VIEWPOINTS Mathematical Perspective and Fractal Geometry in Art

VIEWPOINTS Mathematical Perspective and Fractal Geometry in Art Marc Frantz Indiana University Annalisa Crannell Franklin & Marshall College Draft, May 19, 2007 Contents Preface v 1 Introduction to Perspective and Space Coordinates 1 Artist Vignette: Sherry Stone Clifton 9 2 Perspective by the Numbers 15 Artist Vignette: Peter Galante 26 3 Vanishing Points and Viewpoints 33 Artist Vignette: Jim Rose 43 4 Rectangles in One-Point Perspective 51 What’s My Line: A Perspective Game 62 5 Two-Point Perspective 67 Artist Vignette: Robert Bosch 85 6 Three-Point Perspective and Beyond 93 Artist Vignette: Dick Termes 116 7 Anamorphic Art 123 Artist Vignette: Teri Wagner 128 8 Fractal Geometry 135 Artist Vignette: Kerry Mitchell 150 iii Preface Origin of the Text Viewpoints is an undergraduate text in mathematics and art suit- able for math-for-liberal-arts courses, mathematics courses for fine art majors, and introductory art classes. Instructors in such courses at more than 25 institutions have already used an earlier online version of the text, called Lessons in Mathematics and Art. The material in these texts evolved from courses in mathematics and art which we developed in collaboration and taught at our respective institutions. In addition, this material has been tested at, and influenced by, a series of week-long Viewpoints faculty development workshops (see php.indiana.edu/˜mathart/viewpoints). The initial course development and the first Viewpoints workshops were supported by the Indiana University Mathematics Through- out the Curriculum project, the Indiana University Strategic Direc- tions Initiative, Franklin & Marshall College, and the National Sci- ence Foundation (NSF-DUE 9555408). The 2001 workshop was also supported by the Professional Enhancement Program of the Math- ematical Association of America. We are continuing to expand and revise the text as part of a new collaboration between our two institutions, supported by an NSF Educational Materials Development grant (NSF-DUE 0439891). This new project will continue the View- points workshops and result in an expanded text and an instructor’s manual to be published by Princeton University Press. Pedagogical Approach The approach of Viewpoints is highly activity-based, much like the approach of teaching in art school. As many of our workshop graduates will attest, the true value of the material can only be fully appreciated by engaging in these activities, and not by merely read- ing the book. We have endeavored to include problems and activities of genuine interest and value to art students—problems that go sig- v vi Preface nificantly beyond what students normally learn in art school. In our experience the authenticity of the problems makes them genuinely interesting, not only to art majors but to students from a broad range of disciplines. We have also endeavored to make sure that the problems are real math problems. Happily, there is a wealth of problems at the bound- ary of mathematics and art having a number excellent pedagogical properties: (1) the problems are natural and easily understood; (2) the problems have multiple solutions of varying difficulty and ap- plicability; (3) the problems admit multiple proofs, both geometric and algebraic; (4) once arrived at, the solutions are easy to remem- ber and rewarding to use; (5) the search for solutions captures the essence of mathematical research and discovery. We have sought to make the problems in this book embody each of these characteristics, at a level that is accessible to every undergraduate student. Special Features As mentioned above, we are developing this text and a separate instructor’s manual for eventual publication by Princeton University Press. Although the manuscript is not near final form for publication, we are already formatting the current version to approximate the look of the finished book. The outline around each page is to help us visualize the planned trim size of 8 × 10 inches. More importantly, we have already started to include a unique fea- ture of the text which we call Artist Vignettes. During the course of our project we have been fortunate to meet a number of professional artists willing to contribute to the book. Each Vignette contains a short biography, an artist’s statement, and some images of the artist’s work. Color images of the artists’ work will also appear in the color plate section. We feel that the input of practicing artists informed about mathematics will be an exciting and attractive addition to the text. Marc Frantz Indiana University Annalisa Crannell Franklin & Marshall College CHAPTER 1 Introduction to Perspective and Space Coordinates ur first perspective activity involves using masking tape O to make a perspective picture of a building on a window (Fig- ure 1.1). It’s tricky! One person (the Art Director) must stand rooted to the spot, with one eye closed. Using the one open eye, the Art Director directs one or more people (the Artists), telling them where to place masking tape in order to outline architectural features as seen from the Director’s unique viewpoint. In Figure 1.1, this process resulted in a simple but fairly respectable perspective drawing of the University Library at Indiana University Purdue Uni- versity Indianapolis. Figure 1.1. Making a masking tape drawing on a window. If no windows with views of architecture are available, then a portable “window” made of plexiglass will do just as well. In Fig- ure 1.2, workshop participants at the Indianapolis Museum of Art are making masking tape pictures of interior architectural details in a hallway. 1 2 Chapter 1 Figure 1.2. Plexiglas will do the job indoors. Finally, if a sheet of plexiglass is not available, the window of a Figure 1.3. Using a display case. display case will also work. In this case, the Art Director directs the Artists in making a picture of the interior of the case (Figure 1.3). If the masking tape picture from Figure 1.1 is put in digital form (either by photographing and scanning, or by photographing with a digital camera) it can be drawn on in a computer program, and some interesting patterns emerge. (Figure 1.4). V2 V1 Figure 1.4. Analyzing the library drawing. Architectural lines that are parallel to one another, but not parallel to the plane* of the window, have images that converge to a *A line (extended infinitely in both di- common point called a vanishing point. Two such vanishing points, rections) is parallel to a plane if the V1 and V2, are indicated in Figure 1.4. The correct use of vanishing line does not intersect the plane. points and other geometric devices can greatly enhance not only one’s ability to draw realistically, but also one’s ability to appreciate and enjoy art. To properly understand such things, we need a geometric interpretation of our perspective experiment (Figure 1.5). As you can see from Figure 1.5, we’re going to be using some mathematical objects called points, planes, and lines. To begin describing these objects, let’s start with points. Introduction to Perspective and Space Coordinates 3 path of light ray (a line) eye of viewer Figure 1.5. Mathematical description (a point) image of the window-taping experiment. object window (picture plane) It’s assumed that you’re familiar with the idea of locating points in a plane using the standard xy-coordinate system. To locate points in 3-dimensional space (3-space), we need to introduce a third coordinate called a z-coordinate. The standard arrangement of the xyz-coordinate axes looks like Figure 1.6; the positive x-axis points towards you. For a point P (x,y,z) in 3-space, we can think of the x, y, and z- coordinates as “out,” “over,” and “up,” respectively. For instance, in Figure 1.6, the point P (4, 5, 6) can be located by starting at the origin (0, 0, 0) and going out towards you 4 units along the x-axis (you’d go back if the x-coordinate were negative), then over 5 units to the right (you’d go to the left if the y-coordinate were negative), and finally 6 units up (you’d go down if the z-coordinate were negative). z x y z 6 P(4,5,6) the origin 6 units up (0,0,0) Figure 1.6. The standard y xyz-coordinate system. 4 units out 5 4 5 units over x We took a look at the standard xyz-system in Figure 1.6 simply because it is the standard system, and you may see it again in another course. However, it will be convenient for our purposes to use the 4 Chapter 1 y slightly different xyz-coordinate system in Figure 1.7—it’s the one we’ll be using from now on. In Figure 1.7 we have included sketches (1,3,7) of three special planes called the coordinate planes. In this case, we have to think of the x, y, and z-coordinates as “out,” “up,” and (4,2,3) z “over,” respectively, as indicated in the figure. y x The xz-plane 5 x y z (contains the x- and z-axes). P(4,5,6) Also called "the plane y=0" 6 units over What are the missing vertex coordi- because all y-coordinates nates of this block whose faces are par- on this plane are 0. 5 units up z 6 allel to the coordinate planes? 4 units out 4 x The yz-plane (contains the y- and z-axes). Also called "the plane x=0" The xy-plane because all x-coordinates (contains the x- and y-axes).

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