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Geometry in Nature

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

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

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

  4. Floral SymmetryFlower 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 Magnolia x kewensis “Wada’s Memory” Wada's Memory Kew magnolia  www.hort.net/gallery/view/nym/nymph www.hort.net/gallery/view/mag/magkewm

  5. Floral SymmetryFlower 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

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

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

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

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

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

  11. 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.

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

  13. Common trees with Fibonacci leaf arrangement

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

  15. 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

  16. 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/

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

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

  19. Strength of Hive Wax Cell Wall 0.05mm thick

  20. Golden Ratio

  21. Golden Ratio = 1.618

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

  23. Golden Ratio Nautilus Shell First Dimension Linear Spiral

  24. Golden Ratio Nautilus Shell Second Dimension Golden Proportional Rectangle

  25. Golden Ratio Nautilus Shell

  26. Golden Ratio Nautilus Shell Third Dimension Chamber size is 1.618x larger than the previous

  27. Golden Ratio Human Embryo Logarithmic Spiral

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

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

  30. Golden Ratio Gazelle

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

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

  33. Golden Ratio Starfish Tentacles have ratio of 1.618

  34. Five-Fold Symmetry

  35. Five-Fold Symmetry Sand-Dollar & Starfish are structured similarly to the Icosahedron.

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

  37. Phyllotaxis: phyllos = leaf taxis = order http://ccins.camosun.bc.ca www.ams.org http://members.tripod.com

  38. Whorled Pattern Spiral Pattern Patterns of Phyllotaxis: http://members.tripod.com http://members.tripod.com

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

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

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

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

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

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

  45. Phyllotaxis and the Fibonacci Series: Tapered or Rounded www.ams.org http://ccins.camosun.bc.ca