value, meaning how light or dark it is.[4] The terms shade and tint refer to value changes in colors. In painting, shades are created by adding black to a color, while tints are created by adding white to a color.[2]

1.2.4 Space

Space is an area that an artist provides for a particular purpose.[1] Space includes the background, foreground and middle ground, and refers to the distances or area(s) around, between, and within things. There are two kinds of space: negative space and positive space.[5] Negative space is the area in between, around, through or within an object. Positive spaces are the areas that are occupied by an object and/or form.

1.2.5 Texture

Texture, another element of art, is used to describe either the way a work actually feels when touched, or the depiction of textures in works, as for example in a painter’s rendering of fur. The feeling or look of the art work.

1.2.6 See also

• Design elements and principles

• Style (visual arts)

1.2.7 References

[1] “Understanding Formal Analysis”. Getty. Retrieved 9 May 2014.

[2] “Elements and Principles of Design”. IncredibleArt.org. Retrieved 9 May 2014.

[3] “What Are the Elements of Art?". About.com. Retrieved 9 May 2014.

[4] “What is Value in Art?".

[5] “Vocabulary: Elements of Art, Principles of Art” (PDF). Oberlin. Retrieved 9 May 2014. 1.2. ELEMENTS OF ART 11

1.2.8 External links

• The elements of art at getty.edu •

• Line and Form (1900) by Walter Crane at Project Gutenberg Chapter 2

Day 2

2.1 Visual design elements and principles

Visual design elements and principles describe fundamental ideas about the practice of good visual design. As William Lidwell stated in Universal Principles of Design:

The best designers sometimes disregard the principles of design. When they do so, however, there is usually some compensating merit attained at the cost of the violation. Unless you are certain of doing as well, it is best to abide by the principles.[1]

2.1.1 Design elements

See also: Elements of art

Design elements are the basic units of a painting, drawing, design or other visual piece[2] and include: Author Maitland Graves in The Art of Color and Design, 1951, McGraw-Hill Book Co., Inc. deﬁnes the elements of design as: Line, Direction, Shape, Size, Texture, Value and Color - in that order. “These elements are the materials from which all designs are built.”

Color

• Colors play a large role in the elements of design[3] with the color wheel being used as a tool, and color theory providing a body of practical guidance to color mixing and the visual impacts of speciﬁc color combination.

Uses

• Color can aid organization to develop a color strategy and stay consistent with those colors.[3] • It can give emphasis to create a hierarchy to the piece of art. • It is also important to note that color choices in design change meaning within cultural contexts. For example, white is associated with purity in some cultures while it is associated with death in others.

Attributes

• Hue[3] • Values, tints and shades of colors that are created by adding black to a color for a shade and white for a tint. Creating a tint or shade of color reduces the saturation.[3]

12 2.1. VISUAL DESIGN ELEMENTS AND PRINCIPLES 13

Red Red- Red- violet orange Violet Orange

Blue- Yellow- violet orange

Blue Yellow

Blue- Yellow- green green Green

Color star containing primary, secondary, and tertiary colors.

• Saturation gives a color brightness or dullness, and by doing this it makes the color more vibrant than before.[3]

The three primary hues which cannot be created by mixing are red, yellow and blue. In practice, however, a more practical set of “double primaries” is utilized to allow for creating more intense saturation of colors. One author recommending this double primary system of color mixing is Michael Wilcox in his book: BLUE AND YELLOW DON'T MAKE GREEN.

Shape

A shape is defined as a two or more dimensional area that stands out from the space next to or around it due to a defined or implied boundary, or because of differences of value, color, or texture.[4] All objects are composed of shapes and all other 'Elements of Design' are shapes in some way.[5]

Categories

• Mechanical Shapes or Geometric Shapes are the shapes that can be drawn using a ruler or compass. Mechanical shapes, whether simple or complex, produce a feeling of control or order.[5] 14 CHAPTER 2. DAY 2

• Organic Shapes are freehand drawn shapes that are complex and normally found in nature. Organic shapes produce a natural feel.[5]

Texture

The tree’s visual texture is represented here in this image.

Meaning the way a surface feels or is perceived to feel. Texture can be added to attract or repel interest to an element, depending on the pleasantness of the texture.[5] 2.1. VISUAL DESIGN ELEMENTS AND PRINCIPLES 15

Types of texture

• Tactile texture is the actual three-dimension feel of a surface that can be touched. Painter can use impasto to build peaks and create texture.[5]

• Visual texture is the illusion of the surfaces peaks and valleys, like the tree pictured. Any texture shown in a photo is a visual texture, meaning the paper is smooth no matter how rough the image perceives it to be.[5]

Most textures have a natural touch but still seem to repeat a motif in some way. Regularly repeating a motif will result in a texture appearing as a pattern.[5]

Space

In design, space is concerned with the area deep within the moment of designated design, the design will take place on. For a two-dimensional design, space concerns creating the illusion of a third dimension on a ﬂat surface:[5]

• Overlap is the eﬀect where objects appear to be on top of each other. This illusion makes the top element look closer to the observer. There is no way to determine the depth of the space, only the order of closeness.

• Shading adds gradiation marks to make an object of a two-dimensional surface seem three-dimensional.

• Highlight, Transitional Light, Core of the Shadow, Reﬂected Light, and Cast Shadow give an object a three- dimensional look.[5]

• Linear Perspective is the concept relating to how an object seems smaller the farther away it gets.

• Atmospheric Perspective is based on how air acts as a ﬁlter to change the appearance of distant objects.

Form

Form may be described as any three-dimensional object. Form can be measured, from top to bottom (height), side to side (width), and from back to front (depth). Form is also deﬁned by light and dark. It can be deﬁned by the presence of shadows on surfaces or faces of an object. There are two types of form, geometric (man-made) and natural (organic form). Form may be created by the combining of two or more shapes. It may be enhanced by tone, texture and color. It can be illustrated or constructed.

2.1.2 Principles of design

Principles applied to the elements of design that bring them together into one design. How one applies these principles determines how successful a design may be.[2]

Unity/harmony

According to Alex White, author of The Elements of Graphic Design, to achieve visual unity is a main goal of graphic design. When all elements are in agreement, a design is considered uniﬁed. No individual part is viewed as more important than the whole design. A good balance between unity and variety must be established to avoid a chaotic or a lifeless design.[3]

Methods

• Perspective: sense of distance between elements.

• Similarity: ability to seem repeatable with other elements.

• Continuation: the sense of having a line or pattern extend.

• Repetition: elements being copied or mimicked numerous times. 16 CHAPTER 2. DAY 2

• Rhythm: is achieved when recurring position, size, color, and use of a graphic element has a focal point interruption.

• Altering the basic theme achieves unity and helps keep interest.

Balance

It is a state of equalized tension and equilibrium, which may not always be calm.[3]

Types

• Symmetry

• Asymmetrical balance produces an informal balance that is attention attracting and dynamic.

• Radial balance is arranged around a central element. The elements placed in a radial balance seem to 'radiate' out from a central point in a circular fashion.

• Overall is a mosaic form of balance which normally arises from too many elements being put on a page. Due to the lack of hierarchy and contrast, this form of balance can look noisy but sometimes quiet.

Hierarchy

A good design contains elements that lead the reader through each element in order of its signiﬁcance. The type and images should be expressed starting from most important to the least important.

Scale/proportion

Using the relative size of elements against each other can attract attention to a focal point. When elements are designed larger than life, scale is being used to show drama.[3]

Dominance/emphasis

Dominance is created by contrasting size, positioning, color, style, or shape. The focal point should dominate the design with scale and contrast without sacriﬁcing the unity of the whole.[3]

Similarity and contrast

Planning a consistent and similar design is an important aspect of a designer’s work to make their focal point visible. Too much similarity is boring but without similarity important elements will not exist and an image without contrast is uneventful so the key is to ﬁnd the balance between similarity and contrast.[3]

Similar environment There are several ways to develop a similar environment:[3]

• Build a unique internal organization structure.

• Manipulate shapes of images and text to correlate together.

• Express continuity from page to page in publications. Items to watch include headers, themes, borders, and spaces.

• Develop a style manual and adhere to it. 2.1. VISUAL DESIGN ELEMENTS AND PRINCIPLES 17

Contrasts

• Space • Filled / Empty • Near / Far • 2-D / 3-D • Position • Left / Right • Isolated / Grouped • Centered / Oﬀ-Center • Top / Bottom • Form • Simple / Complex • Beauty / Ugly • Whole / Broken • Direction • Stability / Movement • Structure • Organized / Chaotic • Mechanical / Hand-Drawn • Size • Large / Small • Deep / Shallow • Fat / Thin • Color • Grey scale / Color • Black & White / Color • Light / Dark • Texture • Fine / Coarse • Smooth / Rough • Sharp / Dull • Density • Transparent / Opaque • Thick / Thin • Liquid / Solid • Gravity • Light / Heavy • Stable / Unstable

Movement is the path the viewer’s eye takes through the artwork, often to focal areas. Such movement can be directed along lines edges, shape and color within the artwork, and more. 18 CHAPTER 2. DAY 2

2.1.3 See also

• Composition (visual arts) • Interior design

• Landscape design • Pattern language

• Elements of art • Principles of art

• Color theory

2.1.4 Notes

[1] Lidwell, William; Kritina Holden; Jill Butler (2010). Universal Principles of Design (2nd ed.). Beverly, Massachusetts: Rockport Publishers. ISBN 978-1-59253-587-3.

[2] Lovett, John. “Design and Color”. Retrieved 3 April 2012.

[3] White, Alex (2011). The Elements of Graphic Design. New York, NY: Allworth Press. pp. 81–105. ISBN 978-1-58115- 762-8.

[4] Cindy Kovalik, Ph.D. and Peggy King, M.Ed. “Visual Literacy”. Retrieved 2010-03-27.

[5] Saw, James. “Design Notes”. Palomar College. Retrieved 3 April 2012.

2.1.5 References

• Kilmer, R., & Kilmer, W. O. (1992). Designing Interiors. Orland, FL: Holt, Rinehart and Winston, Inc. ISBN 978-0-03-032233-4.

• Nielson, K. J., & Taylor, D. A. (2002). Interiors: An Introduction. New York: McGraw-Hill Companies, Inc. ISBN 978-0-07-296520-9

• Pile, J.F. (1995; fourth edition, 2007). Interior Design. New York: Harry N. Abrams, Inc. ISBN 978-0-13- 232103-7

• Sully, Anthony (2012). Interior Design: Theory and Process. London: Bloomsbury. ISBN 978-1-4081-5202- 7.

2.1.6 External links

• Art, Design, and Visual Thinking • The 6 Principles of Design 2.1. VISUAL DESIGN ELEMENTS AND PRINCIPLES 19

The top image has symmetrical balance and the bottom image has asymmetrical balance Chapter 3

Day 3

3.1 Shape

This article is about describing the shape of an object e.g. shapes like a triangle. For common shapes, see List of geometric shapes. For other uses, see Shape (disambiguation). A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties

An example of the diﬀerent deﬁnitions of shape. The two triangles on the left are congruent, while the third is similar to them. The last triangle is neither similar nor congruent to any of the others, but it is homeomorphic. such as color, texture, or material composition. Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons.[1] Examples of geons include cones and spheres.

3.1.1 Classiﬁcation of simple shapes

Main article: Lists of shapes Some simple shapes can be put into broad categories. For instance, polygons are classiﬁed according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares, etc. Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, and parabolas. Among the most common 3-dimensional shapes are polyhedra, which are shapes with ﬂat faces; ellipsoids, which are egg-shaped or sphere-shaped objects; cylinders; and cones. If an object falls into one of these categories exactly or even approximately, we can use it to describe the shape of

20 3.1. SHAPE 21

Simple

Convex Concave Cyclic

Equilateral Equiangular

Regular convex Regular star

A variety of polygonal shapes. the object. Thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk.

3.1.2 Shape in geometry

There are several ways to compare the shapes of two objects:

• Congruence: Two objects are congruent if one can be transformed into the other by a sequence of rotations, translations, and/or reﬂections.

• Similarity: Two objects are similar if one can be transformed into the other by a uniform scaling, together with a sequence of rotations, translations, and/or reﬂections.

• Isotopy: Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it.

Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the letters "b" and "d" are a reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, an hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have the same shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. 22 CHAPTER 3. DAY 3

Simple shapes can often be classified into basic geometric objects such as a point, a line, a curve, a plane, a plane figure (e.g. square or circle), or a solid figure (e.g. cube or sphere). However, most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals.

Equivalence of shapes

In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings. In other words, the shape of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an equivalence relation, and accordingly a precise mathematical deﬁnition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape. Mathematician and statistician David George Kendall writes:[2]

In this paper ‘shape’ is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale[3] and rotational effects are filtered out from an object.’

Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on the size and placement in space of the object. For instance, a "d" and a "p" have the same shape, as they can be perfectly superimposed if the "d" is translated to the right by a given distance, rotated upside down and magnified by a given factor (see Procrustes superimposition for details). However, a mirror image could be called a different shape. For instance, a "b" and a "p" have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written. Even though they have the same size, there’s no way to perfectly superimpose them by translating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if the object is scaled non-uniformly. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal directions. In other words, preserving axes of symmetry (if they exist) is important for preserving shapes. Also, shape is determined by only the outer boundary of an object.

Congruence and similarity

Main articles: Congruence (geometry) and Similarity (geometry)

Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are congruent. An object is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes, and have the same size. Objects that have the same shape or mirror image shapes are called geometrically similar, whether or not they have the same size. Thus, objects that can be transformed into each other by rigid transformations, mirroring, and uniform scaling are similar. Similarity is preserved when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have diﬀerent size.

Homeomorphism

Main article: Homeomorphism

A more flexible definition of shape takes into consideration the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions. One way of modeling non-rigid movements is by homeomorphisms. Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated mathematical joke is that topologists can't tell their coffee cup from their donut,[4] since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup’s handle. 3.1. SHAPE 23

3.1.3 Shape analysis

Main article: Statistical shape analysis

The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of statistical shape analysis. In particular Procrustes analysis, which is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example Spectral shape analysis).

3.1.4 Similarity classes

All similar triangles have the same shape. These shapes can be classiﬁed using complex numbers in a method advanced by J.A. Lester[5] and Rafael Artzy. For example, an equilateral triangle can be expressed by the complex numbers 0, 1, (1 + i √3)/2 representing its vertices. Lester and Artzy call the ratio

u−w S(u, v, w) = u−v the shape of triangle (u, v, w). Then the shape of the equilateral triangle is (0–(1+ √3)/2)/(0–1) = ( 1 + i √3)/2 = cos(60°) + i sin(60°) = exp( i π/3).

For any aﬃne transformation of the complex plane, z 7→ az + b, a ≠ 0, a triangle is transformed but does not change its shape. Hence shape is an invariant of aﬃne geometry. The shape p = S(u,v,w) depends on the order of the arguments of function S, but permutations lead to related values. For instance,

1 − p = 1 − (u − w)/(u − v) = (w − v)/(u − v) = (v − w)/(v − u) = S(v, u, w). Also p−1 = S(u, w, v).

Combining these permutations gives S(v, w, u) = (1 − p)−1. Furthermore,

p(1 − p)−1 = S(u, v, w)S(v, w, u) = (u − w)/(v − w) = S(w, v, u).

The shape of a quadrilateral is associated with two complex numbers p,q. If the quadrilateral has vertices u,v,w,x, then p = S(u,v,w) and q = S(v,w,x). Artzy proves these propositions about quadrilateral shapes:

1. If p = (1 − q)−1, then the quadrilateral is a parallelogram.

2. If a parallelogram has |arg p| = |arg q|, then it is a rhombus.

3. When p = 1 + i and q = (1 + i)/2, then the quadrilateral is square.

4. If p = r(1 − q−1) and sgn r = sgn(Im p), then the quadrilateral is a trapezoid.

A polygon (z1, z2, ...zn) has a shape deﬁned by n – 2 complex numbers S(zj, zj+1, zj+2), j = 1, ..., n − 2. The polygon bounds a convex set when all these shape components have imaginary components of the same sign.[6]

3.1.5 See also

• Shape factor

• Solid geometry

• Glossary of shapes with metaphorical names 24 CHAPTER 3. DAY 3

3.1.6 References

[1] Marr, D., & Nishihara, H. (1978). Representation and recognition of the spatial organization of three-dimensional shapes. Proceedings of the Royal Society of London, 200, 269-294.

[2] Kendall, D.G. (1984). “Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces”. Bulletin of the London Mathematical Society. 16 (2): 81–121. doi:10.1112/blms/16.2.81.

[3] Here, scale means only uniform scaling, as non-uniform scaling would change the shape of the object (e.g., it would turn a square into a rectangle).

[4] Hubbard, John H.; West, Beverly H. (1995). Diﬀerential Equations: A Dynamical Systems Approach. Part II: Higher- Dimensional Systems. Texts in Applied Mathematics. 18. Springer. p. 204. ISBN 978-0-387-94377-0.

[5] J.A. Lester (1996) “Triangles I: Shapes”, Aequationes Mathematicae 52:30–54

[6] Rafael Artzy (1994) “Shapes of Polygons”, Journal of Geometry 50(1–2):11–15

3.1.7 External links Chapter 4

Day 4

4.1 Color

For other uses of “Color” and “Colour”, see Color (disambiguation). For editing Wikipedia, see Help:Using color. See also, Colorful (disambiguation) and List of colors.

Colored pencils

Color (American English) or colour (Commonwealth English) is the characteristic of human visual perception described through color categories, with names such as red, yellow, purple, or bronze. This perception of color derives from the stimulation of cone cells in the human eye by electromagnetic radiation in the spectrum of light. Color categories and physical specifications of color are associated with objects through the wavelength of the light that is reflected from them. This reflection is governed by the object’s physical properties such as light absorption, emission spectra, etc. Human trichromacy is the basis for all modern color spaces that assign colors numerical coordinates and associate corresponding distances between colors. The photo-receptivity of the “eyes” of other species also varies considerably from our own and so results in pre-

25 26 CHAPTER 4. DAY 4

Color eﬀect – Sunlight shining through stained glass onto carpet (Nasir ol Molk Mosque located in Shiraz, Iran)

Colors can look diﬀerently depending on their surrounding colors and shapes. The two small squares have exactly the same color, but the right one looks slightly darker.

sumably different color perceptions that cannot readily be compared to one another. The mere presence of “extra” photoreceptor types does not directly imply that they are being used functionally in an animal. Demonstrating im- proved spectral discrimination in any animal can be difficult since complex sets of neurons affect color perception in ways that are generally difficult to interrogate.[1] 4.1. COLOR 27

The science of color is sometimes called chromatics, colorimetry, or simply color science. It includes the perception of color by the human eye and brain, the origin of color in materials, color theory in art, and the physics of electromagnetic radiation in the visible range (that is, what is commonly referred to simply as light).

4.1.1 Physics of color

Continuous optical spectrum rendered into the sRGB color space.

Electromagnetic radiation is characterized by its wavelength (or frequency) and its intensity. When the wavelength is within the visible spectrum (the range of wavelengths humans can perceive, approximately from 390 nm to 700 nm), it is known as “visible light”. Most light sources emit light at many different wavelengths; a source’s spectrum is a distribution giving its intensity at each wavelength. Although the spectrum of light arriving at the eye from a given direction determines the color sensation in that direction, there are many more possible spectral combinations than color sensations. In fact, one may formally define a color as a class of spectra that give rise to the same color sensation, although such classes would vary widely among different species, and to a lesser extent among individuals within the same species. In each such class the members are called metamers of the color in question.

Spectral colors

The familiar colors of the rainbow in the spectrum – named using the Latin word for appearance or apparition by Isaac Newton in 1671 – include all those colors that can be produced by visible light of a single wavelength only, the pure spectral or monochromatic colors. The table at right shows approximate frequencies (in terahertz) and wavelengths (in nanometers) for various pure spectral colors. The wavelengths listed are as measured in air or vacuum (see refractive index). The color table should not be interpreted as a deﬁnitive list – the pure spectral colors form a continuous spectrum, and how it is divided into distinct colors linguistically is a matter of culture and historical contingency (although people everywhere have been shown to perceive colors in the same way[3]). A common list identiﬁes six main bands: red, orange, yellow, green, blue, and violet. Newton’s conception included a seventh color, indigo, between blue and violet. It is possible that what Newton referred to as blue is nearer to what today is known as cyan, and that indigo was simply the dark blue of the indigo dye that was being imported at the time.[4] The intensity of a spectral color, relative to the context in which it is viewed, may alter its perception considerably; for example, a low-intensity orange-yellow is brown, and a low-intensity yellow-green is olive-green.

Color of objects

The color of an object depends on both the physics of the object in its environment and the characteristics of the perceiving eye and brain. Physically, objects can be said to have the color of the light leaving their surfaces, which normally depends on the spectrum of the incident illumination and the reflectance properties of the surface, as well as potentially on the angles of illumination and viewing. Some objects not only reflect light, but also transmit light or emit light themselves, which also contribute to the color. A viewer’s perception of the object’s color depends not only on the spectrum of the light leaving its surface, but also on a host of contextual cues, so that color differences between objects can be discerned mostly independent of the lighting spectrum, viewing angle, etc. This effect is known as color constancy. Some generalizations of the physics can be drawn, neglecting perceptual effects for now:

• Light arriving at an opaque surface is either reflected "specularly" (that is, in the manner of a mirror), scattered (that is, reflected with diffuse scattering), or absorbed – or some combination of these.

• Opaque objects that do not reﬂect specularly (which tend to have rough surfaces) have their color determined by which wavelengths of light they scatter strongly (with the light that is not scattered being absorbed). If 28 CHAPTER 4. DAY 4

The upper disk and the lower disk have exactly the same objective color, and are in identical gray surroundings; based on context differences, humans perceive the squares as having different reflectances, and may interpret the colors as different color categories; see checker shadow illusion.

objects scatter all wavelengths with roughly equal strength, they appear white. If they absorb all wavelengths, they appear black.

• Opaque objects that specularly reflect light of different wavelengths with different efficiencies look like mirrors tinted with colors determined by those differences. An object that reflects some fraction of impinging light and absorbs the rest may look black but also be faintly reflective; examples are black objects coated with layers of enamel or lacquer.

• Objects that transmit light are either translucent (scattering the transmitted light) or transparent (not scattering the transmitted light). If they also absorb (or reflect) light of various wavelengths differentially, they appear tinted with a color determined by the nature of that absorption (or that reflectance).

• Objects may emit light that they generate from having excited electrons, rather than merely reﬂecting or transmitting light. The electrons may be excited due to elevated temperature (incandescence), as a result of chemical reactions (chemoluminescence), after absorbing light of other frequencies ("ﬂuorescence" or "phosphorescence") or from electrical contacts as in light emitting diodes, or other light sources.

To summarize, the color of an object is a complex result of its surface properties, its transmission properties, and its emission properties, all of which contribute to the mix of wavelengths in the light leaving the surface of the object. The perceived color is then further conditioned by the nature of the ambient illumination, and by the color properties of other objects nearby, and via other characteristics of the perceiving eye and brain. 4.1. COLOR 29

When viewed in full size, this image contains about 16 million pixels, each corresponding to a diﬀerent color on the full set of RGB colors. The human eye can distinguish about 10 million diﬀerent colors.[5]

4.1.2 Perception

Development of theories of color vision

Main article: Color theory

Although Aristotle and other ancient scientists had already written on the nature of light and color vision, it was not until Newton that light was identified as the source of the color sensation. In 1810, Goethe published his comprehensive Theory of Colors in which he ascribed physiological effects to color that are now understood as psychological. In 1801 Thomas Young proposed his trichromatic theory, based on the observation that any color could be matched with a combination of three lights. This theory was later refined by James Clerk Maxwell and Hermann von Helmholtz. As Helmholtz puts it, “the principles of Newton’s law of mixture were experimentally confirmed by Maxwell in 1856. Young’s theory of color sensations, like so much else that this marvelous investigator achieved in advance of his time, remained unnoticed until Maxwell directed attention to it.”[6] At the same time as Helmholtz, Ewald Hering developed the opponent process theory of color, noting that color blindness and afterimages typically come in opponent pairs (red-green, blue-orange, yellow-violet, and black-white). 30 CHAPTER 4. DAY 4

Ultimately these two theories were synthesized in 1957 by Hurvich and Jameson, who showed that retinal processing corresponds to the trichromatic theory, while processing at the level of the lateral geniculate nucleus corresponds to the opponent theory.[7] In 1931, an international group of experts known as the Commission internationale de l'éclairage (CIE) developed a mathematical color model, which mapped out the space of observable colors and assigned a set of three numbers to each.

Color in the eye

Main article: Color vision The ability of the human eye to distinguish colors is based upon the varying sensitivity of diﬀerent cells in the retina

1.0 S ML 0.8

0.6

0.4

0.2

0 400 450 500 550 600 650 700

Normalized typical human cone cell responses (S, M, and L types) to monochromatic spectral stimuli to light of different wavelengths. Humans being trichromatic, the retina contains three types of color receptor cells, or cones. One type, relatively distinct from the other two, is most responsive to light that is perceived as blue or blue- violet, with wavelengths around 450 nm; cones of this type are sometimes called short-wavelength cones, S cones, or blue cones. The other two types are closely related genetically and chemically: middle-wavelength cones, M cones, or green cones are most sensitive to light perceived as green, with wavelengths around 540 nm, while the long-wavelength cones, L cones, or red cones, are most sensitive to light is perceived as greenish yellow, with wavelengths around 570 nm. Light, no matter how complex its composition of wavelengths, is reduced to three color components by the eye. For each location in the visual field, the three types of cones yield three signals based on the extent to which each is stimulated. These amounts of stimulation are sometimes called tristimulus values. The response curve as a function of wavelength varies for each type of cone. Because the curves overlap, some tristimulus values do not occur for any incoming light combination. For example, it is not possible to stimulate only the mid-wavelength (so-called “green”) cones; the other cones will inevitably be stimulated to some degree at the same time. The set of all possible tristimulus values determines the human color space. It has been estimated that humans can distinguish roughly 10 million different colors.[5] 4.1. COLOR 31

The other type of light-sensitive cell in the eye, the rod, has a diﬀerent response curve. In normal situations, when light is bright enough to strongly stimulate the cones, rods play virtually no role in vision at all.[8] On the other hand, in dim light, the cones are understimulated leaving only the signal from the rods, resulting in a colorless response. (Furthermore, the rods are barely sensitive to light in the “red” range.) In certain conditions of intermediate illumination, the rod response and a weak cone response can together result in color discriminations not accounted for by cone responses alone. These eﬀects, combined, are summarized also in the Kruithof curve, that describes the change of color perception and pleasingness of light as function of temperature and intensity.

Color in the brain

Main article: Color vision While the mechanisms of color vision at the level of the retina are well-described in terms of tristimulus values, color

The visual dorsal stream (green) and ventral stream (purple) are shown. The ventral stream is responsible for color perception.

processing after that point is organized diﬀerently. A dominant theory of color vision proposes that color information is transmitted out of the eye by three opponent processes, or opponent channels, each constructed from the raw output of the cones: a red–green channel, a blue–yellow channel, and a black–white “luminance” channel. This theory has been supported by neurobiology, and accounts for the structure of our subjective color experience. Speciﬁcally, it explains why humans cannot perceive a “reddish green” or “yellowish blue”, and it predicts the color wheel: it is the collection of colors for which at least one of the two color channels measures a value at one of its extremes. The exact nature of color perception beyond the processing already described, and indeed the status of color as a feature of the perceived world or rather as a feature of our perception of the world – a type of qualia – is a matter of complex and continuing philosophical dispute.

Nonstandard color perception

Color deﬁciency Main article: Color blindness 32 CHAPTER 4. DAY 4

If one or more types of a person’s color-sensing cones are missing or less responsive than normal to incoming light, that person can distinguish fewer colors and is said to be color deficient or color blind (though this latter term can be misleading; almost all color deficient individuals can distinguish at least some colors). Some kinds of color deficiency are caused by anomalies in the number or nature of cones in the retina. Others (like central or cortical achromatopsia) are caused by neural anomalies in those parts of the brain where visual processing takes place.

