Geometry in nature
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Geometry in Nature. Michele Hardwick Alison Gray Beth Denis Amy Perkins. Floral Symmetry Flower Type: Actinomorphic. ~Flowers with radial symmetry and parts arranged at one level; with definite number of parts and size. Anemone pulsatilla Pasque Flower.

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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


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

Logarithmic Spiral &

Geometric sequence

Red= length of Segment

Green= radii

Dots= create 85 degree spiral


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

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


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