Geometry in Nature Michele Hardwick Alison Gray Beth Denis Amy Perkins

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

www.hort.net/gallery/view/ran/anepu

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

http://www.hort.net/gallery/view/ran/aqucahttp://www.hort.net/gallery/view/amy/narif

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


First Dimension

Linear Spiral


Second Dimension

Golden Proportional Rectangle



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

http://members.tripod.co

m

www.ams.orghttp://ccins.camosun.bc.ca

Patterns of Phyllotaxis: Whorled Pattern Spiral Pattern

http://members.tripod.co

m

http://members.tripod.com

Whorled Pattern:

2 leaves at each node n = 2


Whorled Pattern:

The number of leaves may vary in the same stem

n = vary


Spiral Pattern:

Single phyllotaxis at each node


Phyllotaxis and the Fibonacci Series:

Observed in 3 spiral arrangements:

Vertically

Horizontally

Tapered or Rounded


Vertically



Horizontally



Tapered or Rounded

www.ams.org http://ccins.camosun.bc.ca

Documents

Geometry in Nature Michele Hardwick Alison Gray Beth Denis Amy Perkins