Tetrachromacy Main article: Tetrachromacy

While most humans are trichromatic (having three types of color receptors), many animals, known as tetrachromats, have four types. These include some species of spiders, most marsupials, birds, reptiles, and many species of fish. Other species are sensitive to only two axes of color or do not perceive color at all; these are called dichromats and monochromats respectively. A distinction is made between retinal tetrachromacy (having four pigments in cone cells in the retina, compared to three in trichromats) and functional tetrachromacy (having the ability to make enhanced color discriminations based on that retinal difference). As many as half of all women are retinal tetrachromats.[9]:p.256 The phenomenon arises when an individual receives two slightly different copies of the gene for either the medium- or long-wavelength cones, which are carried on the x-chromosome. To have two different genes, a person must have two x-chromosomes, which is why the phenomenon only occurs in women.[9] There is one scholarly report that confirms the existence of a functional tetrachromat.[10]

Synesthesia In certain forms of synesthesia/ideasthesia, perceiving letters and numbers (grapheme–color synesthesia) or hearing musical sounds (music–color synesthesia) will lead to the unusual additional experiences of seeing colors. Behavioral and functional neuroimaging experiments have demonstrated that these color experiences lead to changes in behavioral tasks and lead to increased activation of brain regions involved in color perception, thus demonstrating their reality, and similarity to real color percepts, albeit evoked through a non-standard route.

Afterimages

After exposure to strong light in their sensitivity range, photoreceptors of a given type become desensitized. For a few seconds after the light ceases, they will continue to signal less strongly than they otherwise would. Colors observed during that period will appear to lack the color component detected by the desensitized photoreceptors. This effect is responsible for the phenomenon of afterimages, in which the eye may continue to see a bright figure after looking away from it, but in a complementary color. Afterimage effects have also been utilized by artists, including Vincent van Gogh.

Color constancy

Main article: Color constancy

When an artist uses a limited color palette, the eye tends to compensate by seeing any gray or neutral color as the color which is missing from the color wheel. For example, in a limited palette consisting of red, yellow, black, and white, a mixture of yellow and black will appear as a variety of green, a mixture of red and black will appear as a variety of purple, and pure gray will appear bluish.[11] The trichromatic theory is strictly true when the visual system is in a fixed state of adaptation. In reality, the visual system is constantly adapting to changes in the environment and compares the various colors in a scene to reduce the effects of the illumination. If a scene is illuminated with one light, and then with another, as long as the difference between the light sources stays within a reasonable range, the colors in the scene appear relatively constant to us. This was studied by Edwin Land in the 1970s and led to his retinex theory of color constancy. It should be noted, that both phenomena are readily explained and mathematically modeled with modern theories of chromatic adaptation and color appearance (e.g. CIECAM02, iCAM).[12] There is no need to dismiss the trichromatic theory of vision, but rather it can be enhanced with an understanding of how the visual system adapts to changes in the viewing environment. 4.1. COLOR 33

Color naming

Main article: Color term See also: Lists of colors and Web colors

Colors vary in several diﬀerent ways, including hue (shades of red, orange, yellow, green, blue, and violet), saturation, brightness, and gloss. Some color words are derived from the name of an object of that color, such as "orange" or "salmon", while others are abstract, like “red”. In the 1969 study Basic Color Terms: Their Universality and Evolution, Brent Berlin and Paul Kay describe a pattern in naming “basic” colors (like “red” but not “red-orange” or “dark red” or “blood red”, which are “shades” of red). All languages that have two “basic” color names distinguish dark/cool colors from bright/warm colors. The next colors to be distinguished are usually red and then yellow or green. All languages with six “basic” colors include black, white, red, green, blue, and yellow. The pattern holds up to a set of twelve: black, gray, white, pink, red, orange, yellow, green, blue, purple, brown, and azure (distinct from blue in Russian and Italian, but not English).

4.1.3 Associations

Individual colors have a variety of cultural associations such as national colors (in general described in individual color articles and color symbolism). The field of color psychology attempts to identify the effects of color on human emotion and activity. Chromotherapy is a form of alternative medicine attributed to various Eastern traditions. Colors have different associations in different countries and cultures.[13] Different colors have been demonstrated to have effects on cognition. For example, researchers at the University of Linz in Austria demonstrated that the color red significantly decreases cognitive functioning in men.[14]

4.1.4 Spectral colors and color reproduction

Most light sources are mixtures of various wavelengths of light. Many such sources can still effectively produce a spectral color, as the eye cannot distinguish them from single-wavelength sources. For example, most computer displays reproduce the spectral color orange as a combination of red and green light; it appears orange because the red and green are mixed in the right proportions to allow the eye’s cones to respond the way they do to the spectral color orange. A useful concept in understanding the perceived color of a non-monochromatic light source is the dominant wavelength, which identifies the single wavelength of light that produces a sensation most similar to the light source. Dominant wavelength is roughly akin to hue. There are many color perceptions that by definition cannot be pure spectral colors due to desaturation or because they are purples (mixtures of red and violet light, from opposite ends of the spectrum). Some examples of necessarily non-spectral colors are the achromatic colors (black, gray, and white) and colors such as pink, tan, and magenta. Two different light spectra that have the same effect on the three color receptors in the human eye will be perceived as the same color. They are metamers of that color. This is exemplified by the white light emitted by fluorescent lamps, which typically has a spectrum of a few narrow bands, while daylight has a continuous spectrum. The human eye cannot tell the difference between such light spectra just by looking into the light source, although reflected colors from objects can look different. (This is often exploited; for example, to make fruit or tomatoes look more intensely red.) Similarly, most human color perceptions can be generated by a mixture of three colors called primaries. This is used to reproduce color scenes in photography, printing, television, and other media. There are a number of methods or color spaces for specifying a color in terms of three particular primary colors. Each method has its advantages and disadvantages depending on the particular application. No mixture of colors, however, can produce a response truly identical to that of a spectral color, although one can get close, especially for the longer wavelengths, where the CIE 1931 color space chromaticity diagram has a nearly straight edge. For example, mixing green light (530 nm) and blue light (460 nm) produces cyan light that is slightly desaturated, because response of the red color receptor would be greater to the green and blue light in the mixture than it would be to a pure cyan light at 485 nm that has the same intensity as the mixture of blue and green. Because of this, and because the primaries in color printing systems generally are not pure themselves, the colors 34 CHAPTER 4. DAY 4

0.9 520

0.8 540

0.7 560 0.6 500 0.5 580 y 0.4 600

620 0.3 490 700

0.2

480 0.1 470 460 0.0 380 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x

The CIE 1931 color space chromaticity diagram. The outer curved boundary is the spectral (or monochromatic) locus, with wavelengths shown in nanometers. The colors depicted depend on the color space of the device on which you are viewing the image, and therefore may not be a strictly accurate representation of the color at a particular position, and especially not for monochromatic colors.

reproduced are never perfectly saturated spectral colors, and so spectral colors cannot be matched exactly. However, natural scenes rarely contain fully saturated colors, thus such scenes can usually be approximated well by these systems. The range of colors that can be reproduced with a given color reproduction system is called the gamut. The CIE chromaticity diagram can be used to describe the gamut. Another problem with color reproduction systems is connected with the acquisition devices, like cameras or scanners. The characteristics of the color sensors in the devices are often very far from the characteristics of the receptors in the human eye. In effect, acquisition of colors can be relatively poor if they have special, often very “jagged”, spectra caused for example by unusual lighting of the photographed scene. A color reproduction system “tuned” to a human with normal color vision may give very inaccurate results for other observers. The different color response of different devices can be problematic if not properly managed. For color information 4.1. COLOR 35 stored and transferred in digital form, color management techniques, such as those based on ICC profiles, can help to avoid distortions of the reproduced colors. Color management does not circumvent the gamut limitations of particular output devices, but can assist in finding good mapping of input colors into the gamut that can be reproduced.

4.1.5 Additive coloring

Additive color mixing: combining red and green yields yellow; combining all three primary colors together yields white.

Additive color is light created by mixing together light of two or more diﬀerent colors. Red, green, and blue are the additive primary colors normally used in additive color systems such as projectors and computer terminals.

4.1.6 Subtractive coloring

Subtractive coloring uses dyes, inks, pigments, or filters to absorb some wavelengths of light and not others. The color that a surface displays comes from the parts of the visible spectrum that are not absorbed and therefore remain visible. Without pigments or dye, fabric fibers, paint base and paper are usually made of particles that scatter white light (all colors) well in all directions. When a pigment or ink is added, wavelengths are absorbed or “subtracted” from white light, so light of another color reaches the eye. If the light is not a pure white source (the case of nearly all forms of artificial lighting), the resulting spectrum will 36 CHAPTER 4. DAY 4

Subtractive color mixing: combining yellow and magenta yields red; combining all three primary colors together yields black appear a slightly diﬀerent color. Red paint, viewed under blue light, may appear black. Red paint is red because it scatters only the red components of the spectrum. If red paint is illuminated by blue light, it will be absorbed by the red paint, creating the appearance of a black object.

4.1.7 Structural color

Further information: Structural coloration and Animal coloration

Structural colors are colors caused by interference effects rather than by pigments. Color effects are produced when a material is scored with fine parallel lines, formed of one or more parallel thin layers, or otherwise composed of microstructures on the scale of the color’s wavelength. If the microstructures are spaced randomly, light of shorter wavelengths will be scattered preferentially to produce Tyndall effect colors: the blue of the sky (Rayleigh scattering, caused by structures much smaller than the wavelength of light, in this case air molecules), the luster of opals, and the blue of human irises. If the microstructures are aligned in arrays, for example the array of pits in a CD, they behave as a diffraction grating: the grating reflects different wavelengths in different directions due to interference phenomena, separating mixed “white” light into light of different wavelengths. If the structure is one or more thin layers then it will reflect some wavelengths and transmit others, depending on the layers’ thickness. 4.1. COLOR 37

Structural color is studied in the field of thin-film optics. A layman’s term that describes particularly the most ordered or the most changeable structural colors is iridescence. Structural color is responsible for the blues and greens of the feathers of many birds (the blue jay, for example), as well as certain butterfly wings and beetle shells. Variations in the pattern’s spacing often give rise to an iridescent effect, as seen in peacock feathers, soap bubbles, films of oil, and mother of pearl, because the reflected color depends upon the viewing angle. Numerous scientists have carried out research in butterfly wings and beetle shells, including Isaac Newton and Robert Hooke. Since 1942, electron micrography has been used, advancing the development of products that exploit structural color, such as "photonic" cosmetics.[15]

4.1.8 Mentions of color in social media

According to Pantone, the top three colors in social media for 2012 were red (186 million mentions; accredited to Taylor Swift's Red album, NASA's landing on Mars, and red carpet coverage), blue (125 million mentions; accredited to the United States presidential election, 2012, Mars rover Curiosity ﬁnding blue rocks, and blue sports teams), and Green (102 million mentions; accredited to “environmental friendliness”, Green Bay Packers, and green eyed girls).[16]

4.1.9 Additional terms

• Color wheel: an illustrative organization of color hues in a circle that shows relationships.

• Colorfulness, chroma, purity, or saturation: how “intense” or “concentrated” a color is. Technical deﬁnitions distinguish between colorfulness, chroma, and saturation as distinct perceptual attributes and include purity as a physical quantity. These terms, and others related to light and color are internationally agreed upon and published in the CIE Lighting Vocabulary.[17] More readily available texts on colorimetry also deﬁne and explain these terms.[12][18]

• Dichromatism: a phenomenon where the hue is dependent on concentration and/or thickness of the absorbing substance.

• Hue: the color’s direction from white, for example in a color wheel or chromaticity diagram.

• Shade: a color made darker by adding black.

• Tint: a color made lighter by adding white.

• Value, brightness, lightness, or luminosity: how light or dark a color is.

4.1.10 See also

• Chromophore

• Color analysis (art)

• Color mapping

• Complementary color

• Impossible color

• International Color Consortium

• International Commission on Illumination

• Lists of colors (compact version)

• Neutral color

• Pearlescent coating including Metal eﬀect pigments

• Primary, secondary and tertiary colors 38 CHAPTER 4. DAY 4

4.1.11 References

[1] Morrison, Jessica (23 January 2014). “Mantis shrimp’s super colour vision debunked”. Nature. doi:10.1038/nature.2014.14578. [2] Craig F. Bohren (2006). Fundamentals of Atmospheric Radiation: An Introduction with 400 Problems. Wiley-VCH. ISBN 3-527-40503-8. [3] Berlin, B. and Kay, P., Basic Color Terms: Their Universality and Evolution, Berkeley: University of California Press, 1969. [4] Waldman, Gary (2002). Introduction to light : the physics of light, vision, and color (Dover ed.). Mineola: Dover Publica- tions. p. 193. ISBN 978-0-486-42118-6. [5] Judd, Deane B.; Wyszecki, Günter (1975). Color in Business, Science and Industry. Wiley Series in Pure and Applied Optics (third ed.). New York: Wiley-Interscience. p. 388. ISBN 0-471-45212-2. [6] Hermann von Helmholtz, Physiological Optics – The Sensations of Vision, 1866, as translated in Sources of Color Science, David L. MacAdam, ed., Cambridge: MIT Press, 1970. [7] Palmer, S.E. (1999). Vision Science: Photons to Phenomenology, Cambridge, MA: MIT Press. ISBN 0-262-16183-4. [8] “Under well-lit viewing conditions (photopic vision), cones ...are highly active and rods are inactive.” Hirakawa, K.; Parks, T.W. (2005). Chromatic Adaptation and White-Balance Problem (PDF). IEEE ICIP. doi:10.1109/ICIP.2005.1530559. Archived from the original (PDF) on November 28, 2006. [9] Jameson, K. A.; Highnote, S. M.,; Wasserman, L. M. (2001). “Richer color experience in observers with multiple pho- topigment opsin genes.” (PDF). Psychonomic Bulletin and Review. 8 (2): 244–261. doi:10.3758/BF03196159. PMID 11495112. [10] Jordan, G.; Deeb, S. S.; Bosten, J. M.; Mollon, J. D. (20 July 2010). “The dimensionality of color vision in carriers of anomalous trichromacy”. Journal of Vision. 10 (8): 12–12. doi:10.1167/10.8.12. PMID 20884587. [11] Depauw, Robert C. “United States Patent”. Retrieved 20 March 2011. [12] M.D. Fairchild, Color Appearance Models Archived May 5, 2011, at the Wayback Machine., 2nd Ed., Wiley, Chichester (2005). [13] “Chart: Color Meanings by Culture”. Retrieved 2010-06-29. [14] Gnambs, Timo; Appel, Markus; Batinic, Bernad (2010). “Color red in web-based knowledge testing”. Computers in Human Behavior. 26: 1625–1631. doi:10.1016/j.chb.2010.06.010. [15] “Economic and Social Research Council – Science in the Dock, Art in the Stocks”. Archived from the original on November 2, 2007. Retrieved 2007-10-07. [16] “Celebrate Color”. pantone.com. Pantone. Retrieved 7 December 2014. [17] CIE Pub. 17-4, International Lighting Vocabulary, 1987. http://www.cie.co.at/publ/abst/17-4-89.html [18] R.S. Berns, Principles of Color Technology, 3rd Ed., Wiley, New York (2001).

4.1.12 External links and sources

• Bibliography Database on Color Theory, Buenos Aires University • Maund, Barry. “Color”. Stanford Encyclopedia of Philosophy. • “Color”. Internet Encyclopedia of Philosophy. • Why Should Engineers and Scientists Be Worried About Color? • Robert Ridgway's A Nomenclature of Colors (1886) and Color Standards and Color Nomenclature (1912) – text-searchable digital facsimiles at Linda Hall Library • Albert Henry Munsell's A Color Notation, (1907) at Project Gutenberg • AIC, International Colour Association • The Eﬀect of Color | OFF BOOK Documentary produced by Oﬀ Book (web series) • Study of the history of colors • The Color of Consciousness Chapter 5

Day 5

5.1 Texture (visual arts)

In the visual arts, texture is the perceived surface quality of a work of art. It is an element of two-dimensional and three-dimensional designs and is distinguished by its perceived visual and physical properties. Use of texture, along with other elements of design, can convey a variety of messages and emotions.

5.1.1 Three varieties of texture

Physical Texture

The bumpy texture of a pavement

39 40 CHAPTER 5. DAY 5

Physical texture, also known as actual texture or tactile texture, are the actual variations upon a surface. This can include, but is not limited to, fur, wood grain, sand, smooth surface of canvas or metal, glass, and leather. It differentiates itself from visual texture by having a physical quality that can be felt by touch. Specific use of a texture can affect the smoothness that an artwork conveys. For instance, use of rough surfaces can be visually active, whilst smooth surfaces can be visually restful. The use of both can give a sense of personality to a design, or utilized to create emphasis, rhythm, contrast, etc.[1] Light is an important factor for physical artwork, because it can affect how a surface is viewed. Strong lights on a smooth surface can obscure the readability of a drawing or photograph, whilst they can create strong contrasts in a highly textural surface such as river rocks and sand.

Visual Texture

Visual texture is the illusion of having physical texture. Every material and every support surface has its own visual texture and needs to be taken into consideration before creating a composition. As such, materials such as canvas and watercolour paper are considerably rougher than, for example, photo-quality computer paper and may not be best suited to creating a ﬂat, smooth texture. Photography, drawings and paintings use visual texture both to portray their own subject matter realistically and with interpretation. Texture in these media is generally created by the repetition of the shape and line.

5.1.2 Hypertexture

Hypertexture can be defined as both the “realistic simulated surface texture produced by adding small distortions across the surface of an object”[2] (as pioneered by Ken Perlin) and as an avenue for describing the fluid morphic nature of texture in the realm of cyber graphics and the tranversally responsive works created in the field of visual arts therein (as described by Lee Klein).[3][4]

5.1.3 Examples of physical texture

• Berlin Green Head, 500BC. Note the smooth texture and mood of the bust.

• Detail of woven ﬁbers of a carpet

• Animals are often deﬁned by their physical texture, such as a fuzzy kitten or this scaly iguana. 5.1. TEXTURE (VISUAL ARTS) 41

• Blades of grass provides a soft texture

• Rough bark on the surface of a tree

• A wall of bricks with raised areas

• Auto Texture created over Clear glass Bricks

5.1.4 Examples of visual texture

• View From the Window at Le Gras, Nicéphore Niépce, 1826. Photography.

• An image that has been digitally altered to show text and paper texture over a photograph

• Ralph’s Diner, Ralph Goings, 1982. The actual physical texture of this painting is smooth, despite its visual textures. 42 CHAPTER 5. DAY 5

5.1.5 See also

• Composition (visual arts) • Design elements and principles

• Elements of art • Texture (painting)

• Texture (computer graphics)

5.1.6 Notes

[1] Gatto. Exploring Visual Design: The Elements and Principles. p. 122–123.

[2] “What does HYPERTEXTURE mean?". Deﬁnitions.net. Retrieved 2013-08-17.

[3] “Hypertexture | A Gathering of the Tribes”. Tribes.org. 2006-10-01. Retrieved 2013-08-17.

[4] http://www.nyartsmagazine.com/?p=8504

5.1.7 References

• Gatto, Porter, and Selleck. Exploring Visual Design: The Elements and Principles. 3rd ed. Worcester: Davis Publications, Inc., 2000. ISBN 0-87192-379-3

• Stewart, Mary, Launching the imagination: a comprehensive guide to basic design. 2nd ed. New York: The McGraw-Hill Companies, Inc., 2006. ISBN 0-07-287061-3 Chapter 6

Day 6

6.1 Lightness

For other uses, see Lightness (disambiguation). In colorimetry and color theory, lightness, also known as value or tone, is a representation of variation in the perception of a color or color space's brightness. It is one of the color appearance parameters of any color appearance model. Lightness is a relative term. Lightness means brightness of an area judged relative to the brightness of a similarly illuminated area that appears to be white or highly transmitting. Lightness should not be confused with brightness.[1] Various color models have an explicit term for this property. The Munsell color model uses the term value, while the HSL color model, HCL color space and Lab color space use the term lightness. The HSV model uses the term value a little diﬀerently: a color with a low value is nearly black, but one with a high value is the pure, fully saturated color. In subtractive color (i.e. paints) value changes can be achieved by adding black or white to the color. However, this also reduces saturation. Chiaroscuro and Tenebrism both take advantage of dramatic contrasts of value to heighten drama in art. Artists may also employ shading, subtle manipulation of value.

6.1.1 Relationship between lightness, value, and relative luminance

The Munsell value has long been used as a perceptually uniform lightness scale. A question of interest is the relationship between the Munsell value scale and the relative luminance. Aware of the Weber–Fechner law, Munsell remarked “Should we use a logarithmic curve or curve of squares?"[2] Neither option turned out to be quite correct; scientists eventually converged on a roughly cube-root curve, consistent with the Stevens’ power law for brightness perception, reflecting the fact that lightness is proportional to the number of nerve impulses per nerve fiber per unit time.[3] The remainder of this section is a chronology of lightness approximations, leading to CIE LAB. Note. – Munsell’s V runs from 0 to 10, while Y typically runs from 0 to 100 (often interpreted as a percentage). Typically, the relative luminance is normalized so that the “reference white” (say, magnesium oxide) has a tristimulus value of Y = 100. Since the reflectance of magnesium oxide (MgO) relative to the perfect reflecting diffuser is 97.5%, V = 10 corresponds to Y = 100/97.5% ≈ 102.6 if MgO is used as the reference.[4]

1920 Priest et al. provide a basic estimate of the Munsell value (with Y running from 0 to 1 in this case):[5] √ V = 10 Y.

1933 Munsell, Sloan, and Godlove launch a study on the Munsell neutral value scale, considering several proposals relating the relative luminance to the Munsell value, and suggest:[6][7] V 2 = 1.474, 2Y − 0.004, 743Y 2.

1943 Newhall, Nickerson, and Judd prepare a report for the Optical Society of America. They suggest a quintic parabola (relating the reﬂectance in terms of the value):[8] Y = 1.221, 9V − 0.231, 11V 2 + 0.239, 51V 3 − 0.021, 009V 4 + 0.000, 840, 4V 5.

43 44 CHAPTER 6. DAY 6

1943 Using Table II of the O.S.A. report, Moon and Spencer express the value in terms of the relative luminance:[9] V = 5(Y /19.77)0.426 = 1.4Y 0.426.

1944 Saunderson and Milner introduce a subtractive constant in the previous expression, for a better ﬁt to the Munsell value.[10] Later, Jameson and Hurvich claim that this corrects for simultaneous contrast eﬀects.[11][12] V = 2.357Y 0.343 − 1.52.

1955 Ladd and Pinney of Eastman Kodak are interested in the Munsell value as a perceptually uniform lightness scale for use in television. After considering one logarithmic and five power-law functions (per Stevens’ power law), they relate value to reflectance by raising the reflectance to the power of 0.352:[13] V = 2.217Y 0.352 − 1.324.

Realizing this is quite close to the cube root, they simplify it to: V = 2.468Y 1/3 − 1.636.

1958 Glasser et al. deﬁne the lightness as ten times the Munsell value (so that the lightness ranges from 0 to 100):[14] L⋆ = 25.29Y 1/3 − 18.38.

1964 Wyszecki simpliﬁes this to:[15] W ⋆ = 25Y 1/3 − 17.

This formula approximates the Munsell value function for 1% < Y < 98% (it is not applicable for Y < 1%) and is used for the CIE 1964 color space.

1976 CIE LAB uses the following formula:

⋆ 1/3 L = 116(Y /Yn) − 16.

where Y is the CIE XYZ Y tristimulus value of the reference white point (the subscript n suggests “normalized”) and is subject to the restriction Y/Y > 0.01. Pauli removes this restriction by computing a linear extrapolation which maps Y/Y = 0 to L* = 0 and is tangent to the formula above at the point at which the linear extension takes eﬀect. First, the transition point is determined to be Y/Y = (6/29)^3 ≈ 0.008,856, then the slope of (29/3)^3 ≈ 903.3 is computed. This gives the two-part function:[16] { ( )3 t1/3 ift > 6 f(t) = ( ) 29 1 29 2 4 3 6 t + 29 otherwise

The lightness is then:

⋆ L = 116f(Y /Yn) − 16.

At first glance, you might approximate the lightness function by a cube root, an approximation that is found in much of the technical literature. However, the linear segment near black is significant, and so the 116 and 16 coefficients. The best-fit pure power function has an exponent of about 0.42, far from 1/3.[17] An approximately 18% grey card, having an exact reflectance of (33/58)3 , has a lightness value of 50. It is called “mid grey” because its lightness is midway between black and white.

6.1.2 Other psychological eﬀects

This subjective perception of luminance in a non-linear fashion is one thing that makes gamma compression of images worthwhile. Beside this phenomenon there are other effects involving perception of lightness. Chromacity can affect perceived lightness as described by the Helmholtz–Kohlrausch effect. Though the CIE LAB space and relatives do not account for this effect on lightness, it may be implied in the Munsell color model. Light levels may also affect perceived chromacity, as with the Purkinje effect. 6.1. LIGHTNESS 45

6.1.3 See also

• Brightness • Tints and shades

6.1.4 References

[1] Brightness vs. Lightness

[2] Kuehni, Rolf G. (February 2002). “The early development of the Munsell system”. Color Research & Application. 27 (1): 20–27. doi:10.1002/col.10002.

[3] Hunt, Robert W. G. (May 18, 1957). “Light Energy and Brightness Sensation”. Nature. 179 (4568): 1026. doi:10.1038/1791026a0.

[4] Valberg, Arne (2006). Light Vision Color. John Wiley and Sons. p. 200. ISBN 0470849029.

[5] Priest, Irwin G.; Gibson, K.S.; McNicholas, H.J. (September 1920). “An examination of the Munsell color system. I: Spectral and total reﬂection and the Munsell scale of Value”. Technical paper 167 (3). United States Bureau of Standards: 27.

[6] Munsell, A.E.O.; Sloan, L.L.; Godlove, I.H. (November 1933). “Neutral value scales. I. Munsell neutral value scale”. JOSA. 23 (11): 394–411. doi:10.1364/JOSA.23.000394. Note: This paper contains a historical survey stretching to 1760.

[7] Munsell, A.E.O.; Sloan, L.L.; Godlove, I.H. (December 1933). “Neutral value scales. II. A comparison of results and equations describing value scales”. JOSA. 23 (12): 419–425. doi:10.1364/JOSA.23.000419.

[8] Newhall, Sidney M.; Nickerson, Dorothy; Judd, Deane B (May 1943). “Final report of the O.S.A. subcommittee on the spacing of the Munsell colors”. Journal of the Optical Society of America. 33 (7): 385–418. doi:10.1364/JOSA.33.000385.

[9] Moon, Parry; Spencer, Domina Eberle (May 1943). “Metric based on the composite color stimulus”. JOSA. 33 (5): 270–277. doi:10.1364/JOSA.33.000270.

[10] Saunderson, Jason L.; Milner, B.I. (March 1944). “Further study of ω space”. JOSA. 34 (3): 167–173. doi:10.1364/JOSA.34.000167.

[11] Hurvich, Leo M.; Jameson, Dorothea (November 1957). “An Opponent-Process Theory of Color Vision”. Psychological Review. 64 (6): 384–404. doi:10.1037/h0041403. PMID 13505974.

[12] Jameson, Dorothea; Leo M. Hurvich (May 1964). “Theory of brightness and color contrast in human vision”. Vision Research. 4 (1-2): 135–154. doi:10.1016/0042-6989(64)90037-9. PMID 5888593.

[13] Ladd, J.H.; Pinney, J.E. (September 1955). “Empirical relationships with the Munsell Value scale”. Proceedings of the Institute of Radio Engineers. 43 (9): 1137. doi:10.1109/JRPROC.1955.277892.

[14] Glasser, L.G.; A.H. McKinney; C.D. Reilly; P.D. Schnelle (October 1958). “Cube-root color coordinate system”. JOSA. 48 (10): 736–740. doi:10.1364/JOSA.48.000736.

[15] Wyszecki, Günther (November 1963). “Proposal for a New Color-Diﬀerence Formula”. JOSA. 53 (11): 1318–1319. doi:10.1364/JOSA.53.001318. Note: The asterisks are not used in the paper.

