Bio E 202 Aesthetics The Golden Section its

Bio. E 202: Aesthetics The Golden Section – its origin and usefulness in engineering ISU CCEE

The Fibonacci Series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … (add the last two to get the next) What is the next number? Ratio between numbers Leonardo Fibonacci c 1175 -1250. ISU CCEE

Fibonacci and plant growth Plant branches could be modeled to grow such that they can branch into two every month once they are two months old. This leads to a Fibonacci series for branch counts ISU CCEE

Fibonacci’s rabbits Rabbits could be modeled to conceive at 1 month of age and have two offspring every month thereafter. This leads to a Fibonacci series for rabbit counts for each subsequent month ISU CCEE

Petals on flowers 3 petals (or sepals) : lily, iris Lilies often have 6 petals formed from two sets of 3 4 petals Very few plants show 4 e. g. fuchsia 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia), orchid 8 petals: delphiniums ISU CCEE

Petals on flowers 13 petals: ragwort, corn marigold, cineraria, some daisies 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, Asteraceae family ISU CCEE

Ratio of Fibonacci numbers Divide each number by the number before it, we will find the following series of numbers: 1/ 2 3 5 8 13 21 1 = 1, /1 = 2, /2 = 1· 5, /3 = 1· 666. . . , /5 = 1· 6, /8 = 1· 625, /13 = 1· 61538. . . These values converge to a constant value, 1. 61803 39887……, the golden section, Dividing a number by the number behind it: 0· 61803 39887. . . 1/ ISU CCEE

The golden section in geometry l l l ISU CCEE The occurrence of the ratio, The meaning of the ratio The use of in engineering

Constructing the golden section ISU CCEE

Geometric ratios involving : Pentagon ISU CCEE

Geometric ratios involving : Decagon ISU CCEE

Golden Spiral Construction Start with a golden rectangle Construct a square inside Construct squares in the remaining rectangles in a rotational sequence Construct a spiral through the corners of the squares ISU CCEE

Golden Spiral Shortcut http: //powerretouche. com/Divine_ proportion_tutorial. htm ISU CCEE

Golden Triangle and Spirals 1 1 1/ 1 ISU CCEE

Golden proportions in humans ISU CCEE

Echinacea – the Midwest Coneflower Note the spirals originating from the center. These can be seen moving out both clockwise and anti-clockwise. These spirals are no mirror images and have different curvatures. These can be shown to be square spirals based on series of golden rectangle constructions. ISU CCEE

Cauliflower and Romanesque (or Romanesco) Brocolli. XCauliflower Note the spiral formation in the florets as well as in the total layout ISU CCEE The spirals are, once again, golden section based

Pine cone spiral arrangements The arrangement here can once more be shown to be spirals based on golden section ratios. ISU CCEE

Pine cone spirals ISU CCEE

Fibonacci Rectangles and Shell Spirals ISU CCEE

Construction: Brick patterns The number of patterns possible in brickwork Increases in a Fibonacci series as the width increases ISU CCEE

Phi in Ancient Architecture A number of lengths can be shown to be related in ratio to each other by Phi ISU CCEE

Golden Ratio in Architecture The Dome of St. Paul, London. ISU CCEE Windsor Castle

Golden Ratio in Architecture Baghdad City Gate ISU CCEE The Great Wall of China

Modern Architecture: Eden project The Eden Project's new Education Building ISU CCEE

Modern Architecture: California Polytechnic Engineering Plaza ISU CCEE

More examples of golden sections ISU CCEE

Mathematical Relationships for Phi The Number Phive 50. 5 x. 5 +. 5 = 1. 61803399 = phive to the power of point phive times point phive plus point phive = phi 1. 61803399 2 = 2. 61803399 = phi +1 1 / 1. 61803399 (the reciprocal) = 0. 61803399 = phi - 1 . 618033992 +. 61803399 = 1 ISU CCEE