Part II THE CONCEPT OF AESTHETICS

1. Part IITHE CONCEPTOF AESTHETICS

2. NATURE OF AESTHETICS The Ancient Greeks The ancient Greeks are known for their pioneering work in Geometry. Constructing rectangles based on geometric projections outside of a square, the Greeks defined what today we call the Golden Rectangle or the Golden Section .

3. NATURE OF AESTHETICS The Golden Rectangle The ancient Greeks considered the golden rectangle to be “beautiful” and applied it to their art and architecture. The Parthenon, illustrated here, represents the ultimate achievement of the classic Greek – and subsequently Western – idea of beauty. Its shape is a golden rectangle.

4. NATURE OF AESTHETICS The Golden Rectangle A law established by the ancient architect, Vitruvius, states: "For a space divided into two parts to be agreeable and aesthetic, between the smallest and largest parts there must exist the same relationship as between the larger part and the whole space.“Vitruvius was describing thegolden rectangle. Construction of a Golden Rectangle: a: Draw a square and find the midpoint of the base. b. Draw an arc with a radius that extends from the midpoint to the corner. Extend the base of the square until it meets the arc. c: The aspect ratio of the resulting rectangle is 0.6180339 to 1, the golden ratio. A golden section can be found where the side of the original square crosses a diagonal. a b c arc Golden Section midpoint

5. √5 – 1 2 Φ = 1 Φ 1 Φ2 = 1 + Φ = 2 + Φ NATURE OF AESTHETICS NAURE Phi, The Golden Ratio, Φ Phi (pronounced “fee,” symbol: Φ) is calculated as: = 0.61803398874989484820458683 436563811772030917980576 … Phi has natural symmetry and the aesthetic beauty of Phi can be seen in these curious mathematical relationships: Φ2 = 1 - Φ

6. NATURE OF AESTHETICS The Italians In a book completed in 1202, an Italian mathematician by the name of Leonardo Fibonacci stumbled upon the golden ratio when he explored a sequence of numbers first mentioned in 1150 by Indian mathematicians. Thereafter called Fibonacci Numbers, each value in the sequence is derived by adding together the two previous values. As the numbers progress, Fibonacci found that the ratio of any number to its larger neighbor closely approaches the golden ratio: 0.6180339. Leonardo Fibonacci 10946 17711 = 0.6180339 Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 …

7. NATURE OF AESTHETICS Fibonacci Numbers Much later, botanists were astonished to discover Fibonacci numbers appearing throughout nature. It seemed as if Fibonacci Numbers were Nature’s Formula. Spirals seen in the arrangement of seeds in the head of this sunflower number 34 in a counterclockwise direction and 55 in a clockwise direction. 34 and 55 are the ninth and tenth Fibonacci numbers respectively. The flower itself has 34 petals. Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 …

8. NATURE OF AESTHETICS Fibonacci Numbers The seed pods form 3 spirals clockwise and 5 counterclockwise in the pine cone of the Giant Sequoia. As the Nautilus grows, its shell structure describes a mathematical curve called a logarithmic spiral, whose radius at each complete turn approximates the "Fibonacci Number" series, or the so-called "golden ratio." This Black-eyed Susan has 21 petals. Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 …

9. NATURE OF AESTHETICS Fibonacci Numbers Number of Petals: 3 petals (or 2 sets of 3) 5 petals 8 petals 13 petals 21 petals 34 petals 55 petals Flower Lily, Iris buttercup, wild rose, larkspur, columbine (aquilegia), vinca delphinium, coreopsis ragwort, marigold, cineraria aster, black-eyed susan, chicory plantain, daisy, pyrethrum, sunflower daisy, the asteraceae family Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 …

10. NATURE OF AESTHETICS The Golden Ratio In Art Leonardo da Vinci used the Golden Ratio in much of his artwork. In what is perhaps his most famous work, The Mona Lisa (seen here with golden section lines drawn in red), da Vinci drew diagonal lines from the paintings corners to its golden section points along the edge. Notice how the subject’s limbs tend to follow these lines and how key stress points (anatomical joints and features) appear at intersections (highlighted by circles). The Mona Lisa 1503-5, Leonardo da Vinci, Oil on Panel

11. NATURE OF AESTHETICS The Golden Ratio In Art The “Rule of Thirds” is a simplification of the Golden Section. In Madonna And Child With St. John The Baptist, Jacopo Bassano placed key focal points (the eyes) along diagonals and their intersections. Renaissance artists and those to follow employed the golden section to stress what they felt were important elements of their work. Some even used it to relay hidden meaning. Madonna And Child With St. John The Baptist 1570, Jacopo Bassano, Oil on Canvas

12. NATURE OF AESTHETICS Summary:THE CONCEPT OF AESTHETICS The ancient Greeks thought the golden rectangle and its corresponding golden section to be the most aesthetically pleasing shape. In a natural series of numbers described by Leonardo Fibonacci in 1202, each value is a golden ratio of its next larger neighbor. Fibonacci numbers and hence, the golden ratio, are found everywhere in the structure of living things. Artists have used the golden ratio in their work to align key elements of their design.

13. In Part I we saw how the brain has evolved neural processes which enhance the perception of objects in the visual field. Honed over hundreds of thousands of years, these processes have helped to ensure our very survival. In Part II we visited the concept of aesthetics and its relationship to the golden section and Phi, a number intimately entwined with nature. Please view Part III