[16] Pauli, Hartmut K.A. (1976). “Proposed extension of the CIE recommendation on “Uniform color spaces, color spaces, and color-diﬀerence equations, and metric color terms"". JOSA. 66 (8): 866–867. doi:10.1364/JOSA.66.000866.

[17] Poynton, Charles; Funt, Brian (Feb 2014). “Perceptual uniformity in digital image representation and display”. Color Research and Application. 39 (1): 6–15. doi:10.1002/col.21768.

6.1.5 External links

Media related to Lightness at Wikimedia Commons 46 CHAPTER 6. DAY 6

9

8

7

6

5

4

3

2

1

0

Three hues in the Munsell color model. Each color diﬀers in value from top to bottom in equal perception steps. The right column undergoes a dramatic change in perceived color. 6.1. LIGHTNESS 47

100

80

60

40 Lightness (L*, 10 x V, etc.) Munsell Renotation Hunter Rational Function 20 Square Root (Priest) Ladd & Pinney Cube Root (truncated)

0 0 0.2 0.4 0.6 0.8 1

Y/Yn

Observe that the lightness is 50% for a relative luminance of around 18% relative to the reference white. Chapter 7

Day 7

7.1 Space

This article is about the general framework of distance and direction. For the space beyond Earth’s atmosphere, see Outer space. For the keyboard key, see Space bar. For other uses, see Space (disambiguation).

Space is the boundless three-dimensional extent in which objects and events have relative position and direction.[1] Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the Timaeus of Plato, or Socrates in his reflections on what the Greeks called khôra (i.e. “space”), or in the Physics of Aristotle (Book IV, Delta) in the definition of topos (i.e. place), or in the later “geometrical conception of place” as “space qua extension” in the Discourse on Place (Qawl fi al-Makan) of the 11th-century Arab polymath Alhazen.[2] Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics. In Isaac Newton's view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space.[3] Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the “visibility of spatial depth” in his Essay Towards a New Theory of Vision. Later, the metaphysician Immanuel Kant said that the concepts of space and time are not empirical ones derived from experiences of the outside world—they are elements of an already given systematic framework that humans possess and use to structure all experiences. Kant referred to the experience of “space” in his Critique of Pure Reason as being a subjective “pure a priori form of intuition”. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space.[4] Experimental tests of general relativity have confirmed that non- Euclidean geometries provide a better model for the shape of space.

7.1.1 Philosophy of space

Leibniz and Newton

In the seventeenth century, the philosophy of space and time emerged as a central issue in epistemology and metaphysics. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, the English physicist- mathematician, set out two opposing theories of what space is. Rather than being an entity that independently exists over and above other matter, Leibniz held that space is no more than the collection of spatial relations between objects in the world: “space is that which results from places taken together”.[5] Unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an idealised abstraction

48 7.1. SPACE 49

y

x

z

A right-handed three-dimensional Cartesian coordinate system used to indicate positions in space.

from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete.[6] Space could be thought of in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people.[7] Leibniz argued that space could not exist independently of objects in the world because that implies a difference between two universes exactly alike except for the location of the material world in each universe. But since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them. According to the principle of sufficient reason, any theory of space that implied that there could be these two possible universes must therefore be wrong.[8] Newton took space to be more than relations between material objects and based his position on observation and experimentation. For a relationist there can be no real difference between inertial motion, in which the object travels with constant velocity, and non-inertial motion, in which the velocity changes with time, since all spatial measurements are relative to other objects and their motions. But Newton argued that since non-inertial motion generates forces, it must be absolute.[9] He used the example of water in a spinning bucket to demonstrate his argument. Water in a bucket is hung from a rope and set to spin, starts with a flat surface. After a while, as the bucket continues to spin, the surface of the water becomes concave. If the bucket’s spinning is stopped then the surface of the water remains concave as it continues to spin. The concave surface is therefore apparently not the result of relative motion between the bucket and the water.[10] Instead, Newton argued, it must be a result of non-inertial motion relative to space itself. For several centuries the bucket argument was considered decisive in showing that space must exist independently of 50 CHAPTER 7. DAY 7

Gottfried Leibniz matter.

Kant

In the eighteenth century the German philosopher Immanuel Kant developed a theory of knowledge in which knowledge about space can be both a priori and synthetic.[11] According to Kant, knowledge about space is synthetic, in that statements about space are not simply true by virtue of the meaning of the words in the statement. In his work, Kant rejected the view that space must be either a substance or relation. Instead he came to the conclusion that space and time are not discovered by humans to be objective features of the world, but imposed by us as part of a framework 7.1. SPACE 51

Isaac Newton

for organizing experience.[12] 52 CHAPTER 7. DAY 7

Immanuel Kant

Non-Euclidean geometry

Main article: Non-Euclidean geometry Euclid’s Elements contained ﬁve postulates that form the basis for Euclidean geometry. One of these, the parallel 7.1. SPACE 53

N

90°90°

S

Spherical geometry is similar to elliptical geometry. On a sphere (the surface of a ball) there are no parallel lines.

postulate, has been the subject of debate among mathematicians for many centuries. It states that on any plane on which there is a straight line L1 and a point P not on L1, there is exactly one straight line L2 on the plane that passes through the point P and is parallel to the straight line L1. Until the 19th century, few doubted the truth of the postulate; instead debate centered over whether it was necessary as an axiom, or whether it was a theory that could be derived from the other axioms.[13] Around 1830 though, the Hungarian János Bolyai and the Russian Nikolai Ivanovich Lobachevsky separately published treatises on a type of geometry that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an inﬁnite number of parallel lines pass through the point P. Consequently, the sum of angles in a triangle is less than 180° and the ratio of a circle's circumference to its diameter is greater than pi. In the 1850s, Bernhard Riemann developed an equivalent theory of elliptical geometry, in which no parallel lines pass through P. In this geometry, triangles have more than 180° and circles have a ratio of circumference-to-diameter that is less than pi. 54 CHAPTER 7. DAY 7

Carl Friedrich Gauss

Gauss and Poincaré

Although there was a prevailing Kantian consensus at the time, once non-Euclidean geometries had been formalised, some began to wonder whether or not physical space is curved. Carl Friedrich Gauss, a German mathematician, was the ﬁrst to consider an empirical investigation of the geometrical structure of space. He thought of making a test of the sum of the angles of an enormous stellar triangle, and there are reports that he actually carried out a test, on a small scale, by triangulating mountain tops in Germany.[14] 7.1. SPACE 55

Henri Poincaré

Henri Poincaré, a French mathematician and physicist of the late 19th century, introduced an important insight in which he attempted to demonstrate the futility of any attempt to discover which geometry applies to space by experiment.[15] He considered the predicament that would face scientists if they were conﬁned to the surface of an imaginary large sphere with particular properties, known as a sphere-world. In this world, the temperature is taken to vary in such a way that all objects expand and contract in similar proportions in diﬀerent places on the sphere. 56 CHAPTER 7. DAY 7

With a suitable falloﬀ in temperature, if the scientists try to use measuring rods to determine the sum of the angles in a triangle, they can be deceived into thinking that they inhabit a plane, rather than a spherical surface.[16] In fact, the scientists cannot in principle determine whether they inhabit a plane or sphere and, Poincaré argued, the same is true for the debate over whether real space is Euclidean or not. For him, which geometry was used to describe space was a matter of convention.[17] Since Euclidean geometry is simpler than non-Euclidean geometry, he assumed the former would always be used to describe the 'true' geometry of the world.[18]

Einstein

In 1905, Albert Einstein published his special theory of relativity, which led to the concept that space and time can be viewed as a single construct known as spacetime. In this theory, the speed of light in a vacuum is the same for all observers—which has the result that two events that appear simultaneous to one particular observer will not be simultaneous to another observer if the observers are moving with respect to one another. Moreover, an observer will measure a moving clock to tick more slowly than one that is stationary with respect to them; and objects are measured to be shortened in the direction that they are moving with respect to the observer. Subsequently, Einstein worked on a general theory of relativity, which is a theory of how gravity interacts with spacetime. Instead of viewing gravity as a force field acting in spacetime, Einstein suggested that it modifies the geometric structure of spacetime itself.[19] According to the general theory, time goes more slowly at places with lower gravitational potentials and rays of light bend in the presence of a gravitational field. Scientists have studied the behaviour of binary pulsars, confirming the predictions of Einstein’s theories, and non-Euclidean geometry is usually used to describe spacetime.

7.1.2 Mathematics

Main article: Three-dimensional space Not to be confused with Space (mathematics).

In modern mathematics spaces are defined as sets with some added structure. They are frequently described as different types of manifolds, which are spaces that locally approximate to Euclidean space, and where the properties are defined largely on local connectedness of points that lie on the manifold. There are however, many diverse mathematical objects that are called spaces. For example, vector spaces such as function spaces may have infinite numbers of independent dimensions and a notion of distance very different from Euclidean space, and topological spaces replace the concept of distance with a more abstract idea of nearness.

7.1.3 Physics

Many of the laws of physics, such as the various inverse square laws, depend on dimension three.[20] In physics, our three-dimensional space is viewed as embedded in four-dimensional spacetime, called Minkowski space (see special relativity). The idea behind space-time is that time is hyperbolic-orthogonal to each of the three spatial dimensions.

Classical mechanics

Main article: Classical mechanics

Space is one of the few fundamental quantities in physics, meaning that it cannot be deﬁned via other quantities because nothing more fundamental is known at the present. On the other hand, it can be related to other fundamental quantities. Thus, similar to other fundamental quantities (like time and mass), space can be explored via measurement and experiment. 7.1. SPACE 57

Albert Einstein

Relativity

Main article: Theory of relativity

Before Einstein's work on relativistic physics, time and space were viewed as independent dimensions. Einstein’s discoveries showed that due to relativity of motion our space and time can be mathematically combined into one 58 CHAPTER 7. DAY 7

object–spacetime. It turns out that distances in space or in time separately are not invariant with respect to Lorentz coordinate transformations, but distances in Minkowski space-time along space-time intervals are—which justifies the name. In addition, time and space dimensions should not be viewed as exactly equivalent in Minkowski space-time. One can freely move in space but not in time. Thus, time and space coordinates are treated differently both in special relativity (where time is sometimes considered an imaginary coordinate) and in general relativity (where different signs are assigned to time and space components of spacetime metric). Furthermore, in Einstein’s general theory of relativity, it is postulated that space-time is geometrically distorted- curved -near to gravitationally significant masses.[21] One consequence of this postulate, which follows from the equations of general relativity, is the prediction of moving ripples of space-time, called gravitational waves. While indirect evidence for these waves has been found (in the motions of the Hulse–Taylor binary system, for example) experiments attempting to directly measure these waves are ongoing.

Cosmology

Main article: Shape of the universe

Relativity theory leads to the cosmological question of what shape the universe is, and where space came from. It appears that space was created in the Big Bang, 13.8 billion years ago[22] and has been expanding ever since. The overall shape of space is not known, but space is known to be expanding very rapidly due to the Cosmic Inﬂation.

7.1.4 Spatial measurement

Main article: Measurement

The measurement of physical space has long been important. Although earlier societies had developed measuring systems, the International System of Units, (SI), is now the most common system of units used in the measuring of space, and is almost universally used. Currently, the standard space interval, called a standard meter or simply meter, is defined as the distance traveled by light in a vacuum during a time interval of exactly 1/299,792,458 of a second. This definition coupled with present definition of the second is based on the special theory of relativity in which the speed of light plays the role of a fundamental constant of nature.

7.1.5 Geographical space

See also: Spatial analysis

Geography is the branch of science concerned with identifying and describing the Earth, utilizing spatial awareness to try to understand why things exist in speciﬁc locations. Cartography is the mapping of spaces to allow better navigation, for visualization purposes and to act as a locational device. Geostatistics apply statistical concepts to collected spatial data to create an estimate for unobserved phenomena. Geographical space is often considered as land, and can have a relation to ownership usage (in which space is seen as property or territory). While some cultures assert the rights of the individual in terms of ownership, other cultures will identify with a communal approach to land ownership, while still other cultures such as Australian Aboriginals, rather than asserting ownership rights to land, invert the relationship and consider that they are in fact owned by the land. Spatial planning is a method of regulating the use of space at land-level, with decisions made at regional, national and international levels. Space can also impact on human and cultural behavior, being an important factor in architecture, where it will impact on the design of buildings and structures, and on farming. Ownership of space is not restricted to land. Ownership of airspace and of waters is decided internationally. Other forms of ownership have been recently asserted to other spaces—for example to the radio bands of the electromagnetic spectrum or to cyberspace. 7.1. SPACE 59

Public space is a term used to deﬁne areas of land as collectively owned by the community, and managed in their name by delegated bodies; such spaces are open to all, while private property is the land culturally owned by an individual or company, for their own use and pleasure. Abstract space is a term used in geography to refer to a hypothetical space characterized by complete homogeneity. When modeling activity or behavior, it is a conceptual tool used to limit extraneous variables such as terrain.

7.1.6 In psychology

Psychologists ﬁrst began to study the way space is perceived in the middle of the 19th century. Those now concerned with such studies regard it as a distinct branch of psychology. Psychologists analyzing the perception of space are concerned with how recognition of an object’s physical appearance or its interactions are perceived, see, for example, visual space. Other, more specialized topics studied include amodal perception and object permanence. The perception of surroundings is important due to its necessary relevance to survival, especially with regards to hunting and self preser- vation as well as simply one’s idea of personal space. Several space-related phobias have been identiﬁed, including agoraphobia (the fear of open spaces), astrophobia (the fear of celestial space) and claustrophobia (the fear of enclosed spaces). The understanding of three-dimensional space in humans is thought to be learned during infancy using unconscious inference, and is closely related to hand-eye coordination. The visual ability to perceive the world in three dimensions is called depth perception.

7.1.7 See also

• Absolute space and time

• Aether theories

• Cosmology

• General relativity

• Personal space

• Shape of the universe

• Space exploration

• Spatial-temporal reasoning

• Spatial analysis

7.1.8 References

[1] “space - physics and metaphysics”. Encyclopedia Britannica.

[2] Refer to Plato’s Timaeus in the Loeb Classical Library, Harvard University, and to his reflections on khora. See also Aristotle’s Physics, Book IV, Chapter 5, on the definition of topos. Concerning Ibn al-Haytham’s 11th century conception of “geometrical place” as “spatial extension”, which is akin to Descartes' and Leibniz’s 17th century notions of extensio and analysis situs, and his own mathematical refutation of Aristotle’s definition of topos in natural philosophy, refer to: Nader El-Bizri, “In Defence of the Sovereignty of Philosophy: al-Baghdadi’s Critique of Ibn al-Haytham’s Geometrisation of Place”, Arabic Sciences and Philosophy (Cambridge University Press), Vol. 17 (2007), pp. 57-80.

[3] French and Ebison, Classical Mechanics, p. 1

[4] Carnap, R. An introduction to the Philosophy of Science

[5] Leibniz, Fifth letter to Samuel Clarke

[6] Vailati, E, Leibniz & Clarke: A Study of Their Correspondence p. 115 60 CHAPTER 7. DAY 7

[7] Sklar, L, Philosophy of Physics, p. 20

[8] Sklar, L, Philosophy of Physics, p. 21

[9] Sklar, L, Philosophy of Physics, p. 22

[10] “Newton’s bucket”. st-and.ac.uk.

[11] Carnap, R, An introduction to the philosophy of science, p. 177-178

[12] Lucas, John Randolph. Space, Time and Causality. p. 149. ISBN 0-19-875057-9.

[13] Carnap, R, An introduction to the philosophy of science, p. 126

[14] Carnap, R, An introduction to the philosophy of science, p. 134-136

[15] Jammer, M, Concepts of Space, p. 165

[16] A medium with a variable index of refraction could also be used to bend the path of light and again deceive the scientists if they attempt to use light to map out their geometry

[17] Carnap, R, An introduction to the philosophy of science, p. 148

[18] Sklar, L, Philosophy of Physics, p. 57

[19] Sklar, L, Philosophy of Physics, p. 43

[20] Greene, Brian (2003). The Fabric of the Cosmos. New York: Random House. ISBN 0-375-72720-5.

[21] chapters 8 and 9- John A. Wheeler “A Journey Into Gravity and Spacetime” Scientiﬁc American ISBN 0-7167-6034-7

[22] “Cosmic Detectives”. The European Space Agency (ESA). 2013-04-02. Retrieved 2013-04-26.

7.1.9 External links

• Seth Shostak on Space Exploration Chapter 8

Day 8

8.1 Rule of thirds

This article is about the visual arts rule. For the scuba diving rule, see Rule of thirds (diving). For the rule of thumb used in military organisation, see Rule of thirds (military). For similar concepts, see Rule of three (disambiguation). The rule of thirds is a "rule of thumb" or guideline which applies to the process of composing visual images such as designs, ﬁlms, paintings, and photographs.[1] The guideline proposes that an image should be imagined as divided into nine equal parts by two equally spaced horizontal lines and two equally spaced vertical lines, and that important compositional elements should be placed along these lines or their intersections.[2] Proponents of the technique claim that aligning a subject with these points creates more tension, energy and interest in the composition than simply centering the subject. The photograph to the right demonstrates the application of the rule of thirds. The horizon sits at the horizontal line dividing the lower third of the photo from the upper two-thirds. The tree sits at the intersection of two lines, sometimes called a power point or a crash point. Points of interest in the photo do not have to actually touch one of these lines to take advantage of the rule of thirds. For example, the brightest part of the sky near the horizon where the sun recently set does not fall directly on one of the lines, but does fall near the intersection of two of the lines, close enough to take advantage of the rule.

8.1.1 Use

The rule of thirds is applied by aligning a subject with the guide lines and their intersection points, placing the horizon on the top or bottom line, or allowing linear features in the image to flow from section to section. The main reason for observing the rule of thirds is to discourage placement of the subject at the center, or prevent a horizon from appearing to divide the picture in half. Michael Ryan and Melissa Lenos, authors of the book An Introduction to Film Analysis: Technique and Meaning in Narrative Film state that the use of rule of thirds is “favored by cinematographers in their effort to design balanced and unified images” (page 40).[3] When filming or photographing people, it is common to line the body up to a vertical line and the person’s eyes to a horizontal line. If filming a moving subject, the same pattern is often followed, with the majority of the extra room being in front of the person (the way they are moving).[4] Likewise, when photographing a still subject who is not directly facing the camera, the majority of the extra room should be in front of the subject with the vertical line running through their perceived center of mass.

8.1.2 History

The rule of thirds was ﬁrst written down[5] by John Thomas Smith in 1797. In his book Remarks on Rural Scenery, Smith quotes a 1783 work by Sir Joshua Reynolds, in which Reynolds discusses, in unquantiﬁed terms, the balance of dark and light in a painting.[6] Smith then continues with an expansion on the idea, naming it the “Rule of thirds":

Two distinct, equal lights, should never appear in the same picture : One should be principal, and the rest sub-ordinate, both in dimension and degree : Unequal parts and gradations lead the attention

61 62 CHAPTER 8. DAY 8

This photograph demonstrates the principles of the rule of thirds

easily from part to part, while parts of equal appearance hold it awkwardly suspended, as if unable to determine which of those parts is to be considered as the subordinate. “And to give the utmost force and solidity to your work, some part of the picture should be as light, and some as dark as possible : These two extremes are then to be harmonized and reconciled to each other.” (Reynolds’ Annot. on Du Fresnoy.)

Analogous to this “Rule of thirds”, (if I may be allowed so to call it) I have presumed to think that, in connecting or in breaking the various lines of a picture, it would likewise be a good rule to do it, in general, by a similar scheme of proportion; for example, in a design of landscape, to determine the sky at about two-thirds ; or else at about one-third, so that the material objects might occupy the other two : Again, two thirds of one element, (as of water) to one third of another element (as of land); and then both together to make but one third of the picture, of which the two other thirds should go for the sky and aerial perspectives. This rule would likewise apply in breaking a length of wall, or any other too great continuation of line that it may be found necessary to break by crossing or hiding it with some other object : In short, in applying this invention, generally speaking, or to any other case, whether of light, shade, form, or color, I have found the ratio of about two thirds to one third, or of one to two, a much better and more harmonizing proportion, than the precise formal half, the too-far-extending four- ﬁfths—and, in short, than any other proportion whatever. I should think myself honored by the opinion of any gentleman on this point; but until I shall by better informed, shall conclude this general proportion of 8.1. RULE OF THIRDS 63

A typical usage of the rule of thirds

A picture cropped without and with the rule of thirds

two and one to be the most pictoresque medium in all cases of breaking or otherwise qualifying straight lines and masses and groupes [sic], as Hogarth’s line is agreed to be the most beautiful, (or, in other words, the most pictoresque) medium of curves.[7]

Writing in 1845, in his book Chromatics, George Field notes (perhaps erroneously) that Sir Joshua Reynolds gives the ratio 2:1 as a rule for the proportion of warm to cold colors in a painting, and attributes to Smith the expansion of that rule to all proportions in painting:

Sir Joshua has given it as a rule, that the proportion of warm to cold colour in a picture should be as two to one, although he has frequently deviated therefrom; and Smith, in his “Remarks on Rural Scenery,” would extend a like rule to all the proportions of painting, begging for it the term of the “rule of thirds,” according to which, a landscape, having one third of land, should have two thirds of water, and these together, forming about one-third of the picture, the remaining two-thirds to be for air and sky; and he applies the same rule to the crossing and breaking of lines and objects, &c. [8] 64 CHAPTER 8. DAY 8

Excerpt from John Thomas Smith’s illustrated book, published in 1797, deﬁning a compositional “rule of thirds”

Even at this early date, there was skepticism over the universality of such a rule, at least in regards to color, for Field continues:

This rule, however, does not supply a general law, but universalises a particular, the invariable obser- vance of which would produce a uniform and monotonous practice. But, however occasionally useful, it is neither accurate nor universal, the true mean of nature requiring compensation, which, in the case of warmth and coolness, is in about equal proportions, while, in regard to advancing and retiring colours, the true balance of effect is, approximately, three of the latter to one of the former; nevertheless, the proportions in both cases are to be governed by the predominance of light or shade, and the required effect of a picture, in which, and other species of antagonism, the scale of equivalents affords a guide.

Smith’s conception of the rule is meant to apply more generally than the version commonly explained today, as he recommends it not just for dividing the frame, but also for all division of straight lines, masses, or groups. On the other hand, he does not discuss the now-common idea that intersections of the third-lines of the frame are particularly strong or interesting for composition.

8.1.3 See also

• Golden ratio (in aesthetics)

• Headroom (photographic framing)

• Lead room

• Rabatment of the rectangle 8.1. RULE OF THIRDS 65

8.1.4 References

[1] Sandra Meech (2007). Contemporary Quilts: Design, Surface and Stitch. Sterling Publishing. ISBN 0-7134-8987-1.

[2] Bryan F. Peterson (2003). Learning to see creatively. Amphoto Press. ISBN 0-8174-4181-6.

[3] Bert P. Krages (2005). The Art of Composition. Allworth Communications, Inc. ISBN 1-58115-409-7.

[4] leadroom

[5] Caplin, Steve (2008). Art and Design in Photoshop. Focal Press. p. 35.

[6] Reynolds, Sir Joshua (1783). Annotations on The art of painting of Charles Alphonse Du Fresnoy. Printed by A. Ward, and sold by J. Dodsley. p. 103.

[7] Smith, John Thomas (1797). Remarks on rural scenery; with twenty etchings of cottages, from nature; and some observations and precepts relative to the pictoresque. printed for, and sold by Nathaniel Smith ancient Print seller at Rembrandts-Head May’s Buildings, St. Martin’s Lane, and I. T. Smith, at No 40 Trith Street Soho. pp. 15–17.

[8] Field, George (1845). Chromatics; or, The analogy, harmony, and philosophy of colours. David Bogue, Fleet Street. Chapter 9

Day 9

9.1 Golden ratio

This article is about the number. It is not to be confused with the pop music album or the calendar dates.

is to as is to

Line segments in the golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The ﬁgure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, a + b a = def= φ, a b where the Greek letter phi ( φ or ϕ ) represents the golden ratio. Its value is:

√ 1+ 5 φ = 2 = 1.6180339887 .... A001622

The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[1][2][3] Other names include extreme and mean ratio,[4] medial section, divine proportion, divine section (Latin: sectio divina), golden

66 9.1. GOLDEN RATIO 67

A golden rectangle (in pink) with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will a+b a ≡ produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship a = b φ .

proportion, golden cut,[5] and golden number.[6][7][8] Some twentieth-century artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as ﬁnancial markets, in some cases based on dubious ﬁts to data.[9]

9.1.1 Calculation

Two quantities a and b are said to be in the golden ratio φ if

a + b a = = φ. a b One method for ﬁnding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, 68 CHAPTER 9. DAY 9

a + b b 1 = 1 + = 1 + . a a φ

Therefore,

1 1 + = φ. φ

Multiplying by φ gives

φ + 1 = φ2

which can be rearranged to

φ2 − φ − 1 = 0.

Using the quadratic formula, two solutions are obtained:

√ 1 + 5 φ = = 1.61803 39887 ... 2 and

√ 1 − 5 φ = = −0.6180 339887 ... 2 Because φ is the ratio between positive quantities φ is necessarily positive:

√ 1 + 5 φ = = 1.61803 39887 ... 2 This derivation can also be found with a compass-and-straightedge construction:

The initial situation is the dividing a line segment by exterior division with the additions a = 1 and therefore a + b = φ. First, the line segment AB is about doubled and then the semicircle with the radius AS around the point S is drawn, thus the intersection point D is obtained. Now the semicircle is drawn with the radius AB around the point B. The arising intersection point E corresponds 2φ. Next up, the perpendicular on the line segment AE from the point D will be establish. The subsequent parallel FS to the line segment CM , produces, as it were, the hypotenuse of the right triangle SDF. It is well recognizable, this triangle and the triangle MSC are similar to each other. The hypotenuse FS has due to the cathetuses√ SD = 1 and DF = 2 according the Pythagorean theorem, a length that is equal to the value of 5. √ Finally, the circle arc is drawn with the radius 5 around the point F. Because of SD√= 2 · MS the circular arc meets the point E respectively 2φ , and thus leads to the result 2φ = 1 + 5 , from this it follows that √ 1+ 5 φ = 2 = 1.61803 39887 ... . 9.1. GOLDEN RATIO 69

Golden ratio, derivation of the numerical value ( Φ ≡ φ ) animation 1 min 15 s

9.1.2 History

Further information: Mathematics and art The golden ratio has been claimed to have held a special fascination for at least 2,400 years, although without reliable evidence.[11] According to Mario Livio:

“Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.”[12]

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into “extreme and mean ratio” (the golden section) is important in the geometry of regular pentagrams and pentagons. Euclid's Elements (Greek: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio:

“A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.”[13]

Euclid explains a construction for cutting (sectioning) a line “in extreme and mean ratio” (i.e., the golden ratio).[14] Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.[15] 70 CHAPTER 9. DAY 9

Mathematician Mark Barr proposed using the ﬁrst letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. Usually, the lowercase form (φ) is used. Sometimes the uppercase form ( Φ ) is used for the reciprocal of the golden ratio, 1/ φ.[10]

The golden ratio is explored in Luca Pacioli's book De divina proportione (1509).[8] The ﬁrst known approximation of the (inverse) golden ratio by a decimal fraction, stated as “about 0.6180340”, was written in 1597 by Michael Maestlin of the University of Tübingen in a letter to his former student Johannes Kepler.[16] Since the 20th century, the golden ratio has been represented by the Greek letter φ (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the ﬁrst letter of the ancient Greek root τομή—meaning cut).[1][17]

Timeline

Timeline according to Priya Hemenway:[18]

• Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.

• Plato (427–347 BC), in his Timaeus, describes ﬁve possible regular solids (the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron), some of which are related to the golden ratio.[19]

• Euclid (c. 325–c. 265 BC), in his Elements, gave the ﬁrst recorded deﬁnition of the golden ratio, which he called, as translated into English, “extreme and mean ratio” (Greek: ἄκρος καὶ μέσος λόγος).[4]

• Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci; the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.

