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

Geometry in Nature

Michele Hardwick

Alison Gray

Beth Denis

Amy Perkins

Floral symmetry flower type actinomorphic

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

Floral symmetry flower type stereomorphic

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

Floral symmetry flower type haplomorphic

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

Floral symmetry flower type zygomorphic

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

Geometry in nature

Tulip : Haplomorphic

Rose Garden in Washington D.C.

Smithsonian Castle in D.C. (pansies in foreground)

My Backyard

Geometry in nature

Pansy: Haplomorphic

Butterfly Garden D.C. (grape hyacenths in arrangment)

Modern Sculpture Garden D.C.

Butterfly Garden D.C.

Geometry in nature

Azalea: Actinomorphic

National Art Gallery D.C.

Smithsonian Castle D.C.

Hyacinth: Zygomorphic

Biography of leonardo fibonacci

Biography of Leonardo Fibonacci

  • Born in Pisa, Italy

    Around 1770

    He worked on his own

    Mathematical compositions.

    He died around 1240.

Fibonacci numbers

Fibonacci Numbers

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

For example

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.



  • 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

Common trees with Fibonacci leaf arrangement

This is a puzzle to show why fibonacci numbers are the solution

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



  • 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



  • 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…


Why the hexagonal pattern

Why the Hexagonal Pattern?

Cross cut of a bee hive shows a mathematical pattern



Equillateral Triangle Area


Area of Square


Area of hexagon


Strength of hive

Strength of Hive

Wax Cell Wall

0.05mm thick

Golden ratio

Golden Ratio

Golden ratio 1 618

Golden Ratio = 1.618

Golden ratio nautilus shell

Golden Ratio Nautilus Shell

1,2,3 Dimensional Planes

Golden ratio nautilus shell1

Golden Ratio Nautilus Shell

First Dimension

Linear Spiral

Golden ratio nautilus shell2

Golden Ratio Nautilus Shell

Second Dimension

Golden Proportional Rectangle

Golden ratio nautilus shell3

Golden Ratio Nautilus Shell

Golden ratio nautilus shell4

Golden Ratio Nautilus Shell

Third Dimension

Chamber size is 1.618x larger than the previous

Golden ratio human embryo

Golden Ratio Human Embryo

Logarithmic Spiral

Golden ratio logarithmic spiral

Golden Ratio Logarithmic Spiral

Repeated Squares and Rectangles create the Logarithmic Spiral

Golden ratio spider web

Golden Ratio Spider Web

Logarithmic Spiral &

Geometric sequence

Red= length of Segment

Green= radii

Dots= create 85 degree spiral

Golden ratio gazelle

Golden Ratio Gazelle

Golden ratio butterflies

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 vs. Radial Symmetry

Bilateral: single plane divides organism into two mirror images

Radial: many planes divide organism into two mirror images

Golden ratio starfish

Golden Ratio Starfish

Tentacles have ratio of 1.618

Five fold symmetry

Five-Fold Symmetry

Five fold symmetry1

Five-Fold Symmetry

Sand-Dollar & Starfish are structured similarly to the Icosahedron.

Five fold symmetry2

Five-Fold Symmetry

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



phyllos = leaf

taxis = order

Patterns of phyllotaxis

Whorled Pattern

Spiral Pattern

Patterns of Phyllotaxis:

Whorled pattern

Whorled Pattern:

  • 2 leaves at each node

  • n = 2

Whorled pattern1

Whorled Pattern:

  • The number of leaves may vary in the same stem

  • n = vary

Spiral pattern

Spiral Pattern:

Single phyllotaxis at each node

Phyllotaxis and the fibonacci series

Phyllotaxis and the Fibonacci Series:

Observed in 3 spiral arrangements:



Tapered or Rounded

Phyllotaxis and the fibonacci series1

Phyllotaxis and the Fibonacci Series:


Phyllotaxis and the fibonacci series2

Phyllotaxis and the Fibonacci Series:


Phyllotaxis and the fibonacci series3

Phyllotaxis and the Fibonacci Series:

Tapered or Rounded

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