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## Geometry in Nature

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**Geometry in Nature**Michele Hardwick Alison Gray Beth Denis Amy Perkins**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**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**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**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**Tulip : Haplomorphic**Rose Garden in Washington D.C. Smithsonian Castle in D.C. (pansies in foreground) My Backyard**Pansy: Haplomorphic**Butterfly Garden D.C. (grape hyacenths in arrangment) Modern Sculpture Garden D.C. Butterfly Garden D.C.**Azalea: Actinomorphic**National Art Gallery D.C. Smithsonian Castle D.C. Hyacinth: Zygomorphic**Biography of Leonardo Fibonacci**• Born in Pisa, Italy Around 1770 He worked on his own Mathematical compositions. He died around 1240.**Fibonacci Numbers**• This is a brief introduction to Fibonacci and how his numbers are used in nature.**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.**Fibonacci**• Top plant can be written as a 3/5 rotation • The lower plant can be written as a 5/8 rotation**This is a puzzle to show why Fibonacci numbers are the**solution**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**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/**Why the Hexagonal Pattern?**Cross cut of a bee hive shows a mathematical pattern**Efficiency**Equillateral Triangle Area 0.048 Area of Square 0.063 Area of hexagon 0.075**Strength of Hive**Wax Cell Wall 0.05mm thick**Golden Ratio Nautilus Shell**1,2,3 Dimensional Planes**Golden Ratio Nautilus Shell**First Dimension Linear Spiral**Golden Ratio Nautilus Shell**Second Dimension Golden Proportional Rectangle**Golden Ratio Nautilus Shell**Third Dimension Chamber size is 1.618x larger than the previous**Golden Ratio Human Embryo**Logarithmic Spiral**Golden Ratio Logarithmic Spiral**Repeated Squares and Rectangles create the Logarithmic Spiral**Golden Ratio Spider Web**Logarithmic Spiral & Geometric sequence Red= length of Segment Green= radii Dots= create 85 degree spiral**Golden Ratio Butterflies**Height Of Butterfly Is Divided By The Head Total Height Of Body Is Divided By The Border Between Thorax & Abdomen**Bilateral vs. Radial Symmetry**Bilateral: single plane divides organism into two mirror images Radial: many planes divide organism into two mirror images**Golden Ratio Starfish**Tentacles have ratio of 1.618**Five-Fold Symmetry**Sand-Dollar & Starfish are structured similarly to the Icosahedron.**Five-Fold Symmetry**Design of Five-Fold Symmetry is very strong and flexible, allowing for the virus to be resilient to antibodies.**Phyllotaxis:**phyllos = leaf taxis = order http://ccins.camosun.bc.ca www.ams.org http://members.tripod.com**Whorled Pattern**Spiral Pattern Patterns of Phyllotaxis: http://members.tripod.com http://members.tripod.com**Whorled Pattern:**• 2 leaves at each node • n = 2 http://members.tripod.com**Whorled Pattern:**• The number of leaves may vary in the same stem • n = vary http://members.tripod.com**Spiral Pattern:**Single phyllotaxis at each node http://members.tripod.com**Phyllotaxis and the Fibonacci Series:**Observed in 3 spiral arrangements: Vertically Horizontally Tapered or Rounded**Phyllotaxis and the Fibonacci Series:**Vertically http://members.tripod.com**Phyllotaxis and the Fibonacci Series:**Horizontally http://members.tripod.com**Phyllotaxis and the Fibonacci Series:**Tapered or Rounded www.ams.org http://ccins.camosun.bc.ca