• Luca Pacioli (1445–1517) deﬁnes the golden ratio as the “divine proportion” in his Divina Proportione.

• Michael Maestlin (1550–1631) publishes the ﬁrst known approximation of the (inverse) golden ratio as a decimal fraction.

• Johannes Kepler (1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers,[20] and describes the golden ratio as a “precious jewel": “Geometry has two great treasures: one is 9.1. GOLDEN RATIO 71

the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the ﬁrst we may compare to a measure of gold, the second we may name a precious jewel.” These two treasures are combined in the Kepler triangle.

• Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter- clockwise were frequently two successive Fibonacci series.

• Martin Ohm (1792–1872) is believed to be the ﬁrst to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.[21]

• Édouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.

• Mark Barr (20th century) suggests the Greek letter phi (φ), the initial letter of Greek sculptor Phidias’s name, as a symbol for the golden ratio.[22]

• Roger Penrose (b. 1931) discovered in 1974 the Penrose tiling, a pattern that is related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.[23] This in turn led to new discoveries about quasicrystals.[24]

9.1.3 Applications and observations

Aesthetics

See also: History of aesthetics (pre-20th-century) and Mathematics and art

De Divina Proportione, a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. De Divina Proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio’s application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[1] Pacioli also saw Catholic religious signiﬁcance in the ratio, which led to his work’s title. De Divina Proportione contains illustrations of regular solids by Leonardo da Vinci, Pacioli’s longtime friend and collaborator; these are not directly linked to the golden ratio.

Architecture

Further information: Mathematics and architecture

The Parthenon's façade as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles.[25] Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazalé says, “It was not until Euclid, however, that the golden ratio’s mathematical properties were studied. In the Elements (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties.”[26] And Keith Devlin says, “Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 BC, showed how to calculate its value.”[27] Later sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions. A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.[28] They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. The authors note, however, that the areas where ratios close to the golden ratio were found are not part of the original construction, and theorize that these elements were added in a reconstruction. 72 CHAPTER 9. DAY 9

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier’s faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as “rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned.”[29] Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci’s "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body’s height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier’s 1927 Villa Stein in Garches exemplified the Modulor system’s application. The villa’s rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.[30] Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.[31] In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.[32] From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher has concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries.[33]

Painting

The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram inside a circle, implying a relationship to the golden ratio.[2] Leonardo da Vinci's illustrations of polyhedra in De divina proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings.[34] But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by anything in Leonardo’s own writings.[35] Similarly, although the Vitruvian Man is often[36] shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.[37] Salvador Dalí, influenced by the works of Matila Ghyka,[38] explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[1][39] Mondrian has been said to have used the golden section extensively in his geometrical paintings,[40] though other experts (including critic Yve-Alain Bois) have disputed this claim.[1] A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).[41] On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.[42]

Book design

Main article: Canons of page construction

According to Jan Tschichold,[44]

There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden 9.1. GOLDEN RATIO 73

Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.

Design

Some sources claim that the golden ratio is commonly used in everyday design, for example in the shapes of postcards, playing cards, posters, wide-screen televisions, photographs, light switch plates and cars.[45][46][47][48][49]

Music

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[50] though other music scholars reject that analysis.[1] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which “the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position.”[51] The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section.[52] Trezise finds the intrinsic evidence “remarkable,” but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[53] Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.[54] Though Heinz Bohlen proposed the non-octave-repeating 833 cents scale based on combination tones, the tuning features relations based on the golden ratio. As a musical interval the ratio 1.618... is 833.090... cents ( Play ).[55]

Nature

Main article: Patterns in nature

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of parts such as leaves and branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these patterns in nature he saw the golden ratio operating as a universal law.[56][57] In connection with his scheme for golden-ratio-based human body proportions, Zeising wrote in 1854 of a universal law “in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which ﬁnds its fullest realization, however, in the human form.”[58] In 2010, the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.[59] Since 1991, several researchers have proposed connections between the golden ratio and human genome DNA.[60][61][62] However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are ﬁctitious.[63]

Optimization

The golden ratio is key to the golden section search.

Perceptual studies

Studies by psychologists, starting with Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.[1][64] 74 CHAPTER 9. DAY 9

9.1.4 Mathematics

Irrationality

The golden ratio is an irrational number. Below are two short proofs of irrationality:

Contradiction from an expression in lowest terms Recall that:

the whole is the longer part plus the shorter part; the whole is to the longer part as the longer part is to the shorter part.

If we call the whole n and the longer part m, then the second statement above becomes

n is to m as m is to n − m,

or, algebraically

n m = . (∗) m n − m To say that φ is rational means that φ is a fraction n/m where n and m are integers. We may take n/m to be in lowest terms and n and m to be positive. But if n/m is in lowest terms, then the identity labeled (*) above says m/(n − m) is in still lower terms. That is a contradiction that follows from the assumption that φ is rational.

Derivation from irrationality of √5 Another short proof—perhaps more commonly known—of the√ irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If 1+ 5 is rational, ( √ ) √ 2 1+ 5 − then 2 2 1 = 5 is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational.

Minimal polynomial

The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial

x2 − x − 1

Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate.

Golden ratio conjugate

The conjugate root to the minimal polynomial x2 - x - 1 is

√ 1 1 − 5 − = 1 − φ = = −0.61803 39887 .... φ 2 The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate.[10] It is denoted here by the capital Phi ( Φ ):

1 Φ = = φ−1 = 0.61803 39887 .... φ Alternatively, Φ can be expressed as 9.1. GOLDEN RATIO 75

Φ = φ − 1 = 1.61803 39887 ... − 1 = 0.61803 39887 .... This illustrates the unique property of the golden ratio among positive numbers, that

1 = φ − 1, φ or its inverse:

1 = Φ + 1. Φ This means 0.61803...:1 = 1:1.61803....

Alternative forms

The formula φ = 1 + 1/φ can be expanded recursively to obtain a continued fraction for the golden ratio:[65]

1 φ = [1; 1, 1, 1,... ] = 1 + 1 1 + 1 1 + . 1 + .. and its reciprocal:

1 φ−1 = [0; 1, 1, 1,... ] = 0 + 1 1 + 1 1 + . 1 + .. The convergents of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci numbers. The equation φ2 = 1 + φ likewise produces the continued square root, or inﬁnite surd, form:

√ √ √ √ φ = 1 + 1 + 1 + 1 + ···.

An inﬁnite series can be derived to express phi:[66]

∞ 13 ∑ (−1)(n+1)(2n + 1)! φ = + . 8 (n + 2)!n!4(2n+3) n=0 Also:

φ = 1 + 2 sin(π/10) = 1 + 2 sin 18◦ 1 1 φ = csc(π/10) = csc 18◦ 2 2 φ = 2 cos(π/5) = 2 cos 36◦ φ = 2 sin(3π/10) = 2 sin 54◦. These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a pentagram. 76 CHAPTER 9. DAY 9

Geometry

The number φ turns up frequently in geometry, particularly in ﬁgures with pentagonal symmetry. The length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several deﬁnitions of even distribution (see, for example, Thomson problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ ≅ 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[67]

Dividing a line segment by interior division

1. Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the hypotenuse AC.

2. Draw an arc with center C and radius BC. This arc intersects the hypotenuse AC at point D.

3. Draw an arc with center A and radius AD. This arc intersects the original line segment AB at point S. Point S divides the original line segment AB into line segments AS and SB with lengths in the golden ratio.

Dividing a line segment by exterior division

1. Construct on line segment AS oﬀ the point S, a vertical length of AS with the endpoint C.

2. Do bisect the line segment AS with M.

3. The circular arc around M with the radius MC divides the extension AS in point B. Point S divides the constructed line segment AB into line segments AS and SB with lengths in the golden ratio.

Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length. The both above displayed diﬀerent algorithms produce geometric constructions that divides a line segment into two line segments where the ratio of the longer to the shorter line segment is the golden ratio.

Golden triangle, pentagon and pentagram

Golden triangle The golden triangle can be characterized as an isosceles triangle ABC with the property that bisecting the angle C produces a new triangle CXB which is a similar triangle to the original. If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°−72°−72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°−36°−108°. Suppose XB has length 1, and we call BC length φ. Because of the isosceles triangles XC=XA and BC=XC, so these are also length φ. Length AC = AB, therefore equals φ + 1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, AC/φ = φ/1, and so AC also equals φ2. Thus φ2 = φ + 1, conﬁrming that φ is indeed the golden ratio. Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to φ, while the inverse ratio is φ − 1.

Pentagon In a regular pentagon the ratio between a side and a diagonal is Φ (i.e. 1/φ), while intersecting diagonals section each other in the golden ratio.[8] 9.1. GOLDEN RATIO 77

Odom’s construction George Odom has given a remarkably simple construction for φ involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the intersecting chords theorem and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer H. S. M. Coxeter who published it in Odom’s name as a diagram in the American Mathematical Monthly accompanied by the single word “Behold!" [68]

Pentagram The golden ratio plays an important role in the geometry of pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram’s center) is φ, as the four-color illustration shows. The pentagram includes ten isosceles triangles: ﬁve acute and ﬁve obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.

Ptolemy’s theorem The golden ratio properties of a regular pentagon can be conﬁrmed by applying Ptolemy’s theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral’s long edge and diagonals are b, and short edges are a, then Ptolemy’s theorem gives b2 = a2 + ab which yields

√ b 1 + 5 = . a 2

Scalenity of triangles Consider a triangle with sides of lengths a, b, and c in decreasing order. Deﬁne the “scalenity” of the triangle to be the smaller of the two ratios a/b and b/c. The scalenity is always less than φ and can be made as close as desired to φ.[69]

Triangle whose sides form a geometric progression If the side lengths of a triangle form a geometric progression and are in the ratio 1 : r : r2, where r is the common ratio, then r must lie in the range φ−1 < r < φ, which is a consequence of the triangle inequality (the sum of any two sides of a triangle must be strictly bigger than the length of the third side). If r = φ then the shorter two sides are 1 and φ but their sum is φ2, thus r < φ. A similar calculation shows that r > φ−1. A triangle whose sides are in the ratio 1 : √φ : φ is a right triangle (because 1 + φ = φ2) known as a Kepler triangle.[70]

Golden triangle, rhombus, and rhombic triacontahedron A golden rhombus is a rhombus whose diagonals are in the golden ratio. The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi. In the rhombic triacontahedron the dihedral angle between any two adjacent rhombi is 144°, which is twice the isosceles angle of a golden triangle and four times its most acute angle.[71]

Relationship to Fibonacci sequence

The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....

The closed-form expression for the Fibonacci sequence involves the golden ratio:

φn − (1 − φ)n φn − (−φ)−n F (n) = √ = √ . 5 5 The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by Kepler:[20] 78 CHAPTER 9. DAY 9

F (n + 1) lim = φ. n→∞ F (n) Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:

∑∞ |F (n)φ − F (n + 1)| = φ. n=1 More generally:

F (n + a) lim = φa, n→∞ F (n) where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when a = 1 . Furthermore, the successive powers of φ obey the Fibonacci recurrence:

φn+1 = φn + φn−1.

This identity allows any polynomial in φ to be reduced to a linear expression. For example:

3φ3 − 5φ2 + 4 = 3(φ2 + φ) − 5φ2 + 4 = 3[(φ + 1) + φ] − 5(φ + 1) + 4 = φ + 2 ≈ 3.618. The reduction to a linear expression can be accomplished in one step by using the relationship

k φ = Fkφ + Fk−1,

th where Fk is the k Fibonacci number. However, this is no special property of φ, because polynomials in any solution x to a quadratic equation can be reduced in an analogous manner, by applying:

x2 = ax + b

for given coefficients a, b such that x satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible nth-degree polynomial over the rationals can be reduced to a polynomial of degree n ‒ 1. Phrased in terms of field theory, if α is a root of an irreducible nth-degree polynomial, then Q(α) has degree n over Q , with basis {1, α, . . . , αn−1} .

Symmetries √ The golden ratio and inverse golden ratio φ = (1 5)/2 have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations x, 1/(1−x), (x−1)/x, – this fact corresponds to the identity and the deﬁnition quadratic equation. Further, they are interchanged by the three maps 1/x, 1 − x, x/(x − 1) – they are reciprocals, symmetric about 1/2 , and (projectively) symmetric about 2. More deeply, these maps form a subgroup of the modular group PSL(2, Z) isomorphic to the symmetric group on 3 letters, S3, corresponding to the stabilizer of the set {0, 1, ∞} of 3 standard points on the projective line, and the symmetries correspond to the quotient map S3 → S2 – the subgroup C3 < S3 consisting of the 3-cycles and the identity ()(01∞)(0∞1) ﬁxes the two numbers, while the 2-cycles interchange these, thus realizing the map. 9.1. GOLDEN RATIO 79

Other properties

The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see Alternate forms above). It is, for that reason, one of the worst cases of Lagrange’s approximation theorem and it is an extremal case of the Hurwitz inequality for Diophantine approximations. This may be why angles close to the golden ratio often show up in phyllotaxis (the growth of plants).[72] The deﬁning quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with φ:

φ2 = φ + 1 = 2.618 ... 1 = φ − 1 = 0.618 .... φ The sequence of powers of φ contains these values 0.618..., 1.0, 1.618..., 2.618...; more generally, any power of φ is equal to the sum of the two immediately preceding powers:

n n−1 n−2 φ = φ + φ = φ · Fn + Fn−1 .

As a result, one can easily decompose any power of φ into a multiple of φ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of φ: If ⌊n/2 − 1⌋ = m , then:

φn = φn−1 + φn−3 + ··· + φn−1−2m + φn−2−2m

φn − φn−1 = φn−2. When the golden ratio is used as the base of a numeral system (see Golden ratio base, sometimes dubbed phinary or φ-nary), every integer has a terminating representation, despite φ being irrational, but every fraction has a non- terminating representation. √ Q [73] The golden ratio√ is a fundamental unit of the√ algebraic number ﬁeld ( 5) and is a Pisot–Vijayaraghavan number. Q n Ln+Fn 5 In the ﬁeld ( 5) we have φ = 2 , where Ln is the n -th Lucas number. The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is 4 log(φ) .[74]

Decimal expansion

The golden ratio’s decimal expansion can be calculated directly from the expression

√ 1 + 5 φ = 2

with √5 ≈ 2.2360679774997896964 A002163. The square root of 5 can be calculated with the Babylonian method, starting with an initial estimate such as xφ = 2 and iterating

(x + 5/x ) x = n n n+1 2 for n = 1, 2, 3, ..., until the diﬀerence between xn and xn₋₁ becomes zero, to the desired number of digits. The Babylonian algorithm for √5 is equivalent to Newton’s method for solving the equation x2 − 5 = 0. In its more general form, Newton’s method can be applied directly to any algebraic equation, including the equation x2 − x − 1 = 0 that deﬁnes the golden ratio. This gives an iteration that converges to the golden ratio itself, 80 CHAPTER 9. DAY 9

2 xn + 1 xn+1 = , 2xn − 1

for an appropriate initial estimate xφ such as xφ = 1. A slightly faster method is to rewrite the equation as x − 1 − 1/x = 0, in which case the Newton iteration becomes

2 xn + 2xn xn+1 = 2 . xn + 1

These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster than known algorithms for the transcendental numbers π and e. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F 25001 and F 25000, each over 5000 digits, yields over 10,000 signiﬁcant digits of the golden ratio. The decimal expansion of the golden ratio φ ( A001622) has been calculated to an accuracy of two trillion (2×1012 = 2,000,000,000,000) digits.[75]

9.1.5 Pyramids

Further information: mathematics and art Both Egyptian pyramids and the regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios.

Mathematical pyramids and triangles

A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of√ size semi-base by apothem), joining the medium-length√ edges to make the apothem. The height of this pyramid is φ times the semi-base (that is, the slope of the face is φ ); the square of the height is equal to the area of a face, φ times the square of the semi-base. √ The medial right triangle of this “golden” pyramid (see diagram), with sides 1 : φ : φ is interesting in its own √ √ √ right, demonstrating via the Pythagorean theorem the relationship φ = φ2 − 1 or φ = 1 + φ . This Kepler [76] [70] triangle is the only right triangle proportion with edge lengths in geometric progression, just as the 3–4–5√ triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent φ corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827... degrees (51° 49' 38”).[77] A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle;[78] the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes).[79] The slant height or apothem is 5/3 or 1.666... times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers,[80] and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.[78] Another mathematical pyramid with proportions almost identical to the “golden” one is the one with perimeter equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51'), very√ close to the 51.827° of the Kepler triangle. This pyramid relationship corresponds to the coincidental relationship φ ≈ 4/π . Egyptian pyramids very close in proportion to these mathematical pyramids are known.[79] 9.1. GOLDEN RATIO 81

Egyptian pyramids

In the mid-nineteenth century, Röber studied various Egyptian pyramids including Khafre, Menkaure and some of the Giza, Sakkara, and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identiﬁed as the Kepler triangle; many other mathematical theories of the shape of the pyramids have also been explored.[70] One Egyptian pyramid is remarkably close to a “golden pyramid”—the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the “golden” pyramid inclination of 51° 50' and the π-based pyramid inclination of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47')[78] are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains open to speculation.[81] Several other Egyptian pyramids are very close to the rational 3:4:5 shape.[79] Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ.[82] Michael Rice[83] asserts that principal authorities on the history of Egyptian architecture have argued that the Egyp- tians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citing Giedon (1957).[84] Historians of science have always debated whether the Egyptians had any such knowledge or not, con- tending rather that its appearance in an Egyptian building is the result of chance.[85] In 1859, the pyramidologist John Taylor claimed that, in the Great Pyramid of Giza, the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle θ.[86] The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, Howard Vyse, according to Matila Ghyka,[87] reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.

9.1.6 Disputed observations

Examples of disputed observations of the golden ratio include the following:

• Historian John Man states that the pages of the Gutenberg Bible were “based on the golden section shape”. However, according to Man’s own measurements, the ratio of height to width was 1.45.[88]

• Some specific proportions in the bodies of many animals (including humans[89][90]) and parts of the shells of mollusks[3] are often claimed to be in the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.[89] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[90] The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one;[91] however, measurements of nautilus shells do not support this claim.[92]

• In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after signiﬁcant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.[93] The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.[94]

9.1.7 See also

• Golden angle

• Section d'Or 82 CHAPTER 9. DAY 9

• List of works designed with the golden ratio

• Plastic number

• Sacred geometry

• Silver ratio

9.1.8 References and footnotes

[1] Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number. New York: Broadway Books. ISBN 0-7679-0815-5.

[2] Piotr Sadowski (1996). The knight on his quest: symbolic patterns of transition in Sir Gawain and the Green Knight. University of Delaware Press. p. 124. ISBN 978-0-87413-580-0.

[3] Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientiﬁc Publishing, 1997

[4] Euclid, Elements, Book 6, Deﬁnition 3.

[5] Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) p. 37. “And the same applies in architecture, to the rectangles representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.”

[6] Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920

[7] William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers, 2003

[8] Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.

[9] Strogatz, Steven (September 24, 2012). “Me, Myself, and Math: Proportion Control”. New York Times.

[10] Weisstein, Eric W. “Golden Ratio Conjugate”. MathWorld.

[11] Markowsky, George (January 1992). “Misconceptions about the Golden Ratio” (PDF). The College Mathematics Journal. 23 (1).

[12] Mario Livio,The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number, p.6

[13] "῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν" as translated in Richard Fitzpatrick (translator) (2007). Euclid’s Elements of Geometry. ISBN 978-0615179841., p. 156

[14] Euclid, Elements, Book 6, Proposition 30. Retrieved from http:/.aleph0.clarku.edu/~{}djoyce/java/elements/toc.html.

[15] Euclid, Elements, Book 2, Proposition 11; Book 4, Propositions 10–11; Book 13, Propositions 1–6, 8–11, 16–18.

[16] “The Golden Ratio”. The MacTutor History of Mathematics archive. Retrieved 2007-09-18.

[17] Weisstein, Eric W. “Golden Ratio”. MathWorld.

[18] Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21. ISBN 1-4027-3522-7.

[19] Plato. “Timaeus”. Translated by Benjamin Jowett. The Internet Classics Archive. Retrieved 30 May 2006.

[20] James Joseph Tattersall (2005). Elementary number theory in nine chapters (2nd ed.). Cambridge University Press. p. 28. ISBN 978-0-521-85014-8.

[21] Underwood Dudley (1999). Die Macht der Zahl: Was die Numerologie uns weismachen will. Springer. p. 245. ISBN 3-7643-5978-1.

[22] Cook, Theodore Andrea (1979) [1914]. The Curves of Life. New York: Dover Publications. ISBN 0-486-23701-X.

[23] Gardner, Martin (2001), The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Inﬁnity, and Other Topics of Recreational Mathematics, W. W. Norton & Company, p. 88, ISBN 9780393020236. 9.1. GOLDEN RATIO 83

[24] Jaric, Marko V. (2012), Introduction to the Mathematics of Quasicrystals, Elsevier, p. x, ISBN 9780323159470, Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major inﬂuence on the new ﬁeld.

[25] Van Mersbergen, Audrey M., “Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic”, Communication Quarterly, Vol. 46 No. 2, 1998, pp 194-213.

[26] Midhat J. Gazalé , Gnomon, Princeton University Press, 1999. ISBN 0-691-00514-1

[27] Keith J. Devlin The Math Instinct: Why You're A Mathematical Genius (Along With Lobsters, Birds, Cats, And Dogs), p. 108. New York: Thunder’s Mouth Press, 2005, ISBN 1-56025-672-9

[28] Boussora, Kenza and Mazouz, Said, The Use of the Golden Section in the Great Mosque of Kairouan, Nexus Network Journal, vol. 6 no. 1 (Spring 2004),

[29] Le Corbusier, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6

[30] Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: “Both the paintings and the architectural designs make use of the golden section”.

[31] Urwin, Simon. Analysing Architecture (2003) pp. 154-5, ISBN 0-415-30685-X

[32] Jason Elliot (2006). Mirrors of the Unseen: Journeys in Iran. Macmillan. pp. 277, 284. ISBN 978-0-312-30191-0.

[33] Patrice Foutakis, “Did the Greeks Build According to the Golden Ratio?", Cambridge Archaeological Journal, vol. 24, n° 1, February 2014, p. 71-86.

[34] Leonardo da Vinci’s Polyhedra, by George W. Hart

[35] Livio, Mario. “The golden ratio and aesthetics”. Retrieved 2008-03-21.

[36] “Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the ﬁrst mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.” Keith Devlin (May 2007). “The Myth That Will Not Go Away”. Retrieved September 26, 2013.

[37] Donald E. Simanek. “Fibonacci Flim-Flam”. Retrieved April 9, 2013.

[38] Salvador Dalí (2008). The Dali Dimension: Decoding the Mind of a Genius (DVD). Media 3.14-TVC-FGSD-IRL-AVRO.

[39] Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X

[40] Bouleau, Charles, The Painter’s Secret Geometry: A Study of Composition in Art (1963) pp.247-8, Harcourt, Brace & World, ISBN 0-87817-259-9

[41] Olariu, Agata, Golden Section and the Art of Painting Available online

[42] Tosto, Pablo, La composición áurea en las artes plásticas – El número de oro, Librería Hachette, 1969, p. 134–144

[43] Jan Tschichold. The Form of the Book, pp.43 Fig 4. “Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is ﬁxed by a diagonal as well.”

[44] Jan Tschichold, The Form of the Book, Hartley & Marks (1991), ISBN 0-88179-116-4.

[45] Jones, Ronald (1971). “The golden section: A most remarkable measure”. The Structurist. 11: 44–52. Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?

[46] Art Johnson (1999). Famous problems and their mathematicians. Libraries Unlimited. p. 45. ISBN 978-1-56308-446-1. The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.

[47] Alexey Stakhov; Scott Olsen; Scott Anthony Olsen (2009). The mathematics of harmony: from Euclid to contemporary mathematics and computer science. World Scientiﬁc. p. 21. ISBN 978-981-277-582-5. A credit card has a form of the golden rectangle. 84 CHAPTER 9. DAY 9

[48] Simon Cox (2004). Cracking the Da Vinci code: the unauthorized guide to the facts behind Dan Brown’s bestselling novel. Barnes & Noble Books. ISBN 978-0-7607-5931-8. The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.

[49] “THE NEW RAPIDE S : Design”. The ‘Golden Ratio’ sits at the heart of every Aston Martin.

[50] Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.

[51] Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) pp 83, ISBN 0-415- 30010-X

[52] Roy Howat (1983). Debussy in Proportion: A Musical Analysis. Cambridge University Press. ISBN 0-521-31145-4.

[53] Simon Trezise (1994). Debussy: La Mer. Cambridge University Press. p. 53. ISBN 0-521-44656-2.

[54] “Pearl Masters Premium”. Pearl Corporation. Retrieved December 2, 2007.

[55] "An 833 Cents Scale: An experiment on harmony", Huygens-Fokker.org. Accessed December 1, 2012.

[56] Richard Padovan (1999). Proportion. Taylor & Francis. pp. 305–306. ISBN 978-0-419-22780-9.

[57] Padovan, Richard (2002). “Proportion: Science, Philosophy, Architecture”. Nexus Network Journal. 4 (1): 113–122. doi:10.1007/s00004-001-0008-7.

[58] Zeising, Adolf (1854). Neue Lehre van den Proportionen des meschlischen Körpers. preface.

[59] “Golden ratio discovered in a quantum world”. Eurekalert.org. 2010-01-07. Retrieved 2011-10-31.

[60] J.C. Perez (1991), “Chaos DNA and Neuro-computers: A Golden Link”, in Speculations in Science and Technology vol. 14 no. 4, ISSN 0155-7785.

[61] Yamagishi, Michel E.B., and Shimabukuro, Alex I. (2007), “Nucleotide Frequencies in Human Genome and Fibonacci Numbers”, in Bulletin of Mathematical Biology, ISSN 0092-8240 (print), ISSN 1522-9602 (online). PDF full text

[62] Perez, J.-C. (September 2010). “Codon populations in single-stranded whole human genome DNA are fractal and ﬁne- tuned by the Golden Ratio 1.618”. Interdisciplinary Sciences: Computational Life Science. 2 (3): 228–240. doi:10.1007/s12539- 010-0022-0. PMID 20658335. PDF full text

[63] Pommersheim, James E., Tim K. Marks, and Erica L. Flapan, eds. 2010. “Number Theory: A Lively Introduction with Proofs, Applications, and Stories”. John Wiley and Sons: 82.

[64] The golden ratio and aesthetics, by Mario Livio.

[65] Max. Hailperin; Barbara K. Kaiser; Karl W. Knight (1998). Concrete Abstractions: An Introduction to Computer Science Using Scheme. Brooks/Cole Pub. Co. ISBN 0-534-95211-9.

[66] Brian Roselle, “Golden Mean Series”

[67] “A Disco Ball in Space”. NASA. 2001-10-09. Retrieved 2007-04-16.

[68] Chris and Penny. “Quandaries and Queries”. Math Central. Retrieved 23 October 2011.

[69] American Mathematical Monthly, pp. 49-50, 1954.

[70] Roger Herz-Fischler (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5.

[71] Koca, Mehmet; Koca, Nazife Ozdes; Koç, Ramazan (2010), “Catalan solids derived from three-dimensional-root systems and quaternions”, Journal of Mathematical Physics, 51: 043501, arXiv:0908.3272 , doi:10.1063/1.3356985.

[72] Fibonacci Numbers and Nature - Part 2 : Why is the Golden section the “best” arrangement?, from Dr. Ron Knott’s Fibonacci Numbers and the Golden Section, retrieved 2012-11-29.

[73] Weisstein, Eric W. “Pisot Number”. MathWorld.

[74] Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved 2009-07-21.

[75] Yee, Alexander J. (17 August 2015). “Golden Ratio”. numberword.org. Independent computations done by Ron Watkins and Dustin Kirkland.

[76] Radio, Astraea Web (2006). The Best of Astraea: 17 Articles on Science, History and Philosophy. Astrea Web Radio. ISBN 1-4259-7040-0. 9.1. GOLDEN RATIO 85

[77] Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton Univ. Press, 1999

[78] Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000

[79] “The Great Pyramid, The Great Discovery, and The Great Coincidence”. Retrieved 2007-11-25.

