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BioE 202: Aesthetics

BioE 202: Aesthetics. The Golden Section – its origin and usefulness in engineering. The Fibonacci Series. 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 c1175-1250.

robert-butler
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BioE 202: Aesthetics

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  1. BioE 202: Aesthetics The Golden Section – its origin and usefulness in engineering

  2. The Fibonacci Series 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 c1175-1250.

  3. 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

  4. 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

  5. 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

  6. 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

  7. Ratio of Fibonacci numbers Divide each number by the number before it, we will find the following series of numbers: 1/1 = 1,   2/1 = 2,   3/2 = 1·5,   5/3 = 1·666...,   8/5 = 1·6,   13/8 = 1·625,   21/13 = 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/

  8. The golden section in geometry • The occurrence of the ratio,  • The meaning of the ratio  • The use of  in engineering

  9. Constructing the golden section

  10. Geometric ratios involving : Pentagon

  11. Geometric ratios involving : Decagon

  12. 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

  13. Golden Spiral Shortcut http://powerretouche.com/Divine_proportion_tutorial.htm

  14. Golden Triangle and Spirals 1  1 1/ 1

  15. Golden proportions in humans

  16. 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.

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

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

  19. Pine cone spirals

  20. Fibonacci Rectangles and Shell Spirals

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

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

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

  24. Golden Ratio in Architecture Baghdad City Gate The Great Wall of China

  25. Modern Architecture: Eden project The Eden Project's new Education Building

  26. Modern Architecture: California Polytechnic Engineering Plaza

  27. More examples of golden sections

  28. Mathematical Relationships for Phi The Number Phive  50.5 x .5 + .5 = 1.61803399  = phi 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

  29. Golden Ratio in the ArtsAztec Ornament

  30. Golden Ratio in the Arts

  31. Golden Ratio in the Arts Piet Mondrian’s Rectangles

  32. Design Applications of Phi

  33. Design Applications of Phi

  34. Phi in Design

  35. Phi in Advertising

  36. Three-dimensional symmetry:the Platonic solids

  37. Octahedron 8-sided solid

  38. Dodecahedron This 12-sided regular solid is the 4th Platonian figure

  39. Icosahedron 20-sided solid Note the three mutually orthogonal golden rectangles that could be constructed 

  40. Three-dimensional near-symmetry http://www.mathconsult.ch/showroom/unipoly/list-graph.html

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