2. Geometry in Nature Michele Hardwick Alison Gray Beth Denis Amy Perkins

3. Floral Symmetry Flower Type: Actinomorphic ~Flowers with radial symmetry and parts arranged at one level; with definite number of parts and size www.hort.net/gallery/view/ran/anepu Anemone pulsatilla Pasque Flower Caltha introloba Marsh Marigold http://www.anbg.gov.au/stamps/stamp.983.html

4. Floral Symmetry Flower Type: Stereomorphic ~Flowers are three dimensional with basically radial symmetry; parts many o reduced, and usually regular http://www.hort.net/gallery/view/ran/aqucahttp://www.hort.net/gallery/view/amy/narif Narcissus “Ice Follies” Ice Follies Daffodil Aquilegia canadensis Wild Columbine

5. Floral Symmetry Flower Type: Haplomorphic ~Flowers with parts spirally arranged at a simple level in a semispheric or hemispheric form; petals or tepals colored; parts numerous Nymphaea spp Water Lilly www.hort.net/gallery/view/nym/nymph Magnolia x kewensis “Wada’s Memory” Wada's Memory Kew magnolia www.hort.net/gallery/view/mag/magkewm

6. Floral Symmetry Flower Type: Zygomorphic ~ Flowers with bilateral symmetry; parts usually reduced in number and irregular Cypripedium acaule Stemless lady's- slipper Pink lady's- slipper Moccasin flower http://www.hort.net/gallery/view/orc/cypac

7. Tulip : Haplomorphic Rose Garden in Washington D.C. My Backyard Smithsonian Castle in D.C. (pansies in foreground)

8. Pansy: Haplomorphic Butterfly Garden D.C. Modern Sculpture Garden D.C. Butterfly Garden D.C. (grape hyacenths in arrangment)

9. Azalea: Actinomorphic Hyacinth: Zygomorphic National Art Gallery D.C. Smithsonian Castle D.C.

10. Biography of Leonardo Fibonacci Born in Pisa, Italy Around 1770 He worked on his own Mathematical compositions. He died around 1240.

11. Fibonacci Numbers This is a brief introduction to Fibonacci and how his numbers are used in nature.

12. For Example Many Plants show Fibonacci numbers in the arrangement of leaves around their stems. The Fibonacci numbers occur when counting both the number of times we go around the stem.

13. Fibonacci Top plant can be written as a 3/5 rotation The lower plant can be written as a 5/8 rotation

14. Common trees with Fibonacci leaf arrangement

15. This is a puzzle to show why Fibonacci numbers are the solution

16. Answer Fibonacci numbers: Fibonacci series is formed by adding the latest 2 numbers to get the next one, starting from 0 and 1 0 1 0+1=1 so the series is now 0 1 1 1+1=2 so the series continues

17. Fibonacci This is just a snapshot of Fibonacci numbers and a very small introduction, if you would like more information on Fibonacci.Check out this website… www.mcs.surrey.ac.uk/personal/r.knott/

18. Why the Hexagonal Pattern? Cross cut of a bee hive shows a mathematical pattern

19. Efficiency Equillateral Triangle Area 0.048 Area of Square 0.063 Area of hexagon 0.075

20. Strength of Hive Wax Cell Wall 0.05mm thick

21. Golden Ratio

22. Golden Ratio = 1.618

23. Golden Ratio Nautilus Shell 1,2,3 Dimensional Planes

24. Golden Ratio Nautilus Shell First Dimension Linear Spiral

25. Golden Ratio Nautilus Shell Second Dimension Golden Proportional Rectangle

26. Golden Ratio Nautilus Shell

27. Third Dimension Chamber size is 1.618x larger than the previous

28. Golden Ratio Human Embryo Logarithmic Spiral

29. Golden Ratio Logarithmic Spiral Repeated Squares and Rectangles create the Logarithmic Spiral

30. Golden Ratio Spider Web Red= length of Segment Green= radii Dots= create 85 degree spiral Logarithmic Spiral & Geometric sequence

31. Golden Ratio Gazelle

32. Golden Ratio Butterflies Height Of Butterfly Is Divided By The Head Total Height Of Body Is Divided By The Border Between Thorax & Abdomen

33. Bilateral vs. Radial Symmetry Bilateral: single plane divides organism into two mirror images Radial: many planes divide organism into two mirror images

34. Golden Ratio Starfish Tentacles have ratio of 1.618

35. Five-Fold Symmetry

36. Sand-Dollar & Starfish are structured similarly to the Icosahedron.

37. Five-Fold Symmetry Design of Five-Fold Symmetry is very strong and flexible, allowing for the virus to be resilient to antibodies.

38. Phyllotaxis: phyllos = leaf taxis = order http://members.tri pod.com www.ams.orghttp://ccins.camosun.b c.ca

39. Patterns of Phyllotaxis: Whorled Pattern Spiral Pattern http://members.trip od.com

40. Whorled Pattern: 2 leaves at each node n = 2 http://members.tripod.com

41. Whorled Pattern: The number of leaves may vary in the same stem n = vary http://members.tripod.com

42. Spiral Pattern: Single phyllotaxis at each node http://members.tripod.com

43. Phyllotaxis and the Fibonacci Series: Observed in 3 spiral arrangements: Vertically Horizontally Tapered or Rounded

44. Phyllotaxis and the Fibonacci Series: Vertically http://members.tripod.com

45. Phyllotaxis and the Fibonacci Series: Horizontally http://members.tripod.com

46. Phyllotaxis and the Fibonacci Series: Tapered or Rounded www.ams.orghttp://ccins.camosun.bc.ca