[80] Lancelot Hogben, Mathematics for the Million, London: Allen & Unwin, 1942, p. 63., as cited by Dick Teresi, Lost Discoveries: The Ancient Roots of Modern Science—from the Babylonians to the Maya, New York: Simon & Schuster, 2003, p.56

[81] Burton, David M. (1999). The history of mathematics: an introduction (4 ed.). WCB McGraw-Hill. p. 56. ISBN 0-07- 009468-3.

[82] Bell, Eric Temple (1940). The Development of Mathematics. New York: Dover. p. 40.

[83] Rice, Michael, Egypt’s Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C pp. 24 Routledge, 2003, ISBN 0-415-26876-1

[84] S. Giedon, 1957, The Beginnings of Architecture, The A.W. Mellon Lectures in the Fine Arts, 457, as cited in Rice, Michael, Egypt’s Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C pp.24 Routledge, 2003

[85] Markowsky, George (January 1992). “Misconceptions about the Golden Ratio” (PDF). College Mathematics Journal. Mathematical Association of America. 23 (1): 2–19. doi:10.2307/2686193. JSTOR 2686193.

[86] Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859

[87] Matila Ghyka The Geometry of Art and Life, New York: Dover, 1977

[88] Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp. 166–167, Wiley, ISBN 0-471-21823-5. “The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which speciﬁes a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.”

[89] Pheasant, Stephen (1998). Bodyspace. London: Taylor & Francis. ISBN 0-7484-0067-2.

[90] van Laack, Walter (2001). A Better History Of Our World: Volume 1 The Universe. Aachen: van Laach GmbH.

[91] Ivan Moscovich, Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles, New York: Sterling, 2004

[92] Peterson, Ivars. “Sea shell spirals”. Science News.

[93] For instance, Osler writes that “38.2 percent and 61.8 percent retracements of recent rises or declines are common,” in Osler, Carol (2000). “Support for Resistance: Technical Analysis and Intraday Exchange Rates” (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53–68.

[94] Roy Batchelor and Richard Ramyar, "Magic numbers in the Dow,” 25th International Symposium on Forecasting, 2005, p. 13, 31. "Not since the 'big is beautiful' days have giants looked better", Tom Stevenson, The Daily Telegraph, Apr. 10, 2006, and “Technical failure”, The Economist, Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar’s research.

9.1.9 Further reading

• Doczi, György (2005) [1981]. The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala Publications. ISBN 1-59030-259-1.

• Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications. ISBN 0-486-22254-3.

• Joseph, George G. (2000) [1991]. The Crest of the Peacock: The Non-European Roots of Mathematics (New ed.). Princeton, NJ: Princeton University Press. ISBN 0-691-00659-8.

• Livio, Mario (2002) [2002]. The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number (Hard- back ed.). NYC: Broadway (Random House). ISBN 0-7679-0815-5.

• Sahlqvist, Leif (2008). Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design (3rd Rev. ed.). Charleston, SC: BookSurge. ISBN 1-4196-2157-2. 86 CHAPTER 9. DAY 9

• Schneider, Michael S. (1994). A Beginner’s Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperCollins. ISBN 0-06-016939-7. • Scimone, Aldo (1997). La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo: Sigma Edizioni. ISBN 978-88-7231-025-0. • Stakhov, A. P. (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Com- puter Science. Singapore: World Scientiﬁc Publishing. ISBN 978-981-277-582-5. • Walser, Hans (2001) [Der Goldene Schnitt 1993]. The Golden Section. Peter Hilton trans. Washington, DC: The Mathematical Association of America. ISBN 0-88385-534-8.

9.1.10 External links

• Hazewinkel, Michiel, ed. (2001), “Golden ratio”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608- 010-4 • “Golden Section” by Michael Schreiber, Wolfram Demonstrations Project, 2007.

• Golden Section in Photography: Golden Ratio, Golden Triangles, Golden Spiral • Weisstein, Eric W. “Golden Ratio”. MathWorld.

• “Researcher explains mystery of golden ratio”. PhysOrg. December 21, 2009..

• Knott, Ron. “The Golden section ratio: Phi”. Information and activities by a mathematics professor. • The Pentagram & The Golden Ratio. Green, Thomas M. Updated June 2005. Archived November 2007. Geometry instruction with problems to solve.

• Schneider, Robert P. (2011). “A Golden Pair of Identities in the Theory of Numbers”. arXiv:1109.3216 [math.HO]. Proves formulas that involve the golden mean and the Euler totient and Möbius functions.

• The Myth That Will Not Go Away, by Keith Devlin, addressing multiple allegations about the use of the golden ratio in culture. 9.1. GOLDEN RATIO 87

Michael Maestlin, ﬁrst to publish a decimal approximation of the golden ratio, in 1597. 88 CHAPTER 9. DAY 9

Many of the proportions of the Parthenon are alleged to exhibit the golden ratio.

The drawing of a man’s body in a pentagram suggests relationships to the golden ratio.[2] 9.1. GOLDEN RATIO 89

Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: “Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section.”[43] 90 CHAPTER 9. DAY 9

Detail of Aeonium tabuliforme showing the multiple spiral arrangement (parastichy) 9.1. GOLDEN RATIO 91

n–m

m

n

If φ were rational, then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indeﬁnitely because the integers have a lower bound, so φ cannot be rational. 92 CHAPTER 9. DAY 9

Approximations to the reciprocal golden ratio by ﬁnite continued fractions, or ratios of Fibonacci numbers 9.1. GOLDEN RATIO 93

1 1/φ φ-1

1 1/ φ

1/φ3 2 φ

Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of one square divided by that of the next smaller square is the golden ratio.

C

D 1. 2. 3.

A S B

Dividing a line segment by interior division according to the golden ratio 94 CHAPTER 9. DAY 9 C

3. 1. 2. MA S B

Dividing a line segment by exterior division according to the golden ratio A X B φ 1

φ φ φ2 =1+φ

C

Golden triangle 9.1. GOLDEN RATIO 95

Let A and B be midpoints of the sides EF and ED of an equilateral triangle DEF. Extend AB to meet the circumcircle of DEF at C.

|AB| |AC| |BC| = |AB| = ϕ 96 CHAPTER 9. DAY 9

A pentagram colored to distinguish its line segments of diﬀerent lengths. The four lengths are in golden ratio to one another. 9.1. GOLDEN RATIO 97

A B a

b a b b C a

D

The golden ratio in a regular pentagon can be computed using Ptolemy’s theorem. 98 CHAPTER 9. DAY 9

φ

1

One of the rhombic triacontahedron’s rhombi 9.1. GOLDEN RATIO 99

All of the faces of the rhombic triacontahedron are golden rhombi 100 CHAPTER 9. DAY 9

A Fibonacci spiral which approximates the golden spiral, using Fibonacci sequence square sizes up to 34. The spiral is drawn starting from the inner 1×1 square and continues outwards to successively larger squares.

a h

b

A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid’s√ apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Mathematical proportions b:h:a of 1 : φ : φ and 3 : 4 : 5 and 1 : 4/π : 1.61899 are of particular interest in relation to Egyptian pyramids. 9.2. RABATMENT OF THE RECTANGLE 101

9.2 Rabatment of the rectangle

The dotted line represents one of two possible rabatments of the rectangle

Rabatment of the rectangle is a compositional technique used as an aid for the placement of objects or the division of space within a rectangular frame, or as an aid for the study of art. Every rectangle contains two implied squares, each consisting of a short side of the rectangle, an equal length along each longer side, and an imaginary fourth line parallel to the short side. The process of mentally rotating the short sides onto the long ones is called “rabatment”, and often the imaginary fourth line is called “the rabatment”. Also known as rebatement and rabattement, rabatment means the rotation of a plane into another plane about their line of intersection, as in closing an open hinge.[1] In two dimensions, it means to rotate a line about a point until the line coincides with another sharing the same point. The term is used in geometry, art and architecture.[2]

9.2.1 Theory

There is no absolute explanation of the mechanism of this method, but there are various theories.[3] One argument is that squares are such a simple, primal geometric shape that the brain automatically looks for them, mentally com- pleting this rabatment whether it is made explicit or not. When a composition uses elements of the scene to match, the square feels complete in itself, producing a feeling of harmony.[3]

9.2.2 Practice

Renaissance artists used rabatment as a foundation to art and architectural works,[4][5] but the rabatment can be observed in art taken from almost any period.[6] As one of many composition techniques, rabatment of the rectangle can be used to inform the positioning of elements within the rectangle. There is no hard and fast rule regarding such positioning; a composition can have a sense of 102 CHAPTER 9. DAY 9

Animation of a rabatment line within a rectangle dynamic unrest or a sense of equilibrium relative to important lines such as ones taken from rabatment or from the rule of thirds, or from nodal points such as the “eyes of a rectangle”—the four intersections derived from the rule of thirds.[7] Primary image elements can be positioned within one of the two rabatment squares to deﬁne the center of interest, and secondary image elements can be placed outside of a rabatment square.[8] The concept of rabatment can be applied to rectangles of any proportion.[9] For rectangles with a 3:2 ratio (as in 35mm ﬁlm in still photography), it happens that the rabatment lines are exactly matched to the rule of thirds lines.[10] In a horizontally-aligned rectangle, there is one implied square for the left side and one for the right; for a vertically- aligned rectangle, there are upper and lower squares.[3] If the long sides of the rectangle are exactly twice the length of the short, this line is right in the middle. With longer-proportioned rectangles, the squares don't overlap, but with shorter-proportioned ones, they do. In Western cultures that read left to right, attention is often focused inside the left-hand rabatment, or on the line it forms at the right-hand side of the image.[11] When rabatment is used with one side of a golden rectangle, and then iteratively applied to the left-over rectangle, the resulting “whirling rectangles” describe the golden spiral.[12]

9.2.3 Examples

9.2.4 References

[1] Oxford English Dictionary (2 ed.). Oxford University Press. 2003.

[2] Paynter, J.E. (1921). Practical geometry for builders and architects. London: Chapman & Hall.

[3] Mize, Dianne (27 January 2009). “How to Use Rabatment in Your Compositions”. Empty Easel. Retrieved 26 February 2011.

[4] Sriraman, Bharath; Freiman, Viktor; Lirette-Pitre, Nicole (2009). Interdisciplinarity, creativity, and learning: mathematics with literature, paradoxes, history, technology, and modeling. IAP. p. 122. ISBN 1-60752-101-6.

[5] Fett, Birch (2006). “An In-depth Investigation of the Divine Ratio” (PDF). The Montana Mathematics Enthusiast. The Montana Council of Teachers of Mathematics. 3 (2): 157–175. ISSN 1551-3440.

[6] Bouleau, Charles (1963). The painter’s secret geometry: a study of composition in art. New York: Harcourt, Brace. pp. 43–46. 9.3. HEADROOM (PHOTOGRAPHIC FRAMING) 103

[7] Feltus, Alan. “Painting and Composition”. Umbria, Italy: International School of Painting, Drawing, and Sculpture. Retrieved 26 February 2011.

[8] Mize, Dianne (16 July 2008). “Placing Our Images: Rabatment”. Compose. Retrieved 27 February 2011.

[9] Dunstan, Bernard (1979). Composing your paintings. Start To Paint. Taplinger Publishing. pp. 22, 26. ISBN 0-8008- 1803-2.

[10] Brown, Scott (2009). “Glossary”. Watercolorists of Whatcom. Retrieved 27 February 2011.

[11] Nelson, Connie (2010). “Composition in art: Rabatment”. Explore-Drawing-and-Painting.com. Retrieved 1 March 2011.

[12] Fairbanks, Avard T.; Fairbanks, Eugene F. (2005). Human Proportions for Artists. Fairbanks Art and Books. p. 210. ISBN 0-9725841-1-0.

9.2.5 External links

• “The painting’s secret geometry” by François Murez

• “Rabatment as a Compositional Tool” by Judith Reeve

9.3 Headroom (photographic framing)

In photography, headroom or head room is a concept of aesthetic composition that addresses the relative vertical position of the subject within the frame of the image. Headroom refers specifically to the distance between the top of the subject’s head and the top of the frame, but the term is sometimes used instead of lead room, nose room or 'looking room'[1] to include the sense of space on both sides of the image. The amount of headroom that is considered aesthetically pleasing is a dynamic quantity; it changes relative to how much of the frame is filled by the subject. One rule of thumb taken from classic portrait painting techniques,[2] called the "rule of thirds",[3][4] suggests that the subject’s eyes, as a center of interest, are ideally positioned one-third of the way down from the top of the frame.[5] Moving images such as movie cameras and video cameras have the same headroom issues as still photography, but with the added factors of the movement of the subject, the movement of the camera, and the possibility of zooming in or out. Perceptual psychological studies have been carried out with experimenters using a white dot placed in various positions within a frame to demonstrate that observers attribute potential motion to a static object within a frame, relative to its position. The unmoving object is described as 'pulling' toward the center or toward an edge or corner.[6] Proper headroom is achieved when the object is no longer seen to be slipping out of the frame—when its potential for motion is seen to be neutral in all directions. Headroom changes as the camera zooms in or out, and the camera must simultaneously tilt up or down to keep the center of interest approximately one-third of the way down from the top of the frame.[5] The closer the subject, the less headroom needed.[7] In extreme close-ups, the top of the head is out of the frame,[1] but the concept of headroom still applies via the rule of thirds. In television broadcast camera work, the amount of headroom seen by the production crew is slightly greater than the amount seen by home viewers, whose frames are reduced in area by about 5%.[1] To adjust for this, broadcast camera headroom is slightly expanded so that home viewers will see the correct amount of headroom.[1] Professional video camera viewfinders and professional video monitors often include an overscan setting to compare between full screen resolution and “domestic cut-off”[1] as an aid to achieving good headroom and lead room. One of the most common mistakes that casual camera users make is to have too much headroom: too much space above the subject’s head.[8] 104 CHAPTER 9. DAY 9

9.3.1 Examples

• A portrait of guitarist Adrian Legg demonstrates an excessive amount of headroom, with the subject’s nose centered in the frame (a common mistake.)

• A subtle lack of headroom with the subject’s eyes only 28% of the way down from the top, not 33%

• Good composition, with the subject’s eyes one-third of the distance down from the top of the frame, following the rule of thirds

• For moving images, the action of zooming in to ﬁll the frame with the subject requires the simultaneous tilting up of the camera, shown by the red lines, to maintain the correct amount of headroom. Conversely, zooming out requires tilting down.

9.3.2 See also

• Highlight headroom

9.3.3 References

[1] Thompson, Roy. Grammar of the shot, Focal Press, 1998, p. 64. ISBN 0-240-51398-3

[2] Hurter, Bill. The Best of Portrait Photography: Techniques and Images from the Pros, Amherst Media, Inc, 2003, p. 38. ISBN 1-58428-101-4

[3] Camera Terms, Retrieved on June 25, 2009.

[4] MediaCollege.com. Framing, Retrieved on June 25, 2009.

[5] Barbash, Ilisa; Taylor, Lucien. Cross-cultural Filmmaking: A Handbook for Making Documentary and Ethnographic Films and Videos, University of California Press, 1997, p. 97. ISBN 0-520-08760-7

[6] Ward, Peter. Picture composition for ﬁlm and television, Elsevier, 2003, p. 84. ISBN 0-240-51681-8

[7] Millerson, 1994, p. 80.

[8] Trem.ca, digital photography. Basic Rules of Photography, Retrieved on June 25, 2009. 9.3. HEADROOM (PHOTOGRAPHIC FRAMING) 105

9.3.4 Further reading

• Millerson, Gerald. Video camera techniques, Focal Press, 1994, p. 80. ISBN 0-240-51376-2 Chapter 10

Day 10

10.1 Perspective (graphical)

Staircase in two-point perspective

Perspective (from Latin: perspicere to see through) in the graphic arts is an approximate representation, on a ﬂat surface (such as paper), of an image as it is seen by the eye. The two most characteristic features of perspective are that objects are smaller as their distance from the observer increases; and that they are subject to foreshortening, meaning that an object’s dimensions along the line of sight are shorter than its dimensions across the line of sight. Italian Renaissance painters and architects including Filippo Brunelleschi, Masaccio, Paolo Uccello, Piero della Francesca and Luca Pacioli studied linear perspective, wrote treatises on it, and incorporated it into their artworks, thus contributing to the mathematics of art.

106 10.1. PERSPECTIVE (GRAPHICAL) 107

A cube in two-point perspective

10.1.1 Overview

Linear perspective always works by representing the light that passes from a scene through an imaginary rectangle (realized as the plane of the painting), to the viewer’s eye, as if a viewer were looking through a window and painting what is seen directly onto the windowpane. If viewed from the same spot as the windowpane was painted, the painted image would be identical to what was seen through the unpainted window. Each painted object in the scene is thus a flat, scaled down version of the object on the other side of the window.[1] Because each portion of the painted object lies on the straight line from the viewer’s eye to the equivalent portion of the real object it represents, the viewer sees no difference (sans depth perception) between the painted scene on the windowpane and the view of the real scene. All perspective drawings assume the viewer is a certain distance away from the drawing. Objects are scaled relative to that viewer. An object is often not scaled evenly: a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening. Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewer’s eye, represents objects infinitely far away. They have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to (and named after) the Earth’s horizon. Any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a single vanishing point, usually (though not necessarily) directly opposite the viewer’s eye and usually (though not necessarily) on the horizon line. All lines parallel with the viewer’s line of sight recede to the horizon towards this vanishing point. This is the standard “receding railroad tracks” phenomenon. A two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of parallel lines that are at an angle relative to the plane of the drawing. Perspectives consisting of many parallel lines are observed most often when drawing architecture (architecture frequently uses lines parallel to the x, y, and z axes). Because it is rare to have a scene consisting solely of lines parallel to the three Cartesian axes (x, y, and z), it is rare to see perspectives in practice with only one, two, or three vanishing points; even a simple house frequently has a peaked roof which results in a minimum of six sets of parallel lines, in 108 CHAPTER 10. DAY 10

Rays of light travel from the object, through the picture plane, and to the viewer’s eye. This is the basis for graphical perspective.

turn corresponding to up to six vanishing points. In contrast, natural scenes often do not have any sets of parallel lines and thus no vanishing points.

Early history

The earliest art paintings and drawings typically sized many objects and characters hierarchically according to their spiritual or thematic importance, not their distance from the viewer, and did not use foreshortening. The most important figures are often shown as the highest in a composition, also from hieratic motives, leading to the so-called “vertical perspective”, common in the art of Ancient Egypt, where a group of “nearer” figures are shown below the larger figure or figures. The only method to indicate the relative position of elements in the composition was by overlapping, of which much use is made in works like the Parthenon Marbles.

History

Chinese artists made use of oblique perspective from the first or second century until the 18th century. It is not certain how they came to use the technique; some authorities suggest that the Chinese acquired the technique from India, which acquired it from Ancient Rome.[2] Oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga (1752–1815).[2] In the 18th century, Chinese artists began to combine oblique perspective with regular diminution of size of people and objects with distance; no particular vantage point is chosen, but a convincing effect is achieved.[2] Systematic attempts to evolve a system of perspective are usually considered to have begun around the fifth century BC in the art of Ancient Greece, as part of a developing interest in illusionism allied to theatrical scenery. This was detailed within Aristotle’s Poetics as skenographia: using flat panels on a stage to give the illusion of depth.[3] The philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage. 10.1. PERSPECTIVE (GRAPHICAL) 109

15th century illustration from the Old French translation of William of Tyre's Histoire d'Outremer.[lower-alpha 1]

Euclid's Optics introduced a mathematical theory of perspective, but there is some debate over the extent to which Euclid’s perspective coincides with the modern mathematical definition. By the later periods of antiquity, artists, especially those in less popular traditions, were well aware that distant objects could be shown smaller than those close at hand for increased realism, but whether this convention was actually used in a work depended on many factors. Some of the paintings found in the ruins of Pompeii show a remarkable realism and perspective for their time.[4] It has been claimed that comprehensive systems of perspective were evolved in antiquity, but most scholars do not accept this. Hardly any of the many works where such a system would have been used have survived. A passage in Philostratus suggests that classical artists and theorists thought in terms of “circles” at equal distance from the viewer, like a classical semi-circular theatre seen from the stage.[5] The roof beams in rooms in the Vatican Virgil, from about 400 AD, are shown converging, more or less, on a common vanishing point, but this is not systematically related to the rest of the composition.[6] In the Late Antique period use of perspective techniques declined. The art of the new cultures of the Migration Period had no tradition of attempting compositions of large numbers of figures and Early Medieval art was slow and inconsistent in relearning the convention from classical models, though the process can be seen underway in Carolingian art. Various paintings and drawings during the Middle Ages show amateur attempts at projections of furniture, where parallel lines are successfully represented in isometric projection, or by non parallel ones, but without a single vanishing point. Medieval artists in Europe, like those in the Islamic world and China, were aware of the general principle of varying the relative size of elements according to distance, but even more than classical art was perfectly ready to override it for other reasons. Buildings were often shown obliquely according to a particular convention. The use and sophisti- cation of attempts to convey distance increased steadily during the period, but without a basis in a systematic theory. Byzantine art was also aware of these principles, but also had the reverse perspective convention for the setting of principal figures. 110 CHAPTER 10. DAY 10

A Song Dynasty watercolor painting of a mill in an oblique perspective, 12th century

Renaissance: Mathematical basis

Further information: Mathematics and art

In about 1413 a contemporary of Ghiberti, Filippo Brunelleschi, demonstrated the geometrical method of perspective, used today by artists, by painting the outlines of various Florentine buildings onto a mirror. When the building’s outline was continued, he noticed that all of the lines converged on the horizon line. According to Vasari, he then set up a demonstration of his painting of the Baptistery in the incomplete doorway of the Duomo. He had the viewer look through a small hole on the back of the painting, facing the Baptistery. He would then set up a mirror, facing the viewer, which reflected his painting. To the viewer, the painting of the Baptistery and the building itself were nearly indistinguishable. Soon after, nearly every artist in Florence and in Italy used geometrical perspective in their paintings,[10] notably Paolo Uccello, Masolino da Panicale and Donatello. Donatello started sculpting elaborate checkerboard floors into the simple manger portrayed in the birth of Christ. Although hardly historically accurate, these checkerboard floors obeyed the primary laws of geometrical perspective: the lines converged approximately to a vanishing point, and the rate at which the horizontal lines receded into the distance was graphically determined. This became an integral part of Quattrocento art. Melozzo da Forlì first used the technique of upward foreshortening (in Rome, Loreto, Forlì and others), and was celebrated for that. Not only was perspective a way of showing depth, it was also a new method of composing a painting. Paintings began to show a single, unified scene, rather than a combination of several. As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician Toscanelli),[11] but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote De pictura (1435/1436), a treatise on proper methods of showing distance in painting. Alberti’s primary breakthrough was not to show the mathematics in terms of conical projections, as it actually appears to the eye. Instead, he formulated the theory based on planar projections, or how the rays of light, passing from the viewer’s eye to the landscape, would strike the picture plane (the painting). He was then able to calculate the apparent height of a distant object using two similar triangles. The mathematics behind similar triangles is relatively simple, having been long ago formulated by Euclid. In viewing a wall, for instance, the first triangle has a vertex at the user’s eye, and vertices at the top and bottom of the wall. The bottom of this triangle is the distance 10.1. PERSPECTIVE (GRAPHICAL) 111

Codex Amiatinus (7th century). Portrait, of Ezra, from folio 5r at the start of Old Testament from the viewer to the wall. The second, similar triangle, has a point at the viewer’s eye, and has a length equal to the viewer’s eye from the painting. The height of the second triangle can then be determined through a simple ratio, as proven by Euclid. Alberti was also trained in the science of optics through the school of Padua and under the inﬂuence of Biagio Pelacani da Parma who studied Alhazen's Book of Optics [12] (see what was noted above in this regard with respect to Ghiberti). 112 CHAPTER 10. DAY 10

Geometrically incorrect attempt at perspective in a 1614 painting of Old St Paul’s Cathedral.(Society of Antiquaries)

Piero della Francesca elaborated on Della Pittura in his De Prospectiva Pingendi in the 1470s. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti’s. Della Francesca was also the first to accurately draw the Platonic solids as they would appear in perspective. Luca Pacioli's 1509 De divina proportione (On Divine Proportion), illustrated by Leonardo da Vinci, summarized the use 10.1. PERSPECTIVE (GRAPHICAL) 113 of perspective in painting.[13] Perspective remained, for a while, the domain of Florence. Jan van Eyck, among others, was unable to create a consistent structure for the converging lines in paintings, as in London’s The Arnolfini Portrait, because he was unaware of the theoretical breakthrough just then occurring in Italy. However he achieved very subtle effects by manipulations of scale in his interiors. Gradually, and partly through the movement of academies of the arts, the Italian techniques became part of the training of artists across Europe, and later other parts of the world. The culmination of these Renaissance traditions finds its ultimate synthesis in the research of the 17th century architect, geometer, and optician Girard Desargues on perspective, optics and projective geometry. Further advances in projective geometry, in the 19th and 20th centuries, led to the development of analytic geometry, algebraic geometry, relativity and quantum mechanics.

Present: Computer graphics

3-D computer games and ray-tracers often use a modiﬁed version of perspective. Like the painter, the computer program is generally not concerned with every ray of light that is in a scene. Instead, the program simulates rays of light traveling backwards from the monitor (one for every pixel), and checks to see what it hits. In this way, the program does not have to compute the trajectories of millions of rays of light that pass from a light source, hit an object, and miss the viewer. CAD software, and some computer games (especially games using 3-D polygons) use linear algebra, and in particular matrix multiplication, to create a sense of perspective. The scene is a set of points, and these points are projected to a plane (computer screen) in front of the view point (the viewer’s eye). The problem of perspective is simply ﬁnding the corresponding coordinates on the plane corresponding to the points in the scene. By the theories of linear algebra, a matrix multiplication directly computes the desired coordinates, thus bypassing any descriptive geometry theorems used in perspective drawing. .

10.1.2 Types of perspective

Of the many types of perspective drawings, the most common categorizations of artiﬁcial perspective are one-, two- and three-point. The names of these categories refer to the number of vanishing points in the perspective drawing.

One-point perspective

A drawing has one-point perspective when it contains only one vanishing point on the horizon line. This type of perspective is typically used for images of roads, railway tracks, hallways, or buildings viewed so that the front is directly facing the viewer. Any objects that are made up of lines either directly parallel with the viewer’s line of sight or directly perpendicular (the railroad slats) can be represented with one-point perspective. These parallel lines converge at the vanishing point. One-point perspective exists when the picture plane is parallel to two axes of a rectilinear (or Cartesian) scene – a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the picture plane (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the picture plane are drawn as parallel lines. All elements that are perpendicular to the picture plane converge at a single point (a vanishing point) on the horizon.

• Examples of one-point perspective

• 114 CHAPTER 10. DAY 10

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Two-point perspective

A drawing has two-point perspective when it contains two vanishing points on the horizon line. In an illustration, these vanishing points can be placed arbitrarily along the horizon. Two-point perspective can be used to draw the same objects as one-point perspective, rotated: looking at the corner of a house, or at two forked roads shrinking into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Seen from the corner, one wall of a house would recede towards one vanishing point while the other wall recedes towards the opposite vanishing point. 10.1. PERSPECTIVE (GRAPHICAL) 115

Two-point perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the z-axis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no diﬀerence exists in the image of the cylinder between a one-point and two-point perspective. Two-point perspective has one set of lines parallel to the picture plane and two sets oblique to it. Parallel lines oblique to the picture plane converge to a vanishing point, which means that this set-up will require two vanishing points.

Three-point perspective

Three-point perspective is often used for buildings seen from above (or below). In addition to the two vanishing points from before, one for each wall, there is now one for how the vertical lines of the walls recede. For an object seen from above, this third vanishing point is below the ground. For an object seen from below, as when the viewer looks up at a tall building, the third vanishing point is high in space. Three-point perspective exists when the perspective is a view of a Cartesian scene where the picture plane is not parallel to any of the scene’s three axes. Each of the three vanishing points corresponds with one of the three axes of the scene. One, two and three-point perspectives appear to embody different forms of calculated perspective, and are generated by different methods. Mathematically, however, all three are identical; the difference is merely in the relative orientation of the rectilinear scene to the viewer.

