THE RENAISSANCE: MATHS AND ART
The new ideal of beauty that is asserted in the Renaissance is based on the harmonious and proportionate harmony of forms that is inspired by classical art. The aspects on which Renaissance art is founded can be summarized in 2 mathematical concepts
PERSPECTIVE PROPORTION Set of From the Latin pro-portione, projections on a two- “according to the portion” and indicates the correspondence dimensional plane (i.e of measures between two or more characterized by two parts in close relation between dimensions, length and them height) of three- It is the ratio (result of division) dimensional objects between two quantities In addition, Renaissance artists use regular geometric shapes such as the square and other regular polygons and the circle, often repeated in modules
THE PERSPECTIVE
For a realistic representation of reality, practical rules were developed, such as the technique of vision at a glimpse, the apparent convergence of parallel lines, the sizing of the figures on the basis of distance, etc.
CANON AND HARMONIC PROPORTIONS
The Greeks had already studied in depth the mathematical relationships in nature, discovering them in music, and applying them to the construction of buildings and art.
Unìsono (1:1) Diapason (1:2) 1 1 2 1
Diapènte (2:3) 2 Diatèssaron (3:4) 3
3 4
At the base of all was the module, which is an element established as a reference measure for proportioning. The proportions derived from repetition of the form (multiples) or from its division (submultiples) and were expressed by ratios of integer or fractional numbers. THE SET OF ALL THESE MATHEMATICAL RELATIONSHIPS OF THE WORK TAKE THE NAME OF CANON THE "DIVINE" RATIO: THE GOLDEN SECTION
In their search for the harmonious proportion of nature, the ancients came across a very special relationship, because it generates a number that the ancients could not explain and that today we know by the name of ϕ = 1,618033988……. Why the dots? Because this number is IRRATIONAL, that is, its decimal part never ends (the division always generates a change, to infinity !!!) The Greeks obtained this proportion through geometric constructions that incredibly generated figures and compositions of unexpected harmony and beauty. In the Renaissance ϕ takes the name of "divine proportion" Today for us it is the "golden section"
THE GOLDEN SECTION IS THE DIVISION OF A SEGMENT IN TWO PARTS THAT THE LARGEST PART IS THE PROPORTIONAL MEDIUM BETWEEN THE ENTIRE SEGMENT AND THE LESS PART
A P B
AB : AP = AP : PB
φ = AB / AP = 1.618 ... THE GOLDEN NUMBER
THE GOLDEN RECTANGLE The golden rectangle is a rectangle whose proportions are based on the golden proportion. This means that the ratio between the major and minor side, a: b, is identical to that between the minor side and the segment obtained by subtracting the latter from the major side b: (a - b)
a : b = b : (a – b)
both ratios are φ ≅ 1.618
THE GOLDEN SPIRAL
You can continue till infinite to build golden rectangle inside the first one
In the spiral obtained from the golden rectangle, the ratio between two consecutive rays is equal to φ
THE REGULAR PENTAGON
The golden ratio is found in the construction of the regular pentagon; it is, among the regular geometric figures, the richest in relationships with the natural world. In fact, it contains, in the internal structure, the same principles found in the golden progression. If you analyze the structure inside the pentagon you will discover a large number of figures all referable to the golden ratios because the side and the diagonal of this figure are two segments in golden ratio. The pentagon can be built with ruler and compass using ф. GOLDEN TRIANGLE DB : AB = ф
DONALD DUCK AND THE GOLDEN SECTION THE GOLDEN SPIRAL, THE SEQUENCE OF FIBONACCI AND THE HARMONY OF NATURE
The Fibonacci sequence (from Leonardo di Pisa, filius Bonacci - 1175 - c.1250) is a numerical sequence in which each term is the sum of the previous two
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,……..
As you proceed in succession ....
The relationship between a number and its previous one gets closer and closer to φ !!!!
BUT ... DO NATURE KNOW MATHEMATICS?
GOLDEN SECTION AND HUMAN BODY
● Height and navel-to floor distance
● Face height - face width
● Navel-knee and knee-to-floor distance
● Chin-eye distance and eye-to-forehead distance
● Mouth-eye distance and mouth-chin distance
● Nose length and nose width
● Elbow-to-toe distance and elbow-to-wrist distance
● Arm length and elbow-hand distance
● Shoulder-navel distance and shoulder-to-forehead distance