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

2. ISU CCEE The Fibonacci Series Fibonacci Series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,… (add the last two to get the next)add the last two to get the next What is the next number? Ratio between numbers Leonardo Fibonacci c1175-1250.

3. 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

4. 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

5. 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

6. 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

7. ISU CCEE 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... Ratio of Fibonacci numbers 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. ISU CCEE The golden section in geometry l The occurrence of the ratio,  l The meaning of the ratio  l The use of  in engineering

9. ISU CCEE Constructing the golden section

10. ISU CCEE Geometric ratios involving  : Pentagon

11. ISU CCEE Geometric ratios involving  : Decagon

12. 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

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

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

15. ISU CCEE Golden proportions in humans

16. 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.

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

19. ISU CCEE Pine cone spirals

20. ISU CCEE Fibonacci Rectangles and Shell Spirals

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

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

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

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

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

26. ISU CCEE Modern Architecture: California Polytechnic Engineering Plaza

27. ISU CCEE More examples of golden sections

28. ISU CCEE Mathematical Relationships for Phi The Number Phive 5 0.5 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 - 1.61803399 2 +.61803399 = 1

29. ISU CCEE

30. Golden Ratio in the Arts Golden Ratio in the Arts Aztec Ornament

31. ISU CCEE Golden Ratio in the Arts

32. ISU CCEE Golden Ratio in the Arts Piet Mondrian’s Rectangles

33. ISU CCEE Design Applications of Phi

34. ISU CCEE Design Applications of Phi

35. ISU CCEE Phi in Design

36. ISU CCEE Phi in Advertising

37. ISU CCEE Three-dimensional symmetry: the Platonic solids

38. ISU CCEE Octahedron 8-sided solid

39. ISU CCEE DodecahedronDodecahedron This 12-sided regular solid is the 4 th Platonian figure

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

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