Four-point perspective

Four-point perspective, also called infinite-point perspective, is the curvilinear (see curvilinear perspective) variant of two-point perspective. A four-point perspective image can represent a 360° panorama, and even beyond 360° to depict impossible scenes. This perspective can be used with either a horizontal or a vertical horizon line: in the latter configuration it can depict both a worm’s-eye and bird’s-eye view of a scene at the same time. Like all other foreshortened variants of perspective (one-point to six-point perspectives), it starts off with a horizon line, followed by four equally spaced vanishing points to delineate four vertical lines. The vanishing points made to create the curvilinear orthogonals are thus made ad hoc on the four vertical lines placed on the opposite side of the horizon line. The only dimension not foreshortened in this type of perspective is that of the rectilinear and parallel lines perpendicular to the horizon line – similar to the vertical lines used in two-point perspective. One-point, two-point, and three-point perspective are dependent on the structure of the scene being viewed. These only exist for strict Cartesian (rectilinear) scenes. By inserting into a Cartesian scene a set of parallel lines that are not parallel to any of the three axes of the scene, a new distinct vanishing point is created. Therefore, it is possible to have an infinite-point perspective if the scene being viewed is not a Cartesian scene but instead consists of infinite pairs of parallel lines, where each pair is not parallel to any other pair.

Zero-point perspective

Because vanishing points exist only when parallel lines are present in the scene, a perspective with no vanishing points (“zero-point” perspective) occurs if the viewer is observing a non-linear scene.[14] The most common example of a nonlinear scene is a natural scene (e.g., a mountain range) which frequently does not contain any parallel lines. A perspective without vanishing points can still create a sense of depth. A zero-point perspective view is equivalent to an elevation.

Foreshortening

Foreshortening is the visual eﬀect or optical illusion that causes an object or distance to appear shorter than it actually is because it is angled toward the viewer. Additionally, an object is often not scaled evenly: a circle often appears as an ellipse and a square can appear as a trapezoid. Although foreshortening is an important element in art where visual perspective is being depicted, foreshortening occurs in other types of two-dimensional representations of three-dimensional scenes. Some other types where foreshortening can occur include oblique parallel projection drawings. Foreshortening also occurs when imaging rugged terrain using a synthetic aperture radar system. 116 CHAPTER 10. DAY 10

In painting, foreshortening in the depiction of the human ﬁgure was perfected in the Italian Renaissance, and the The Lamentation over the Dead Christ by Andrea Mantegna (1480s) is one of the most famous of a number of works that show oﬀ the new technique, which thereafter became a standard part of the training of artists.

10.1.3 Methods of construction

Several methods of constructing perspectives exist, including:

• Freehand sketching (common in art)

• Graphically constructing (once common in architecture)

• Using a perspective grid

• Computing a perspective transform (common in 3D computer applications)

• Mimicry using tools such as a proportional divider (sometimes called a variscaler)

• Copying a photograph

10.1.4 Example

One of the most common, and earliest, uses of geometrical perspective is a checkerboard floor. It is a simple but striking application of one-point perspective. Many of the properties of perspective drawing are used while drawing a checkerboard. The checkerboard floor is, essentially, just a combination of a series of squares. Once a single square is drawn, it can be widened or subdivided into a checkerboard. Where necessary, lines and points will be referred to by their colors in the diagram. To draw a square in perspective, the artist starts by drawing a horizon line (black) and determining where the vanishing point (green) should be. The higher up the horizon line is, the lower the viewer will appear to be looking, and vice versa. The more off-center the vanishing point, the more tilted the square will be. Because the square is made up of right angles, the vanishing point should be directly in the middle of the horizon line. A rotated square is drawn using two-point perspective, with each set of parallel lines leading to a different vanishing point. The foremost edge of the (orange) square is drawn near the bottom of the painting. Because the viewer’s picture plane is parallel to the bottom of the square, this line is horizontal. Lines connecting each side of the foremost edge to the vanishing point are drawn (in grey). These lines give the basic, one point “railroad tracks” perspective. The closer it is the horizon line, the farther away it is from the viewer, and the smaller it will appear. The farther away from the viewer it is, the closer it is to being perpendicular to the picture plane. A new point (the eye) is now chosen, on the horizon line, either to the left or right of the vanishing point. The distance from this point to the vanishing point represents the distance of the viewer from the drawing. If this point is very far from the vanishing point, the square will appear squashed, and far away. If it is close, it will appear stretched out, as if it is very close to the viewer. A line connecting this point to the opposite corner of the square is drawn. Where this (blue) line hits the side of the square, a horizontal line is drawn, representing the farthest edge of the square. The line just drawn represents the ray of light traveling from the farthest edge of the square to the viewer’s eye. This step is key to understanding perspective drawing. The light that passes through the picture plane obviously can not be traced. Instead, lines that represent those rays of light are drawn on the picture plane. In the case of the square, the side of the square also represents the picture plane (at an angle), so there is a small shortcut: when the line hits the side of the square, it has also hit the appropriate spot in the picture plane. The (blue) line is drawn to the opposite edge of the foremost edge because of another shortcut: since all sides are the same length, the foremost edge can stand in for the side edge. Original formulations used, instead of the side of the square, a vertical line to one side, representing the picture plane. Each line drawn through this plane was identical to the line of sight from the viewer’s eye to the drawing, only rotated around the y-axis ninety degrees. It is, conceptually, an easier way of thinking of perspective. It can be easily shown that both methods are mathematically identical, and result in the same placement of the farthest side. 10.1. PERSPECTIVE (GRAPHICAL) 117

10.1.5 Limitations

Plato was one of the ﬁrst to discuss the problems of perspective.

“Thus (through perspective) every sort of confusion is revealed within us; and this is that weakness of the human mind on which the art of conjuring and of deceiving by light and shadow and other ingenious devices imposes, having an eﬀect upon us like magic... And the arts of measuring and numbering and weighing come to the rescue of the human understanding – there is the beauty of them – and the apparent greater or less, or more or heavier, no longer have the mastery over us, but give way before calculation and measure and weight?"[15]

Perspective images are calculated assuming a particular vanishing point. In order for the resulting image to appear identical to the original scene, a viewer of the perspective must view the image from the exact vantage point used in the calculations relative to the image. This cancels out what would appear to be distortions in the image when viewed from a different point. These apparent distortions are more pronounced away from the center of the image as the angle between a projected ray (from the scene to the eye) becomes more acute relative to the picture plane. In practice, unless the viewer chooses an extreme angle, like looking at it from the bottom corner of the window, the perspective normally looks more or less correct. This is referred to as “Zeeman’s Paradox.”[16] It has been suggested that a drawing in perspective still seems to be in perspective at other spots because we still perceive it as a drawing, because it lacks depth of field cues.[17] For a typical perspective, however, the field of view is narrow enough (often only 60 degrees) that the distortions are similarly minimal enough that the image can be viewed from a point other than the actual calculated vantage point without appearing significantly distorted. When a larger angle of view is required, the standard method of projecting rays onto a flat picture plane becomes impractical. As a theoretical maximum, the field of view of a flat picture plane must be less than 180 degrees (as the field of view increases towards 180 degrees, the required breadth of the picture plane approaches infinity). To create a projected ray image with a large field of view, one can project the image onto a curved surface. To have a large field of view horizontally in the image, a surface that is a vertical cylinder (i.e., the axis of the cylinder is parallel to the z-axis) will suffice (similarly, if the desired large field of view is only in the vertical direction of the image, a horizontal cylinder will suffice). A cylindrical picture surface will allow for a projected ray image up to a full 360 degrees in either the horizontal or vertical dimension of the perspective image (depending on the orientation of the cylinder). In the same way, by using a spherical picture surface, the field of view can be a full 360 degrees in any direction (note that for a spherical surface, all projected rays from the scene to the eye intersect the surface at a right angle). Just as a standard perspective image must be viewed from the calculated vantage point for the image to appear identical to the true scene, a projected image onto a cylinder or sphere must likewise be viewed from the calculated vantage point for it to be precisely identical to the original scene. If an image projected onto a cylindrical surface is “unrolled” into a flat image, different types of distortions occur. For example, many of the scene’s straight lines will be drawn as curves. An image projected onto a spherical surface can be flattened in various ways:

• An image equivalent to an unrolled cylinder

• A portion of the sphere can be ﬂattened into an image equivalent to a standard perspective

• An image similar to a ﬁsheye photograph

10.1.6 See also

• Anamorphosis

• Aerial perspective

• Camera angle

• Curvilinear perspective

• Cutaway drawing 118 CHAPTER 10. DAY 10

• Desargues’ theorem

• Perspective control

• Perspective projection distortion

• Perspective transform

• Projective geometry

• Reverse perspective

• Zograscope

10.1.7 Notes

[1] There is clearly a general attempt to reduce the size of more distant elements, but unsystematically. Sections of the composition are at a similar scale, with relative distance shown by overlapping, foreshortening, and further objects being higher than nearer ones, though the workmen at left do show ﬁner adjustment of size. But this is abandoned on the right where the most important ﬁgure is much larger than the mason. Rectangular buildings and the blocks of stone are shown obliquely.

10.1.8 References

[1] D'Amelio, Joseph (2003). Perspective Drawing Handbook. Dover. p. 19.

[2] Cucker, Felix (2013). Manifold Mirrors: The Crossing Paths of the Arts and Mathematics. Cambridge University Press. pp. 269–278. ISBN 978-0-521-72876-8. Dubery and Willats (1983:33) write that 'Oblique projection seems to have arrived in China from Rome by way of India round about the ﬁrst or second century AD.' Figure 10.9 [Wen-Chi returns home, anon, China, 12th century] shows an archetype of the classical use of oblique perspective in Chinese painting.

[3] “Skenographia in Fifth Century”. CUNY. Retrieved 2007-12-27.

[4] “Pompeii. House of the Vettii. Fauces and Priapus”. SUNY Buﬀalo. Retrieved 2007-12-27.

[5] Panofsky, Erwin (1960). Renaissance and Renascences in Western Art. Stockholm: Almqvist & Wiksell. p. 122, note 1. ISBN 0-06-430026-9.

[6] Vatican Virgil image

[7] “Linear Perspective: Brunelleschi’s Experiment”. Smarthistory at Khan Academy. Retrieved 12 May 2013.

[8] “How One-Point Linear Perspective Works”. Smarthistory at Khan Academy. Retrieved 12 May 2013.

[9] “Empire of the Eye: The Magic of Illusion: The Trinity-Masaccio, Part 2”. National Gallery of Art at ArtBabble. Retrieved 12 May 2013.

[10] "...and these works (of perspective by Brunelleschi) were the means of arousing the minds of the other craftsmen, who afterwords devoted themselves to this with great zeal.” Vasari’s Lives of the Artists Chapter on Brunelleschi

[11] “Messer Paolo dal Pozzo Toscanelli, having returned from his studies, invited Filippo with other friends to supper in a garden, and the discourse falling on mathematical subjects, Filippo formed a friendship with him and learned geometry from him.” Vasarai’s Lives of the Artists, Chapter on Brunelleschi

[12] El-Bizri, Nader (2010). “Classical Optics and the Perspectiva Traditions Leading to the Renaissance”. In Hendrix, John Shannon; Carman, Charles H. Renaissance Theories of Vision (Visual Culture in Early Modernity). Farnham, Surrey: Ashgate. pp. 11–30. ISBN 1-409400-24-7.

[13] O'Connor, J. J.; Robertson, E. F. (July 1999). “Luca Pacioli”. University of St Andrews. Retrieved 23 September 2015.

[14] Basant, Agrawal (2008). Engineering Drawing. New Delhi: Tata McGraw-Hill. p. 17-2. ISBN 978-0-07-066863-8.

[15] Plato’s Republic, Book X, 602d. http://etext.library.adelaide.edu.au/mirror/classics.mit.edu/Plato/republic.11.x.html

[16] Mathographics by Robert Dixon New York: Dover, p. 82, 1991. 10.1. PERSPECTIVE (GRAPHICAL) 119

[17] "...the paradox is purely conceptual: it assumes we view a perspective representation as a retinal simulation, when in fact we view it as a two dimensional painting. In other words, perspective constructions create visual symbols, not visual illusions. The key is that paintings lack the depth of ﬁeld cues created by binocular vision; we are always aware a painting is ﬂat rather than deep. And that is how our mind interprets it, adjusting our understanding of the painting to compensate for our position.” http://www.handprint.com/HP/WCL/perspect1.html Retrieved on 25 December 2006

10.1.9 Further reading

• Andersen, Kirsti (2007). The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge. Springer. • Damisch, Hubert (1994). The Origin of Perspective, Translated by John Goodman. Cambridge, Mass.: MIT Press. • Hyman, Isabelle, comp (1974). Brunelleschi in Perspective. Englewood Cliﬀs, New Jersey: Prentice-Hall.

• Kemp, Martin (1992). The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. Yale University Press.

• Pérez-Gómez, Alberto, and Pelletier, Louise (1997). Architectural Representation and the Perspective Hinge. Cambridge, Mass.: MIT Press.

• Vasari, Giorgio (1568). The Lives of the Artists. Florence, Italy. • Gill, Robert W (1974). Perspective From Basic to Creative. Australia: Thames & Hudson.

10.1.10 External links

• A tutorial covering many examples of linear perspective • Teaching Perspective in Art and Mathematics through Leonardo da Vinci’s Work at Mathematical Association of America • Perspective in Ancient Roman-Wall Painting at Southampton Solent University

• How to Draw a Two Point Perspective Grid at Creating Comics 120 CHAPTER 10. DAY 10

Two painter’s apprentices studying perspective. Drawing by Federico Zuccari, 1609 10.1. PERSPECTIVE (GRAPHICAL) 121

Melozzo's use of upward foreshortening in his frescoes at Loreto

Pietro Perugino's use of perspective in this fresco at the Sistine Chapel (1481–82) helped bring the Renaissance to Rome. 122 CHAPTER 10. DAY 10

One-point perspective 10.1. PERSPECTIVE (GRAPHICAL) 123

Two-Point Perspective 124 CHAPTER 10. DAY 10

A cube drawing using 2-point perspective 10.1. PERSPECTIVE (GRAPHICAL) 125

Walls in 2-point perspective, converging toward two vanishing points. All vertical elements are parallel. Model from 3D Warehouse, rendered in SketchUp. 126 CHAPTER 10. DAY 10

Three-Point Perspective 10.1. PERSPECTIVE (GRAPHICAL) 127

The Palazzo del Lavoro in Mussolini’s Esposizione Universale Roma complex, photographed in 3-point perspective. All three axes are oblique to the picture plane; the three vanishing points are at the zenith, and on the horizon to the right and left.

AB

Two diﬀerent projections of a stack of two cubes, illustrating oblique parallel projection foreshortening (“A”) and perspective foreshortening (“B”) 128 CHAPTER 10. DAY 10

Andrea Mantegna, The Lamentation over the Dead Christ 10.1. PERSPECTIVE (GRAPHICAL) 129

Epimetheus (lower left) and Janus (right). The two moons appear close because of foreshortening; in reality, Janus is about 40,000 km farther from the observer than Epimetheus.

Rays of light travel from the object to the eye, intersecting with a notional picture plane. 130 CHAPTER 10. DAY 10

Determining the geometry of a square ﬂoor tile on the perspective drawing 10.1. PERSPECTIVE (GRAPHICAL) 131

Satire on False Perspective by William Hogarth, 1753 Chapter 11

Text and image sources, contributors, and licenses

11.1 Text

• Composition (visual arts) Source: https://en.wikipedia.org/wiki/Composition_(visual_arts)?oldid=762973762 Contributors: LA2, Ronz, Julesd, Robbot, Moncrief, Lethe, Mark.murphy, Girolamo Savonarola, Tail, Imroy, JohnRDaily, RJHall, Zenohockey, Bobo192, Lu- oShengli, Nk, Mduvekot, Wiki-uk, Clubmarx, Match, RJFJR, Camw, Tony999, Jeff3000, Johan Lont, DL5MDA, Mandarax, Sparkit, Mendaliv, Sjö, Ligulem, Bensin, Jared Preston, DVdm, YurikBot, Hede2000, Stephenb, JohanL, NawlinWiki, ONEder Boy, Moe Ep- silon, Closedmouth, Cmglee, SmackBot, Drummondjacob, Elonka, Stimpy, Septegram, Gilliam, Ohnoitsjamie, Theweekly, Sadads, Can't sleep, clown will eat me, John Kjos, PieRRoMaN, MadManLear, Ligulembot, Dicklyon, Stenaught, MIckStephenson, Claudele- poisson, WeggeBot, Ken Gallager, No1lakersfan, Safalra, Cydebot, Amandajm, Thijs!bot, Epbr123, Qwyrxian, Klausness, Escapologist, Modernist, Salgueiro~enwiki, Drake Wilson, JAnDbot, Niaz, Gcm, PhilKnight, Mrs Scarborough, Bongwarrior, Dekimasu, Pgarcia05, Jim.henderson, Bus stop, J.delanoy, Pharaoh of the Wizards, Trusilver, Nanoamp, Johnbod, Cognita, BluePuddle, Hatsandcats, Bonadea, Jorioux, Lear’s Fool, Argusmom, TXiKiBoT, Planetary Chaos, Someguy1221, Voorlandt, Samandcheese, LeaveSleaves, Gak Blimby, Meters, Lamro, Falcon8765, SalomonCeb, Pjoef, Gamsbart, Ian Glenn, SieBot, Yintan, Soler97, Asfmdvnjwnek, Keilana, JD554, Filos96, Adytum72a, Prof saxx, Escape Orbit, Diser55, Martarius, ClueBot, EoGuy, Hafspajen, Lolliedog2, Erebus Morgaine, Lla- mafur, Mrme44, Redthoreau, Bald Zebra, Edgeloop, XLinkBot, Hotcrocodile, BodhisattvaBot, SilvonenBot, Noctibus, Addbot, Non- dropframe, CL, Bassbonerocks, SpBot, Tide rolls, Luckas-bot, Yobot, Cflm001, Markermo, THEN WHO WAS PHONE?, Wiki1472, AnomieBOT, Lustiger seth, ThaddeusB, Piano non troppo, Пика Пика, Imtheman999, Rory monaghan, Materialscientist, Citation bot, Eumolpo, ArthurBot, JimVC3, Capricorn42, - 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132 11.1. TEXT 133

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Parrot, MarkSutton, Beetstra, Dicklyon, Waggers, Radical53, Lenzar, RMHED, Nbhatla, Frvwfr2, KJS77, Quaeler, Iridescent, Markan~enwiki, Fitzwilliam, Gholam, Wleizero, J Di, Igoldste, DavidHOzAu, Drama freak, Shoshonna, Blehfu, Ewulp, Courcelles, Scarlet Lioness, Srain, Luminaux, Tawker- bot2, Emote, Fvasconcellos, SkyWalker, Slmader, JForget, ScottW, Sakurambo, Tanthalas39, Ale jrb, Earthlyreason, Insanephantom, Yinchongding, NickSpiker, CBM, BeenAroundAWhile, Tschel, Rockruler73, Kylu, DanielRigal, ShelfSkewed, Outriggr (2006-2009), Stebulus, Pnatt, MrFish, ADyuaa, Gregbard, Funnyfarmofdoom, Fox6453, Laura S, Lcguang~enwiki, Chuck Marean, Goldfritha, Gogo Dodo, JFreeman, Flowerpotman, Mannyjr95, ST47, Pgomambo, Muhandis, Studerby, Tawkerbot4, Shirulashem, DumbBOT, Chrislk02, Abtract, Ebyabe, Daniel Olsen, Davis 1188, Thijs!bot, Epbr123, Pajz, Martin Hogbin, Keraunos, Mojo Hand, Mr. Brain, Cool Blue, Adam2288, Leon7, AgentPeppermint, Edhubbard, Dr noire, Luke Smith64, MichaelMaggs, Dawnseeker2000, Natalie Erin, Rhythm droid, PrescitedEntity, Escarbot, Danielfolsom, I already forgot, Mentifisto, Hmrox, Porqin, $5forMe, AntiVandalBot, Cultural Freedom, Widefox, CodeWeasel, Seaphoto, Opelio, QuiteUnusual, Doc Tropics, Autocracy, Ilovewikipedia101, GregML, Jj137, Tmopkisn, Clam- ster5, Dylan Lake, LibLord, Sahkpuppet, Random user 8384993, Feathered serpent, Gökhan, ClassicSC, Kariteh, JAnDbot, Yooy, Leuko, Husond, Barek, MER-C, Nthep, Samuel Webster, Arch dude, Fetchcomms, WmRowan, Amacachi, Ikanreed, Max Hyre, PaleAqua, Dar- iusMonsef, Colordoc, Celemourn, Max pesce, Acroterion, Magioladitis, Karlhahn, Pedro, Bongwarrior, VoABot II, Carlwev, Kuyabribri, Mrld, JNW, JamesBWatson, Yyyikes, Confiteordeo, Steven Walling, Caesarjbsquitti, Stewiegriffin, Animum, LookingGlass, Artist in France~enwiki, Mollwollfumble, DerHexer, JaGa, Greenwoodtree, Wikianon, Bibliopegist, Paliku, Hdt83, MartinBot, Xv8M4g3r, Bet- Bot~enwiki, Vanessaezekowitz, Jim.henderson, Rob Lindsey, AlexiusHoratius, Nono64, Deflagro, J.delanoy, Pharaoh of the Wizards, Kimse, Dingdongalistic, Numbo3, Normankoren, Mrhsj, Skumarlabot, Murphy ernsdorff, Icseaturtles, Gzkn, Bot-Schafter, DarkFalls, Envy69, Skier Dude, HiLo48, Chiswick Chap, Yule-john, Datdirtydon, 05fcrane, NewEnglandYankee, In Transit, Kraftlos, BluePuddle, Cmichael, TottyBot, KylieTastic, Juliancolton, Vanished user 39948282, Micro01, Kat586, Pdcook, Jinsenken, CardinalDan, ABCDER, Idioma-bot, Funandtrvl, ReferFire, JameiLei, Lights, Deor, VolkovBot, Thedjatclubrock, ABF, JGHowes, Uyvsdi, Jeff G., Indubitably, Riskykitty1, Jeromesyroyal, AlnoktaBOT, Paulscholesscoresgoals, Chienlit, Aesopos, Barneca, Capsot, Philip Trueman, Chiros Sun- rider, DoorsAjar, Revilo314, TXiKiBoT, Oshwah, BaronVonchesto, Davehi1, Death-923485`, Hqb, GDonato, Jeff jackson100, Briton- amission, Drew 0123, Anonymous Dissident, ElinorD, Rebornsoldier, Crohnie, Charlesdrakew, Aymatth2, Someguy1221, Onyx the hero, ChestRockwell, Martin451, JhsBot, Leafyplant, Jackfork, LeaveSleaves, Therandomerx3, FuddRucker, M994301009, Dpgtime, Odo1982, Blurpeace, Jamesmarkhetterley, Lerdthenerd, Andy Dingley, Dirkbb, Synthebot, Gorank4, Falcon8765, Turgan, Cjwbrown, MCTales, SMIE SMIE, Scottvn, Fugace, Pjoef, AlleborgoBot, Ratuliut, Arnonel, Stomme, Sauronjim, FlyingLeopard2014, EmxBot, D. Recorder, Bb2hunter, GoddersUK, Newbyguesses, Gil Dekel, SieBot, Whiskey in the Jar, Cobra9988, Nubiatech, YonaBot, Cal- tas, Twinkler4, RJaguar3, Vanished User 8a9b4725f8376, Jingee, Keilana, Vonsche, Flyer22 Reborn, Tiptoety, Aruton, AdamWooten, Lightmouse, Training Centre, MadmanBot, Ozonew4m, HighInBC, Mygerardromance, Krefts, Jimqbob, Denisarona, Sitush, Escape Or- bit, C0nanPayne, Itsyourmom, Martarius, Beeblebrox, Elassint, ClueBot, Yiwahikanak, PipepBot, Snigbrook, The Thing That Should Not Be, Helenabella, Kodia, Jan1nad, Battyface, TheHolyThief, Arakunem, Ukabia, Sony10mfd, Drmies, Braksus, Timberframe, San- jeev.singh3, Piledhigheranddeeper, CharlieRCD, Monster boy1, DragonBot, Excirial, Anonymous101, Jusdafax, Resoru, Bvlax2005, Zaharous, GreenGourd, Rhododendrites, MacedonianBoy, Arjayay, Jfox433, Jotterbot, AnshumanF, JamieS93, Kaiba, Mikaey, Ot- tawa4ever, Thehelpfulone, Thingg, Acabashi, PCHS-NJROTC, Berean Hunter, Mythdon, Apparition11, Runefrost, Gnowor, Burn- ingview, Soopto, Ost316, Rreagan007, WikHead, SilvonenBot, Raso mk, Hess88, Schank1234, Addbot, ERK, Some jerk on the Internet, DOI bot, Beamathan, Fyrael, IXavier, Theleftorium, DaughterofSun, TomTom321, Nuvitauy07, RJWiki27, Ronhjones, Fieldday-sunday, Fluffernutter, Btzkillerv, Glane23, YAKNOWDOUGAL!, Favonian, LinkFA-Bot, Owen martin08, Numbo3-bot, Doug youvan~enwiki, Tide rolls, Taketa, Teles, LuK3, Matt.T, Legobot, Luckas-bot, Yobot, QueenCake, Dylpickleh8, Ayrton Prost, Eric-Wester, Tempodivalse, Synchronism, AnomieBOT, Andrewrp, Asdflollyland, Bouleau, Jim1138, Galoubet, Ellexium, TELunus, Kingpin13, Bird shmestical, Guru27gurmeet, Bueford243, Hallyfamen, Crecy99, Moltenriches, Materialscientist, RobertEves92, 90 Auto, Citation bot, Insomnia175, OllieFury, Elm-39, Alexander.ranson, Maxis ftw, Becca stahl, Rene Hubertus, Googlere, Neurolysis, Rcdc008, Jamhal, Gregcatlin, Edgar- 11.1. TEXT 135

