geometry in nature michele hardwick alison gray beth denis amy perkins
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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
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Anemone pulsatilla
Pasque Flower Caltha introloba
Marsh Marigold
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
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Narcissus “Ice Follies”
Ice Follies Daffodil
Aquilegia canadensis
Wild Columbine
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
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
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 : HaplomorphicRose Garden in
Washington D.C.
My Backyard
Smithsonian
Castle in D.C.
(pansies in
foreground)
Pansy: Haplomorphic
Butterfly Garden D.C.
Modern Sculpture Garden
D.C.
Butterfly Garden D.C. (grape
hyacenths in arrangment)
Azalea: Actinomorphic
Hyacinth: Zygomorphic
National Art Gallery D.C.
Smithsonian Castle D.C.
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
Common trees with Fibonacci leaf arrangement
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
Golden Ratio = 1.618
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
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
Red= length of Segment
Green= radii
Dots= create 85 degree spiral
Logarithmic Spiral &Geometric sequence
Golden Ratio Gazelle
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
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
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m
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Patterns of Phyllotaxis: Whorled Pattern Spiral Pattern
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Whorled Pattern:
2 leaves at each node n = 2
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Whorled Pattern:
The number of leaves may vary in the same stem
n = vary
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Spiral Pattern:
Single phyllotaxis at each node
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Phyllotaxis and the Fibonacci Series:
Observed in 3 spiral arrangements:
Vertically
Horizontally
Tapered or Rounded
Phyllotaxis and the Fibonacci Series:
Vertically
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Phyllotaxis and the Fibonacci Series:
Horizontally
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Phyllotaxis and the Fibonacci Series:
Tapered or Rounded
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