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
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