glen, 1210Poppy, Xqbot, Enterphrase, Yomammaisamartyr, Sionus, JimVC3, Capricorn42, , Tad Lincoln, Mlpearc, AbigailAber- nathy, Oldielowrider, J04n, Abce2, ProtectionTaggingBot, Dogilog, Earlypsychosis, Prunesqualer, Bellerophon, Mayor mt, Tabbycat1596, Doulos Christos, Sophus Bie, I corrected them, In fact, Shadowjams, Ehird, E0steven, Gilligan Skipper, MaryBowser, Ashwinvr96, UVA Astronomer, Ryanbrannonrulez, LucienBOT, Pepper, Sky Attacker, Bkotrous036, Soawsome1, Recognizance, Dcheagle, Segga- surra, Ilikemen6, HJ Mitchell, Fizzypop147, NadeL niB asamO, Stezie12, DivineAlpha, HamburgerRadio, Citation bot 1, Pinethicket, ShadowRangerRIT, I dream of horses, Hard Sin, El estremeñu, Barney The Dino, Captain Virtue, Gbueermann, Jschnur, Impala2009, ,Williame3, Dumbo1540 ,ئاراس نوری ,Bethywethy, Crystal84, OnlineBG, Dac04, Mininessie, Zach1775, Tim1357, Leasnam, FoxBot Dark Lord of the Sith, Lily50, Quartonworks, Grapesoda22, North8000, Barcelona02, Lotje, Callanecc, Vrenator, PaleZoe, Fman937, Duoduoduo, Lolzoutloud, Reaper Eternal, Zsimo30, Aiken drum, No One of Consequence, Zhunn, Weedwhacker128, Jhenderson777, JV Smithy, Noahnoodle, Sammetsfan, Tbhotch, Reach Out to the Truth, Zirgaq, Minimac, Mean as custard, Rainbowgifts, 66sankan, Thesmarty, TjBot, Noommos, NerdyScienceDude, Dannyboy1209, JoseCaivano, Xoristzatziki, Slon02, Mandolinface, DASHBot, Ray- man60, EmausBot, Acather96, WikitanvirBot, GoingBatty, RA0808, Andromedabluesphere440, RenamedUser01302013, NotAnony- mous0, Only777, Tommy2010, Wikipelli, K6ka, Michaelthurgood, Candy4567, SoulWithin, ZéroBot, John Cline, PBS-AWB, Slater- nater, Billy252, Josve05a, Darkmanontop, Jeanluc98, Bryce Carmony, The Nut, Blackphyre, Timakazero, EWikist, Confession0791, Wayne Slam, Frigotoni, TyA, Natbrown, Darktrumpet, L Kensington, Owenmann, Rigley, Kranix, MonoAV, Donner60, Puffin, Chuis- pastonBot, Labargeboy, Matthewrbowker, Fgruifgyuydfgus, TYelliot, Sven Manguard, DASHBotAV, SenecaV, Mjbmrbot, Socialservice, Petrb, Frank Dickman, ClueBot NG, Manusaravanan12, Rich Smith, Nmarill, Nothingisoftensomething, Ulflund, Mjanja, This lousy T-shirt, Satellizer, Gameboycolour64, Gwendal, Movses-bot, Owen4004, Skoot13, Travis rowey, IMrightmitch, O.Koslowski, What- seemstobetheproblemwikipedian, Widr, Ikeisco, Secret of success, Helpful Pixie Bot, Cherndo5, 2015magroan, Shnitzel20, HMSSo- lent, Dookymonkey, Calidum, Fenderbender234, Liorj, Kinaro, Lowercase sigmabot, That70sshowlova, Epicman55, Ereallygoodname, Hz.tiang, Northamerica1000, Ok4ycomputer, Tosh.0 Luver, Cyberpower678, Hallows AG, Mr432scott, MusikAnimal, AvocatoBot, Sei- deacreative, Dameliasy, StuGeiger, Kagundu, Mark Arsten, Mdfpph, Louis Roll Calloway, TheWarmOne, Extralivedeggs, Jgsho, Cg- mellor, Snow Blizzard, Mejoribus, AustinNault, Glacialfox, Hewes, Bananabob1492, Bradman32, Pratik10, Hdgveuy, EricEnfermero, MeanMotherJr, BattyBot, Guanaco55, Justincheng12345-bot, Brodyisnumber1, Mtorgoman, Pratyya Ghosh, Cyberbot II, The Illusive Man, Harm01, Horsesandponies45152, Krystaleen, JYBot, IjonTichyIjonTichy, Cmw255, Harsh 2580, Dexbot, Webclient101, Ruthand- kyle, Jatinag22, Oliviamcnamara, Westykraft, Omegahollow1234, Lugia2453, Ak5791, Frosty, SFK2, EatIcecream2, Kiieu57, Jochen Burghardt, Natasha pokeware, Lucyeuington, Silver Cat Tails, Wywin, NightShadow23, Xxnerdzillaxx, BillyMays232, Mr.woisard, Cadil- lac000, Doomedfella989, Reatlas, Fatjoe23343, Isoccer00, Epicgenius, Msartakov, Wajidkanju, Howicus, Artchick27, Eyesnore, Dl- dude2k14, PhantomTech, Everymorning, Mookers13, Johnscotaus, Natimosquera, CokeHanx, CensoredScribe, Editingeditors, Intro123, Sibekoe, Ugog Nizdast, Ginsuloft, Robinesque, Jagannath mango, Quenhitran, Philcourt94, Axeman225, EverythingGeography, My- nameissachin, Aubreybardo, Lizia7, Krotera, TheOrangePuff, Evan Guthrie, MilmolbelGaming, Jmoctopus, Sontic, 673407john, Perri- cal, Mortifierr, TheStrayDog, JaconaFrere, Writers Bond, James horan, Monkbot, Grade X, Dobwynn, Rachid du 25, Bobbysangar902, 12horsecraze34, Funny Jarryd, Crisa2014, JuanRiley, Amortias, Notjustavictor, Aroojdar, Spellchecker3000, Drake8821, Geordielad49, My name is Jo Blogs, Kjerish, Iurii Komogorov, Jess o7 1d, Hamood Abdu Mahyoub Yahya, Rosierocksforrosie, Rasmuskj, Shader8, Morgan132, Edward Braidwaithe, Rosierocksthatsrightlol, Civicsdan, Zhongguoyingdu, Everha, MacPoli1, Emiliuso, Martimama12, DaKoalaLord45, Minnie3380, Kaitlin233433334434444444443, KasparBot, Anarchyte, That Guy You Know4567, Sir Cumference, MitchellPritchettLSD, Anjali das gupta, Permstrump, Simplexity22, Yimingguo, Baking Soda, InternetArchiveBot, Daniel kenneth, GreenC bot, Clawraich (Dalek), Motivação and Anonymous: 1461 • Texture (visual arts) Source: https://en.wikipedia.org/wiki/Texture_(visual_arts)?oldid=763344222 Contributors: Discospinster, R. S. Shaw, Nihiltres, Rsrikanth05, SmackBot, Gilliam, Ohnoitsjamie, Derek R Bullamore, IronGargoyle, Bridgeplayer, Modernist, Hullaballoo Wolfowitz, Bus stop, Bonadea, Pchackal, CardinalDan, Technopat, Mickeymartin01, Jarble, Materialscientist, Anna Frodesiak, Khruner, Emiraly, Masterknighted, SpaceFlight89, RjwilmsiBot, EmausBot, K6ka, Zenao1, 28bot, ClueBot NG, Jack Greenmaven, Turn685, Widr, BG19bot, Geraldo Perez, Photographer24, Sadiqashaheen93, TejasDiscipulus2, Joshuajoson13, AmazingGold, Sumita Roy Dutta, Vsipuli, Fmadd and Anonymous: 53 • Lightness Source: https://en.wikipedia.org/wiki/Lightness?oldid=754298106 Contributors: Mzajac, Rich Farmbrough, Drbreznjev, Ja- cobolus, Salix alba, Srleffler, Bgwhite, Adoniscik, Pegship, Sardanaphalus, SmackBot, Gilliam, Chris the speller, VMS Mosaic, Just plain Bill, J 1982, JorisvS, Dicklyon, Paul Foxworthy, Blehfu, Gproud, Keraunos, P.B. Pilhet, SharkD, In Transit, Ratuliut, Neparis, The Random Editor, Cpoynton, Hxhbot, Ru dagon, Svick, ClueBot, Alexbot, Staticshakedown, Dthomsen8, Ost316, Libcub, Addbot, DOI bot, Fryed-peach, Adelpine, Aboalbiss, AnomieBOT, McSush, Citation bot, Fatheroftebride, Kevdave, FrescoBot, Citation bot 1, Maggyero, Jauhienij, Trappist the monk, KlappCK, Uli Zappe, HappyLogolover2011, MerlIwBot, BattyBot, YiFeiBot, Ibrahim Husain Meraj, Monkbot, Loraof, Bender the Bot and Anonymous: 19 • Space Source: https://en.wikipedia.org/wiki/Space?oldid=736588575 Contributors: AxelBoldt, Mav, Zundark, The Anome, Seb, Larry Sanger, XJaM, Rmhermen, Toby Bartels, SimonP, Paul Ebermann, Leandrod, Jdlh, Patrick, Michael Hardy, Wshun, Llywrch, Pan- dora, Ixfd64, Goatasaur, Egil, Ahoerstemeier, Docu, Slovakia, Jdforrester, Andres, Evercat, Astudent, Smack, Michael Voytinsky, Alex S, RodC, Timwi, RickK, David J Walker, Fuzheado, CBDunkerson, Hyacinth, Omegatron, Jusjih, Banno, Gromlakh, Nufy8, Robbot, Sander123, Fredrik, Goethean, Academic Challenger, Bkell, Hadal, Roozbeh, Cyrius, Tea2min, Alan Liefting, Enochlau, An- cheta Wis, Giftlite, DocWatson42, Christopher Parham, Mintleaf~enwiki, Wolfkeeper, Sunja, Mark.murphy, Monedula, Peruvianllama, Noone~enwiki, Everyking, Curps, Ssd, Ptk~enwiki, Luigi30, Bgoldenberg, SoWhy, Pamri, Alexf, CryptoDerk, Antandrus, OverlordQ, Jossi, JimWae, Tomruen, Elroch, DarkLordSeth, Gscshoyru, Iantresman, Creidieki, Joyous!, Askewchan, Dcandeto, Zondor, Shotwell, Mike Rosoft, Brianjd, D6, Freakofnurture, Imroy, DanielCD, JTN, Jørgen Friis Bak, Discospinster, Rich Farmbrough, Vsmith, Hhielscher, ESkog, Kbh3rd, RJHall, El C, Rgdboer, Mwanner, RoyBoy, Adambro, Bobo192, Army1987, Marco Polo, 23skidoo, Smalljim, Duk, Viriditas, Rbj, Nesnad, 9SGjOSfyHJaQVsEmy9NS, Nk, Kbir1, PiccoloNamek, Sam Korn, Complex Wisdom, Nsaa, Friviere, Storm Rider, Alansohn, Anthony Appleyard, Atlant, AzaToth, Lectonar, MarkGallagher, Goldom, Cdc, Mysdaao, Bart133, Snowolf, Count Iblis, RainbowOfLight, Grenavitar, Sciurinæ, Bsadowski1, SteinbDJ, Drbreznjev, HenryLi, Ceyockey, Phi beta, Scottbell, Oleg Alexan- drov, Brookie, Vrymor, Mel Etitis, OwenX, Camw, Daniel Case, Madchester, WadeSimMiser, JeremyA, Eaolson, Eras-mus, Wayward, Prashanthns, Christopher Thomas, Pfalstad, Dysepsion, Mandarax, DavidParfitt, Ashmoo, Graham87, Magister Mathematicae, Fre- plySpang, Dpv, Enzo Aquarius, Canderson7, Sjakkalle, Seraphimblade, Krash, Bhadani, Dar-Ape, Matt Deres, FlavrSavr, Sango123, Yamamoto Ichiro, Titoxd, Mathbot, El Cid, Nihiltres, Crazycomputers, Hiding, AJR, RexNL, Gurch, AlexCovarrubias, TeaDrinker, Tedder, CJLL Wright, Chobot, Nagytibi, DVdm, Gwernol, Bartleby, Banaticus, YurikBot, Spacepotato, Personman, Hairy Dude, Charles Gaudette, Phantomsteve, RussBot, Splash, SpuriousQ, Yamara, Stephenb, Gaius Cornelius, CambridgeBayWeather, Rsrikanth05, Pseu- domonas, Bullzeye, NawlinWiki, Wiki alf, Ytcracker, Grafen, Johann Wolfgang, Bmdavll, Raven4x4x, Chichui, Bucketsofg, Syrthiss, DeadEyeArrow, Xkeitarox, Wknight94, The Halo, AlanM2, FF2010, Light current, Enormousdude, Theodolite, Zzuuzz, Mirek256, 136 CHAPTER 11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

Closedmouth, Moogsi, Nemu, Livitup, GraemeL, Vicarious, DaltinWentsworth, Rocketrye12, Kevin, Poonerpoob, Katieh5584, Paul Erik, Kgf0, DVD R W, SmackBot, Looper5920, Moeron, Lestrade, KnowledgeOfSelf, Melchoir, Pgk, Bill3000, Jagged 85, Nickst, Andypens9000, Frymaster, KittenKlub, SmartGuy Old, Gaff, PeterSymonds, Ema Zee, Gilliam, Vassgergely, Andy M. Wang, Chris the speller, Happywaffle, Michbich, Keegan, Polotet, Spilla, Trebor, Glynnmania, Silly rabbit, SchfiftyThree, CrazyGuy75, SEIBasaurus, Kythias, Octahedron80, Baa, Colonies Chris, Dethme0w, Can't sleep, clown will eat me, Shalom Yechiel, Sephiroth BCR, Onorem, Rrburke, MDCollins, Addshore, Ssnseawolf, Khukri, Kronosqq, John D. Croft, Dreadstar, Richard001, Bohunk, Weregerbil, Synthetik space, Jon Awbrey, SpiderJon, DMacks, Drc79, Twir, Shushruth, Andeggs, Salamurai, Ck lostsword, Kukini, Skinnyweed, The un- dertow, Lambiam, BridgeBurner, MagnaMopus, Pthag, Nitromoose, Accurizer, Hope(N Forever), WhiteShark, Ocatecir, RomanSpa, Rizzleboffin, Slakr, Werdan7, SkippyNZ, Beetstra, Avs5221, Xiaphias, Rcowlagi, Mets501, Icez, RichardF, Caiaffa, EricR, Warsmith, Xionbox, Quaeler, Hetar, BranStark, HisSpaceResearch, Iridescent, K, Newone, Cannuck, J Di, IvanLanin, Nebank72, Ytny, Cour- celles, Audiosmurf, Nkayesmith, Derekgillespie, Tawkerbot2, MarylandArtLover, Xammer, JForget, Koolkool, Ale jrb, Insanephan- tom, BoH, CBM, Baiji, Nunquam Dormio, Tjkiesel, NickW557, ShelfSkewed, Deibid, Timtrent, Gregbard, Ufviper, TJDay, Pais, Ryan, Gogo Dodo, Jkokavec, Red Director, JFreeman, Flowerpotman, Tawkerbot4, DumbBOT, Btharper1221, Vanished User jdksfa- jlasd, UberScienceNerd, Bcohea, Max Randor, Epbr123, Barticus88, MangoBird, Willworkforicecream, VKemyss, N5iln, Andyjsmith, Hugo.arg, Headbomb, Marek69, Cool Blue, Dawnseeker2000, Escarbot, Mentifisto, Porqin, AntiVandalBot, BokicaK, Luna Santin, Ju- rgenG, Seaphoto, Chairman Meow, Billscottbob, Jj137, TimVickers, Zacmds, Âme Errante, LibLord, Darklilac, Robbo741, Pichote, Spencer, Chris Larkin, Elaragirl, Jaredroberts, ClassicSC, JAnDbot, Eadlam, MER-C, The Transhumanist, Instinct, Fetchcomms, YK Times, FishHeadAbcd, Yahel Guhan, Bunny-chan, Connormah, Bongwarrior, VoABot II, Sushant gupta, Fusionmix, JNW, Yandman, JamesBWatson, Khalidkhoso, Dep. Garcia, DeMongo, Stijn Vermeeren, Twsx, Avicennasis, Catgut, Animum, Ensign beedrill, MiPe, 28421u2232nfenfcenc, Allstarecho, Emw, ANONYMOUS COWARD0xC0DE, Glen, DerHexer, JaGa, Edward321, Esanchez7587, TheRanger, Jimka, Danielratiu, Hintswen, Sir Intellegence, September 11 suicide bomber, BMRR, Hdt83, MartinBot, Poeloq, Random paige, Roman Zacharij, Smart Arse, Jay Litman, Mschel, Kostisl, R'n'B, AlexiusHoratius, Snozzer, Gunkarta, Marky7474, Wiki Raja, Maneater3033, Cyril12, J.delanoy, Trusilver, Tlim7882, Michael3646263, Uncle Dick, Maurice Carbonaro, Nigholith, Dbiel, Momo117, Politikil, Caseyrules, Darth Mike, SirIamMe, Shawn in Montreal, Katalaveno, TomyDuby, McSly, Crakkpot, Jeepday, Mikael Häg- gström, OAC, Jr Cash432, Gurchzilla, Bilbobee, Berserkerz Crit, Krasniy, Sobeninja9090, NewEnglandYankee, Poloipoloi, Parkercon- rad, Mitchy8822, Agoodname, CompuChip, Biglovinb, Potatoswatter, Juliancolton, Cometstyles, Wilston, Damien draco, Tomwzhang95, Vanished user 39948282, Austen rules, WinterSpw, Pdcook, George34567, Xiahou, CardinalDan, Funandtrvl, Spellcast, KingTheodin, Kkkklll, Rianthas, Wikieditor06, Makewater, Egghead06, X!, TreasuryTag, Ericdn, JohnBlackburne, Alexandria, Classical geographer, Aesopos, Philip Trueman, Ggghggg, BlueCanary9999, Red Act, Sickmyduck123, Sparkzy, RyanB88, Miranda, Dude hi, Mingato, Arnon Chaffin, Qxz, Anna Lincoln, Martin451, Robertovegaperalta, Abdullais4u, LeaveSleaves, Space-meister, Random Hippopota- mus, Wiae, ARUNKUMAR P.R, SpecMode, Kurowoofwoof111, BigDunc, Orgyen108, Mooshuuooo2, Synthebot, Ourterranhome, Falcon8765, Anton Gutsunaev, Monty845, W4chris, FlyingLeopard2014, Runewiki777, Neparis, The Random Editor, Botev, Tiddly Tom, Sahilm, Scarian, Euryalus, WereSpielChequers, Caltas, Nathan, Jason Patton, Jrun, Keilana, Bentogoa, Happysailor, Tiptoety, Radon210, Bbb556, JD554, Mirkoruckels, DirectEdge, JuanFox, Allmightyduck, Aruton, Aperseghin, Oxymoron83, Faradayplank, Avn- jay, Harry-, Steven Crossin, Techman224, MooMooz, Ks0stm, Macy, Svick, Benjest, Nosyt612, Kmaniac2, Liamdanny2, Max13102, Sean.hoyland, Payno, Jons63, Escape Orbit, Explicit, TheCatalyst31, XDanielx, WikipedianMarlith, ClueBot, TransporterMan, The Thing That Should Not Be, Rodhullandemu, Wysprgr2005, Krysta w, Meekywiki, Blakmonkey, Drmies, Razimantv, Mild Bill Hic- cup, Shinpah1, DanielDeibler, CounterVandalismBot, Dandog77, Blanchardb, LizardJr8, Orthoepy, Cirt, Astrognaw, MindstormsKid, Excirial, Jusdafax, M4gnum0n, CrazyChemGuy, Eeekster, Abrech, Zaharous, SpikeToronto, KubenP1234567, Cenarium, JamieS93, What a dumb name, Aiden1120, Heyheyhack, Dekisugi, Ajkarimi, Mikaey, Correctionpatrol4, Dutchartlover, Thehelpfulone, Food- formyxxxlmum, Panos84, Tired time, RenamedUser jaskldjslak901, Thingg, AlEinstein10, Aitias, Cdog1, Versus22, Wiifan13, SoxBot III, HumphreyW, Brandonlovesflasks, Goodvac, Glacier Wolf, DumZiBoT, Press olive, win oil, Denisigor, Joingies, Mattsmith halo master, XLinkBot, Zrgt, Fastily, Bjpeewee1, Nanthees, Arcer4366, Wolf713, Little Mountain 5, Mitch Ames, Masonalger, Alligator100, Mifter, Noctibus, NHJG, Tameamseo, Xhannah montana001x, Me, Myself, and I, Angkiki, Ejosse1, Mike5449, Nate0ate, Sillynanerz, 11fiej, 11harj, Rein Coetzer, HexaChord, Averagejefpet, Powers.cons, Nbm589, Addbot, Treehugger101, Firedude1900, Ryururu, Ex- perimental Hobo Infiltration Droid, Dabdbsadhbsafsajfjsbfjbasf, Montibryn11, Phlamer26, Non-dropframe, Fgnievinski, PandaSaver, Luckyscott, MartinezMD, Njaelkies Lea, Vishnava, CanadianLinuxUser, Fluffernutter, Solatido, Cst17, Smelliot9, Chamal N, Red- heylin, LAAFan, Glane23, Death1929, Janey and Stephan, Chzz, Debresser, Roux, Favonian, Macgrew, Nesssa, Sarjan100, 5 albert square, Allyjudd01, Brinkley32, Stefan520, Aaaaaaabbbbbbccccccc, Theking17825, Tide rolls, Lightbot, Anxietycello, Romanskolduns, Luckas Blade, Bartledan, Bobby no hair, Swarm, Sunshine222, Sanjeevparallel, Chris9999100, Harvs123, Luckas-bot, Kibakun64, Cimbom-fan, II MusLiM 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Citation bot 1, Cvan21, Notedgrant, Tom.Reding, Calmer Waters, Dac04, December21st2012Freak, White Shadows, ActivEx- pression, FoxBot, Kukka9002, Blast73, ItsZippy, Lotje, Dinamik-bot, Vrenator, Nejnejjaja, Zink Dawg, Crouchie99, Bobofari, Iamgriz- zly, Diannaa, Sun.portal, CheapCats, Tbhotch, Astro-addict, Reach Out to the Truth, Minimac, LDP0, Scottdjp, Xtarcted, Keegscee, Bio-addict, DARTH SIDIOUS 2, Skullz752, Basshunter1965, Noommos, Salvio giuliano, Mandolinface, Parker171717, DASHBot, EmausBot, Cuboarding2, Bemar11, Cooly45, Westc55998, Jorge c2010, Trickett rocks, Johhnoo, MichaelRoderick, Farpre, Jamie9257, Noobish456, RA0808, RenamedUser01302013, Vanished user zq46pw21, Legend1234567890987, OOo BaMItZaNonYMoUs oOo, Tommy2010, OOo !LeTMeSPaMItUP! oOo, Forster357591, Trevwilson16, Weleepoxypoo, Wikipelli, Freedaja, Erpert, Fæ, Jenks24, Krisma12234, Netknowle, Gz33, Ke$ha stinks, Naseem Awan, Fubajoe, Leroykap, Coasterlover1994, L Kensington, Donner60, Killersk, CybernautiqueMMX, Naty1232, KHvE, Sunshine4921, Breslandrhys, Forever Dusk, LikeLakers2, Jdr900nf, Bean49Bot, Jrsfk2, Gwest- heimer, TheAckademie, Will Beback Auto, ClueBot NG, Wcherowi, CocuBot, A520, Joefromrandb, Delusion23, Widr, Mudiojas, MerlI- wBot, Helpful Pixie Bot, Malak1man, Seannybhoy1888, Cow25, Rump a okn, Ponpopl, Hengist Pod, Lowercase sigmabot, Mattew12345, Kgcd2011, Jelly gelo, Thefactsphere, Kammyy, Denislearyisgod, TrapJesus017, Huzaifa391, PTJoshua, Thelightbites, Abdul.anifowoshe, Gaga123456789, Heylow, Nospildoh, Hattah, Marsman325, Fowzan ahmed, Rvl456, Skysky321, Tara1112, Ankit Dama, Xenoma, My- NameIsAnonymouse, Burntheskytoday, Imyadad, Cami12323, YVSREDDY, Oskavannisutterlyammazing, Wiki2103, Miles502, Phys- iomath, Dourios, Factfindersonline, Mithoron, Khazar2, Stumink, MadGuy7023, KCyron, Skurlo123, Webclient101, Saehry, Choc2, 11.1. TEXT 137

Ches85, Aftabbanoori, Epicgenius, FallingGravity, Jianluk91, Cmckain14, Samroath1, The Oracle of Etro, Chelblue, Totes amazeballs, Walmart666, Smart&Awesome, SmallArsee142, Prokaryotes, Orangecones, W. P. Uzer, Upshout78, Concord hioz, WillemienH, Google- GlassHuman, Tetra quark, Isambard Kingdom, Jark101, KasparBot, BD2412bot and Anonymous: 1034 • Rule of thirds Source: https://en.wikipedia.org/wiki/Rule_of_thirds?oldid=750381166 Contributors: Dan Koehl, Ahoerstemeier, An- gela, Stefan-S, Charles Matthews, Haukurth, Clngre, Mark.murphy, Histrion, Arasithil, Mike Rosoft, Imroy, Discospinster, Rich Farm- brough, Mattdm, RoyBoy, Bobo192, Teeks99, Hooperbloob, Nsaa, Rray, Diego Moya, Stephan Leeds, Shimeru, Daranz, Mindmatrix, JeremyA, Mandarax, Graham87, Chupon, Joe Decker, Nightscream, Borgx, Stephenb, NawlinWiki, NickBush24, Moe Epsilon, The- Seer, SmackBot, Coatesg, McGeddon, Delldot, Imzadi1979, Ohnoitsjamie, Chris the speller, Bluebot, 1337-n00b, JonHarder, Rrburke, SundarBot, Alton.arts, Cybercobra, SteveHopson, Spiritia, LtPowers, Bjankuloski06en~enwiki, Cornel c ilie, Slakr, Mr Stephen, Dick- lyon, H, Iridescent, MIckStephenson, Benwildeboer, Dgw, Amandajm, Gonzo fan2007, Thijs!bot, Epbr123, Mactographer, Moondig- ger, Marek69, JustAGal, Grayshi, Darev, AntiVandalBot, Spirits19, Joachim Michaelis, Jj137, Modernist, Eddyspeeder, RedCoat10, JAnDbot, Skomorokh, SlamDiego, Dazp1970, Hdt83, MartinBot, Mikr18, Kateshortforbob, WarthogDemon, Yongbojiang, Johnbod, Nemo bis, SuzanneKn, NewEnglandYankee, Renardo la vulpo, VolkovBot, Thedjatclubrock, Philip Trueman, TXiKiBoT, David Daniel Turner, Synthebot, Legoktm, Yintan, Filos96, Necoplay, Svick, Cyfal, Owen the greatest, Dcattell, Tradereddy, ClueBot, Binksternet, The Thing That Should Not Be, Vlaze, Woolters, Hafspajen, LizardJr8, Aitias, DumZiBoT, Jengirl1988, Addbot, Nickenge, Jojhut- ton, Fieldday-sunday, Logster871, CanadianLinuxUser, Ferronier, Cst17, Buster7, Alchemist-hp, Tide rolls, DrFO.Tn.Bot, Luckas- bot, Yobot, Markermo, KamikazeBot, AnomieBOT, Dwayne, ArthurBot, Zad68, GrouchoBot, Redoak58, Xposurepro, Shadowjams, E0steven, A.amitkumar, Ubub92, Redrose64, Pinethicket, I dream of horses, A8UDI, Hasenläufer, Tbhotch, Onel5969, RjwilmsiBot, Ssh00, Born2bgratis, DASHBot, EmausBot, Kpufferfish, Eyebrowpiercingguy, Winner 42, ZéroBot, Akerans, Grunny, Rcsprinter123, Donner60, Lastrachris, Gwen-chan, ClueBot NG, MelbourneStar, Widr, Cutano, Helpful Pixie Bot, Darafsh, Davidiad, Rozza911, Klilidiplomus, Achowat, David.moreno72, Freredaran, Lugia2453, Junkyardsparkle, Frosty, Dragon1501, Jfernandes27, Kharkiv07, Chriszerre, Stantheman13, Equinox, CAPTAIN RAJU, Jmahoney1, Mohibullah.mamun, Elfey1, Bender the Bot and Anonymous: 222 • Golden ratio Source: https://en.wikipedia.org/wiki/Golden_ratio?oldid=760067432 Contributors: AxelBoldt, WojPob, Mav, Bryan Derk- sen, Zundark, The Anome, Tarquin, Josh Grosse, Youssefsan, XJaM, Arvindn, Heron, Lightning~enwiki, Olivier, Patrick, Michael Hardy, Paul Barlow, Dominus, Nixdorf, Kku, Gabbe, SGBailey, Menchi, Wapcaplet, Ixfd64, Eliah, Tango, Sannse, Seav, GTBacchus, Ppareit, Looxix~enwiki, ArnoLagrange, Ellywa, Ahoerstemeier, Cyp, Haakon, Ryan Cable, William M. Connolley, Angela, DropDeadGorgias, Mark Foskey, Glenn, Whkoh, Gisle~enwiki, Dod1, Jacquerie27, Smaffy, Rob Hooft, Tobias Conradi, Raven in Orbit, Etaoin, Ideyal, Stephenw32768, Feedmecereal, Crusadeonilliteracy, Alex S, Charles Matthews, CecilBlade, Dysprosia, Prumpf, Tpbradbury, Furrykef, Hyacinth, AndrewKepert, Kwantus, Johnleemk, Finlay McWalter, Frazzydee, Owen, Denelson83, Phil Boswell, Donarreiskoffer, Rob- bot, KeithH, Jakohn, Fredrik, Chris 73, Mayooranathan, Gandalf61, Sverdrup, Academic Challenger, Rursus, Timrollpickering, Alan De Smet, Sheridan, Miles, HaeB, Jleedev, Mattflaschen, Snobot, Weialawaga~enwiki, Giftlite, Gene Ward Smith, Tom harrison, Jabra, MSGJ, Herbee, Fropuff, Peruvianllama, Anton Mravcek, Everyking, Gus Polly, Mcapdevila, Alison, JeffBobFrank, Sunny256, WHEELER, Thierryc, Daveplot, Guanaco, Eequor, Matthead, Jay Carlson, Bobblewik, Architeuthis, Utcursch, Mike R, Jackcsk, Noe, Antandrus, GeneMosher, OverlordQ, Jossi, Girolamo Savonarola, DragonflySixtyseven, Secfan, Icairns, Karl-Henner, Sam Hocevar, DanielZM, Immanuel Goldstein~enwiki, Neutrality, Uaxuctum~enwiki, Klemen Kocjancic, Chmod007, Kousu, Trevor MacInnis, Grunt, Eisnel, ELApro, Flex, Gazpacho, Fls, Frankchn, Mormegil, AAAAA, Freakofnurture, Imroy, Discospinster, Brianhe, Rich Farmbrough, Rhobite, Guanabot, FranksValli, Vsmith, Max Terry, Deelkar, Paul August, Joblio, Bender235, ESkog, Kaisershatner, Jnestorius, Mattdm, Ground, Elwikipedista~enwiki, TruthSifter, El C, Workster, Shanes, RoyBoy, Triona, Andrewpmack~enwiki, Bobo192, Reinyday, C S, Cmdr- jameson, NickSchweitzer, Daf, Apostrophe, DCEdwards1966, Manu.m, Nsaa, Perceval, Mareino, Hyperdivision, Sigurdhu, Osmosys, Alansohn, Jic, Free Bear, Tek022, Jesset77, Andrewpmk, Arvedui, Anittas, Jnothman, Jamiemichelle, MarkGallagher, BryanD, Sligocki, PAR, Wdfarmer, Hu, Yolgie, DreamGuy, Hohum, Pianoplayerontheroof, Snowolf, Gloworm, ReyBrujo, Suruena, Uffish, Evil Mon- key, Lokeshwaran~enwiki, Jheald, 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RadioFan2 (usurped), Stephenb, Rsrikanth05, Wimt, JohanL, Jaxl, Nad, JocK, Nucleusboy, Brandon, Dimabat, DeadEyeArrow, JMBrust, Tachyon01, TUSHANT JHA, Imgregor, Tetracube, Mike Serfas, 2over0, Ali K, Chesnok, Arthur Rubin, Char- lik, StealthFox, BorgQueen, JoanneB, Nae'blis, HereToHelp, Willbyr, Moonsleeper7, Appleseed, Persept, JDspeeder1, GrinBot~enwiki, Mejor Los Indios, DVD R W, Finell, That Guy, From That Show!, NetRolller 3D, Attilios, SmackBot, RDBury, Looper5920, Mmernex, Unschool, Adam majewski, Jerdwyer, IddoGenuth, Nihonjoe, Tomer yaffe, InverseHypercube, KnowledgeOfSelf, Melchoir, McGeddon, CopperMurdoch, Unyoyega, Pgk, Rokfaith, Jagged 85, Jdmt, Tr0gd0rr, Delldot, Jihiro, Eskimbot, Kaimbridge, BiT, Wittylama, Xaosflux, Richmeister, Youremyjuliet, Skizzik, Cabe6403, ERcheck, Afa86, Durova, Cowman109, Kaiserb, Chris the speller, Pdspatrick, Persian Poet Gal, B00P, Tree Biting Conspiracy, Silly rabbit, BALawrence, ERobson, Adpete, Octahedron80, Nbarth, Baa, ACupOfCoffee, Zven, Gracenotes, Can't sleep, clown will eat me, Joerite, Jamse, Jahiegel, Tamfang, Scray, TheGerm, Berland, LouScheffer, Addshore, Alton.arts, Rrc2002, ConMan, Wen D House, Flyguy649, Napalm Llama, Lhf, Angellcruz, M jurrens, Attasarana, Illnab1024, DMacks, Lisasmall, Wizardman, Just plain Bill, Xiutwel, Kukini, Drunken Pirate, John Reid, Will Beback, Pinktulip, Zchenyu, Lambiam, Esrever, Phi1618~enwiki, ArglebargleIV, Rory096, OliverTwist, BorisFromStockdale, JzG, Kuru, Titus III, Scientizzle, Kumarsenthil, Park3r, Adj08, Aroundthewayboy, Eikern, Minna Sora no Shita, Mgiganteus1, NongBot~enwiki, Jbonneau, Phancy Physicist, Ckatz, Loadmaster, MarkSutton, Stwalkerster, George The Dragon, Alethiophile, Mr Stephen, Childzy, Dicklyon, Cxk271, Jaipuria, Waggers, AdultSwim, Ryulong, MTSbot~enwiki, Inquisitus, Xionbox, Iridescent, Kencf0618, Madmath789, Andreas Rejbrand, Lenoxus, GrammarNut, Tib- bits, Tawkerbot2, Gco, Pi, JRSpriggs, Hum richard, Emote, Lavaka, Owen214, The Haunted Angel, JForget, Tanthalas39, Asteriks, Aypak, Ale jrb, Sjmcfarland, Stmrlbs, Woudloper, CBM, Eric, JohnCD, Collinimhof, Runningonbrains, Dgw, Tac-Tics, Joelholdsworth, Emesghali, MrFish, Myasuda, Cydebot, Scorpi0n, Mblumber, Reywas92, Carifio24, Gogo Dodo, Alanbly, Islander, Muhandis, Julian Mendez, Gtalal, Tawkerbot4, Bsdaemon, Dblanchard, Gimmetrow, KamiLian, Billywestom, Thijs!bot, Epbr123, Wikid77, David from Downunder, Herbys, Headbomb, Marek69, Ronbarton, West Brom 4ever, John254, Frank, James086, Jmelody, Dfrg.msc, Charlot- teWebb, Heroeswithmetaphors, Escarbot, Kazrian, Stannered, Maxhawkins, Mentifisto, Somnabot, John Smythe, AntiVandalBot, Nimo8, Luna Santin, Seivad, Clf23, Seaphoto, Opelio, QuiteUnusual, Czj, Quintote, DomainUnavailable, Pixiebat, Ellie57, NSH001, Manushand, LibLord, Minhtung91, Andrew Parkins, Spencer, Moonraker0022, ChrisLamb, Dhrm77, Gökhan, Michael Tiemann, Leuko, Curmi, Aheyfromhome, Jelloman, MER-C, Ricardo sandoval, Alpinu, PhilKnight, Twospoonfuls, Acroterion, Prof.rick, Bensonchan, Bong- 138 CHAPTER 11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

warrior, VoABot II, 123987456, JNW, Swpb, Ling.Nut, PeterStJohn, Richard Bartholomew, Midgrid, Catgut, Animum, SSZ, Erichas, Powerinthelines, Shocking Blue, Virtlink, Allstarecho, David Eppstein, Styrofoam1994, Madmanguruman, Gomm, Martaclare, Der- Hexer, JaGa, GhostofSuperslum, Lelkesa, A2-computist, EduardoAndrade91, Hbent, TimidGuy, GravityWell, AVRS, Tentacles, Mart- inBot, Lord Ibu, Bbi5291, ExplicitImplicity, Poeloq, Axlq, Math Lover, CommonsDelinker, Pbroks13, The Anonymous One, J.delanoy, Pharaoh of the Wizards, GoatGuy, Svetovid, Wmjohn6217, Personjerry, Eliz81, WarthogDemon, AVX, Bluesquareapple, Vanished user 342562, Deborah A. 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Smith, Robo37, Asheryaqub, Citation bot 1, Tkuvho, DrilBot, Emilu18, Pinethicket, Honeymancr12, Foothiller, Hamtechperson, Achim1999, A8UDI, MasterminderBS, RedBot, MastiBot, Flashharry9, Marsal20, Toolnut, FoxBot, Kapgains, Ambarsande, TobeBot, Pollinosisss, Fox Wilson, Mileswms, Xx3nvyxx, Zvn, Bee- -Gzorg, Tbhotch, FKLS, Minimac, Magic cigam, Spencerpiers, Berg.Heron, RjwilmsiBot, Bento00, Mr ,غلامعلي نوري ,bLee, Suburb 77 Right425, Fiboniverse, Regancy42, Balph Eubank, Going3killu, Salvio giuliano, WikitanvirBot, Broselle, Whalefishfood, Nerissa-Marie, EddyLevin, RA0808, Dooche101, 8bits, Ryan c chase, Kris504, Tommy2010, Wikipelli, Dcirovic, Misscmoody, Slawekb, Chricho, Benchdude, Destiney Kirby, ZéroBot, Jargoness, John Cline, Fæ, Shuipzv3, Fg=phi, Chharvey, Matthewcgirling, Cobaltcigs, H3llBot, Aughost, SporkBot, Aknicholas, Gz33, Didi42brown, Tolly4bolly, Vanished user fijw983kjaslkekfhj45, Sbmeirow, Num Ref, Mayur, Arkaever, Donner60, Smartie2thaMaxXx, DeltaQuad.alt, Dragfiter234, JanetteDoe, GrayFullbuster, DASHBotAV, Nikolas Ojala, Dav- esteadman, ClueBot NG, Fridakahlofan, Wcherowi, MelbourneStar, Nepanothus, Frietjes, half-moon bubba, O.Koslowski, KirbyRider, Widr, Kant66, Devsinghing, Anon5791, Oddbodz, Helpful Pixie Bot, Bxzooo, Rosetheprof, Calabe1992, Vagobot, Gasberian, MusikAn- imal, Metricopolus, Wikimpan, Davidiad, FrostBite683, EspaisNT, Sparkie82, Brad7777, Gdmall88, Klilidiplomus, Irung4, Rob Hurt, Tejasadhate, Bill.D Nguyen, Gmoney123456789, Boeing720, WebFlower1, Cyberbot II, Mauricio1994, Maxronnersjo, Petrus3743, -YFdyh-bot, Giufra9396, Afonfbg, Kennethdjenkins, Dexbot, Luckimg2, Hmainsbot1, Iivanyy89, Webclient101, Lu ,شامخ بشموخ gia2453, Jamesx12345, NealCruco, Limit-theorem, Moony22, RPFigueiredo, D. Philip Cook, Ultimatesecret12, Mre env, Melonkelon, Tentinator, Bzavitz, Goss is super, Ugog Nizdast, Mynameisrichard, Evensteven, MrBearHugger, Dvorak182, Jianhui67, Paul2520, For- tok1, Xenxax, Amrik singh nimbran, AnonymousAuthority, ThatRusskiiGuy, Monkbot, Tunisie98, BethNaught, Tk plus, 400 Lux, KH-1, Loraof, Gabrielcwong, , KasparBot, Martin Peter Clarke, BU Rob13, Lemondoge, Skyllfully, Sgr ganesh, GreenC bot, Earl of Arundel, FRANC85 and Anonymous: 1633 • Rabatment of the rectangle Source: https://en.wikipedia.org/wiki/Rabatment_of_the_rectangle?oldid=745114678 Contributors: Mattdm, Roscelese, Dicklyon, David Eppstein, Johnbod, Binksternet, AnomieBOT, The Interior, Smallman12q, Helpful Pixie Bot, Generaliza- tionsAreBad, Messedupkid, Bender the Bot and Anonymous: 3 • Headroom (photographic framing) Source: https://en.wikipedia.org/wiki/Headroom_(photographic_framing)?oldid=749752693 Con- tributors: Julesd, Mattdm, SmackBot, Johnbod, Binksternet, AnomieBOT, Shubinator, Miracle Pen, ClueBot NG, DanielSimpkinsJPGR, JordoCo, Helpful Pixie Bot, Bender the Bot and Anonymous: 3 • Perspective (graphical) Source: https://en.wikipedia.org/wiki/Perspective_(graphical)?oldid=762251515 Contributors: Daniel C. Boyer, Heron, Michael Hardy, SebastianHelm, Jebba, AugPi, Nikai, Raven in Orbit, Charles Matthews, KRS, Furrykef, Warofdreams, Blood- shedder, Wetman, Robbot, Paranoid, Gwrede, Altenmann, P0lyglut, Fuelbottle, Jleedev, Connelly, Giftlite, Pat Kelso, Mintleaf~enwiki, Siroxo, Vina, DragonflySixtyseven, Allefant, Cwoyte, Perey, AliveFreeHappy, Dmr2, Kenb215, ESkog, Brian0918, Robert P. O'Shea, Rgdboer, Marcok, Adambro, Marco Polo, Pearle, Mdd, Alansohn, Mduvekot, Iothiania, Fritzpoll, Metron4, Shadowolf, Ringbang, Sax- ifrage, Camw, Gaf.arq, Marudubshinki, Mandarax, BD2412, Dpv, Rjwilmsi, Nightscream, Bhadani, Volfy, SchuminWeb, Ewlyahoocom, Gurch, Intgr, OpenToppedBus, King of Hearts, Chobot, YurikBot, Sceptre, Hairy Dude, Zhatt, Thane, Dialectric, Voidxor, Tony1, Syrthiss, Mysid, Fram, Churchh, Tyrenius, DVD R W, That Guy, From That Show!, Luk, Attilios, SmackBot, Bigbluefish, Weges- rand, Jagged 85, Wakuran, Waynem, W!B:, Srnec, Hmains, AndrewKay, Durova, J-wiki, Al Hart, Nbarth, Kasyapa, Konstable, Can't sleep, clown will eat me, Smallbones, JonHarder, Rrburke, VMS Mosaic, SundarBot, DickSummerfield, Invincible Ninja, Insinerate- hymn, Markymarkmagic, Bob Castle, Ceoil, Blahm, ArglebargleIV, Mrwilly123, Richard L. Peterson, T g7, Sir Isaac Lime, Cernun- nos, Lazylaces, Ckatz, Slakr, Stwalkerster, TheHYPO, Jon186, Optakeover, Peter Horn, Dl2000, Eliashc, Hu12, Dontworry, Maestlin, 11.2. IMAGES 139

Fitzwilliam, Shoeofdeath, GDallimore, Bottesini, Scarlet Lioness, Tawkerbot2, Astrubi, CmdrObot, Dycedarg, AlbertSM, Jane023, Þor- sHammer, Andrewsandberg, Pwilon, Saintrain, Epbr123, Sry85, Cwtyler, Marek69, Vertium, James086, Oreo Priest, Widefox, Orionus, Wiki212, DarkAudit, Modernist, Charlie401, Klow, MER-C, VoABot II, JNW, Gammy, David Eppstein, JaGa, Patkelso, I Am The Walrus, Artemis-Arethusa, Middlenamefrank, Anaxial, R'n'B, Hasanisawi, Paranomia, J.delanoy, EscapingLife, Uncle Dick, Jrsnbarn, SharkD, Bookworming, Johnbod, Icehose, AntiSpamBot, Chiswick Chap, Fountains of Bryn Mawr, Sean0987, Jamesontai, Elenseel, Useight, Tkgd2007, Meiskam, Philip Trueman, Red Act, Hobe, Deep Atlantic Blue, Ferengi, Amog, Wiae, Mr. Absurd, RdLight, Ceranthor, Arcfrk, AlleborgoBot, Petergans, Austriacus, SieBot, Hertz1888, Brazzouk, Parhamr, Matthew Yeager, Joe Gatt, Zentek, Tombomp, Filam3nt, Anchor Link Bot, HighInBC, Nic bor, Novas0x2a, Jk13, Martarius, De728631, ClueBot, Snigbrook, The Thing That Should Not Be, Kafka Liz, Jan1nad, Nsk92, Mild Bill Hiccup, Laitche, J8079s, Yamakiri, MCBourne, Niceguyedc, Singingle- mon~enwiki, Bbb2007, Excirial, Gtstricky, Yorkshirian, Primasz, SchreiberBike, Shlishke, Thingg, Burner0718, SoxBot III, XLinkBot, Ivan Akira, Ost316, Sonty567, ProfDEH, Avoided, HexaChord, Iranway, Mimimzi, Addbot, Grayfell, Sakhal, Loosedance, Tide rolls, Lightbot, Balabiot, LuK3, Legobot, Yobot, Tohd8BohaithuGh1, TaBOT-zerem, Burbul, AnomieBOT, IRP, Piano non troppo, King- pin13, Flewis, Materialscientist, Citation bot, Akilaa, LilHelpa, William Peters, Zad68, Unigfjkl, Thomaspendergrast, Anna Frodesiak, The Evil IP address, J04n, Omnipaedista, White rotten rabbit, A.amitkumar, Djcam, Finalius, Tonalone, Biker Biker, Pinethicket, Alonso de Mendoza, Serols, Nukeasylum, Heavyweight Gamer, TheSplane, DARTH SIDIOUS 2, Andrea105, DexDor, Aircorn, Dirpaul, John of Reading, Golfandme, Suznews, RA0808, KHamsun, Solarra, Elee, C0ffeeartc, Dcirovic, Knight1993, Expeditionfilms, Eyadhamid, Ὁ οἶστρος, Access Denied, SporkBot, Maurice Owen, 3A2D50, Factfinderz, CybernautiqueMMX, MacStep, DASHBotAV, Rmash- hadi, Catand, 28bot, ClueBot NG, OperaJoeGreen, Slartibartfastibast, Kin10108953, Frietjes, Widr, MerlIwBot, Gallery-of-art, Dark pitbull, HMSSolent, Northamerica1000, Matt Chase, Khiart, Polmandc, Morning Sunshine, Doomguy10011, Mano06031994, Hmains- bot1, Rabenbunt, Tony Mach, 93, Mmxiicybernaut, Parcheesidude, Nait lem, Philolilo, GeorgeErgo, ȸ, Wellset, CarnivorousBunny, Negative24, Richard Payne16, JellyPatotie, Juan Kis Solt, Sro23, Pollock137, DLKingsbury, Saramaira, Tamyukkkkhing, Akhtar Hus- sain Baltistani, Shekil123, RadiculousJ, AlineXu, Xandi 89 and Anonymous: 405

11.2 Images

• File:'The_Raising_of_Lazarus’,_tempera_and_gold_on_panel_by_Duccio_di_Buoninsegna,_1310–11,_Kimbell_Art_Museum.jpg Source: https://upload.wikimedia.org/wikipedia/commons/0/0b/%27The_Raising_of_Lazarus%27%2C_tempera_and_gold_on_panel_ by_Duccio_di_Buoninsegna%2C_1310%E2%80%9311%2C_Kimbell_Art_Museum.jpg License: Public domain Contributors: Kimbell Art Museum Original artist: Duccio • File:01-Goldener_Schnitt_Formel-Animation.gif Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/01-Goldener_Schnitt_ Formel-Animation.gif License: CC BY-SA 4.0 Contributors: Own work Original artist: Petrus3743 • File:16777216colors.png Source: https://upload.wikimedia.org/wikipedia/commons/e/e9/16777216colors.png License: CC BY-SA 2.5 Contributors: Own work Original artist: Marc Mongenet • File:2-pt-sketchup.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/4e/2-pt-sketchup.jpg License: CC BY-SA 3.0 Con- tributors: Own work (Original text: I(Al Hart (talk)) created this work entirely by myself.) Original artist: Al Hart (talk) • File:3D_coordinate_system.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2c/3D_coordinate_system.svg License: CC- BY-SA-3.0 Contributors: No machine-readable source provided. Own work assumed (based on copyright claims). Original artist: No machine-readable author provided. Sakurambo~commonswiki assumed (based on copyright claims). • File:7234014_Parthenonas_(cropped).jpg Source: https://upload.wikimedia.org/wikipedia/commons/c/c8/7234014_Parthenonas_%28cropped% 29.jpg License: CC0 Contributors: Own work Original artist: C messier • File:AdditiveColor.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c2/AdditiveColor.svg License: Public domain Con- tributors: Transferred from en.wikipedia to Commons. Original artist: SharkD at English Wikipedia Later versions were uploaded by Jacobolus at en.wikipedia. • File:Aeonium_tabuliforme.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/4f/Aeonium_tabuliforme.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: Max Ronnersjö • File:Albert_Einstein_Head.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/d3/Albert_Einstein_Head.jpg License: Pub- lic domain Contributors: This image is available from the United States Library of Congress's Prints and Photographs division under the digital ID cph.3b46036. This tag does not indicate the copyright status of the attached work. A normal copyright tag is still required. See Commons:Licensing for more information. Original artist: Photograph by Orren Jack Turner, Princeton, N.J.

• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public domain Contributors: Own work, based oﬀ of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs) • File:Andrea_Mantegna_-_The_Lamentation_over_the_Dead_Christ_-_WGA13981.jpg Source: https://upload.wikimedia.org/wikipedia/ commons/c/c3/Andrea_Mantegna_-_The_Lamentation_over_the_Dead_Christ_-_WGA13981.jpg License: Public domain Contributors: Web Gallery of Art: Image Info about artwork Original artist: Andrea Mantegna • File:Auto_Texture_created_over_Clear_glass_Bricks.jpg Source: https://upload.wikimedia.org/wikipedia/commons/1/15/Auto_Texture_ created_over_Clear_glass_Bricks.jpg License: CC BY-SA 4.0 Contributors: Own work Original artist: Sumita Roy Dutta 140 CHAPTER 11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

• File:Axonometric_projection.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/48/Axonometric_projection.svg License: Public domain Contributors: This vector image was created with Inkscape. Original artist: Yuri Raysper • File:CIExy1931_fixed.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f3/CIExy1931_fixed.svg License: CC-BY-SA- 3.0 Contributors: • CIExy1931.svg Original artist: CIExy1931.svg: Sakurambo • File:Cams.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/de/Cams.svg License: Public domain Contributors: Own work Original artist: Zureks • File:Carl_Friedrich_Gauss.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/9b/Carl_Friedrich_Gauss.jpg License: Pub- lic domain Contributors: Gauß-Gesellschaft Göttingen e.V. (Foto: A. Wittmann). Original artist: After Christian Albrecht Jensen • File:Carpet_detail.JPG Source: https://upload.wikimedia.org/wikipedia/commons/7/74/Carpet_detail.JPG License: CC BY-SA 3.0 Contributors: Own work Original artist: Mark Ahsmann • File:CodxAmiatinusFolio5rEzra.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/44/CodxAmiatinusFolio5rEzra.jpg License: Public domain Contributors: ? Original artist: ? • File:ColorValue.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/45/ColorValue.svg License: CC-BY-SA-3.0 Contrib- utors: • ColorValue.jpg Original artist: ColorValue.jpg: Mahlum • File:Color_star-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/55/Color_star-en.svg License: CC BY-SA 3.0 Con- tributors: Own work Original artist: Al2 • File:Colour_shift.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/95/Colour_shift.jpg License: CC BY-SA 4.0 Con- tributors: Own work http://www.poeticmind.co.uk/research/organising-information-colours-design-tips/ Original artist: 39james • File:Colouring_pencils.jpg Source: https://upload.wikimedia.org/wikipedia/commons/b/b1/Colouring_pencils.jpg License: CC BY- SA 3.0 Contributors: Own work Original artist: MichaelMaggs • File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: PD Contributors: ? Orig- inal artist: ? • File:Composition_with_cloud.JPG Source: https://upload.wikimedia.org/wikipedia/en/8/8c/Composition_with_cloud.JPG License: CC- BY-SA-3.0 Contributors: self-made Original artist: Niaz (Talk • Contribs) • File:Cones_SMJ2_E.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1e/Cones_SMJ2_E.svg License: CC BY-SA 3.0 Contributors: Based on Dicklyon’s PNG version, itself based on data from Stockman, MacLeod & Johnson (1993) Journal of the Opti- cal Society of America A, 10, 2491-2521d http://psy.ucsd.edu/~{}dmacleod/publications/61StockmanMacLeodJohnson1993.pdf (log E human cone response, via http://www.cvrl.org/database/text/cones/smj2.htm) Original artist: Vanessaezekowitz at en.wikipedia / Later version uploaded by BenRG. • File:Congruent_non-congruent_triangles.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/12/Congruent_non-congruent_ triangles.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Lfahlberg • File:Correct_headroom.png Source: https://upload.wikimedia.org/wikipedia/commons/1/14/Correct_headroom.png License: CC BY- SA 2.0 Contributors: • Far_too_much_headroom.png Original artist: Far_too_much_headroom.png:*Adrian_Legg_at_Legion_Arts.jpg: Mel Andringa • File:Correct_headroom_with_zoom_grid.png Source: https://upload.wikimedia.org/wikipedia/commons/f/f4/Correct_headroom_with_ zoom_grid.png License: CC BY-SA 2.0 Contributors: • Correct_headroom.png Original artist: Correct_headroom.png:*Far_too_much_headroom.png:*Adrian_Legg_at_Legion_Arts.jpg: Mel Andringa • File:Design_portal_logo.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/3b/Design_portal_logo.jpg License: CC BY- SA 2.5 Contributors: Transferred from en.wikinews to Commons. (transferred to commons by Microchip08) Original artist: Alainr345 • File:Drawing_Square_in_Perspective_1.svg Source: https://upload.wikimedia.org/wikipedia/en/5/58/Drawing_Square_in_Perspective_ 1.svg License: PD Contributors: ? Original artist: ? • File:Drawing_Square_in_Perspective_2.svg Source: https://upload.wikimedia.org/wikipedia/en/9/93/Drawing_Square_in_Perspective_ 2.svg License: PD Contributors: ? Original artist: ? • File:Dry_Etosha_Pan.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/60/Dry_Etosha_Pan.jpg License: FAL Con- tributors: Own work Original artist: Alchemist-hp (talk) (www.pse-mendelejew.de) • File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: The Tango! Desktop Project. Original artist: The people from the Tango! project. And according to the meta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (although minimally).” • File:Entrega_de_las_llaves_a_San_Pedro_(Perugino).jpg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Entrega_ de_las_llaves_a_San_Pedro_%28Perugino%29.jpg License: Public domain Contributors: See below. Original artist: Pietro Perugino • File:FWF_Samuel_Monnier_détail.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/99/FWF_Samuel_Monnier_d% C3%A9tail.jpg License: CC BY-SA 3.0 Contributors: Own work (low res file) Original artist: Samuel Monnier 11.2. IMAGES 141

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Original artist: ? • File:Reconstruction_of_the_temple_of_Jerusalem.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/de/Reconstruction_ of_the_temple_of_Jerusalem.jpg License: Public domain Contributors: ? Original artist: ? • File:Rembrandt_-_The_Raising_of_Lazarus_-_WGA19118.jpg Source: https://upload.wikimedia.org/wikipedia/commons/1/1b/Rembrandt_ -_The_Raising_of_Lazarus_-_WGA19118.jpg License: Public domain Contributors: Unknown Original artist: Rembrandt 144 CHAPTER 11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

• File:Rembrandt_The_Artist_in_his_studio_rabatment_study.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/60/ Rembrandt_The_Artist_in_his_studio_rabatment_study.jpg License: Public domain Contributors: • Rembrandt_The_Artist_in_his_studio.jpg Original artist: • derivative work: Binksternet (talk) • File:Rendered_Spectrum.png Source: https://upload.wikimedia.org/wikipedia/commons/c/c4/Rendered_Spectrum.png License: CC BY-SA 3.0 Contributors: Own work Original artist: Spigget • File:Rhombictriacontahedron.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2c/Rhombictriacontahedron.svg License: CC-BY-SA-3.0 Contributors: Vectorisation of Image:Rhombictriacontahedron.jpg Original artist: User:DTR • File:Rivertree_thirds_md.gif Source: https://upload.wikimedia.org/wikipedia/commons/c/ce/Rivertree_thirds_md.gif License: CC BY- SA 2.5 Contributors: Own work Original artist: User:Moondigger • File:Rome_Palace_of_Labor.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/a8/Rome_Palace_of_Labor.jpg License: CC0 Contributors: Transferred from en.wikipedia to Commons. 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