Download Biography of Leonardo Fibonacci

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
Caltha
introloba
Marsh
Marigold
www.hort.net/gallery/view/ran/anepu
http://www.anbg.gov.au/stamps/stamp.983.html
Floral Symmetry
Flower Type: Stereomorphic
~Flowers are three dimensional with basically radial
symmetry; parts many o reduced, and usually regular
Narcissus
“Ice Follies”
Aquilegia
canadensis
Ice Follies
Daffodil
Wild
Columbine
http://www.hort.net/gallery/view/amy/narif
http://www.hort.net/gallery/view/ran/aquca
Floral Symmetry
Flower 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 Symmetry
Flower Type: Zygomorphic
~ Flowers with bilateral symmetry; parts usually
reduced in number and irregular
Cypripedium
acaule
Stemless lady'sslipper
Pink lady'sslipper
Moccasin flower
http://www.hort.net/gallery/view/orc/cypac
Pansy: Haplomorphic
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.
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Fibonacci Numbers
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This is a brief introduction to Fibonacci
and how his numbers are used in nature.
For Example
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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
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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
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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
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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
Height Of Butterfly Is
Divided By The Head
Total Height Of Body Is
Divided By The Border
Between Thorax &
Abdomen
Butterflies
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.b
c.ca
www.ams.org
http://members.tri
pod.com
Patterns of Phyllotaxis:
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Whorled Pattern
http://members.trip
od.com

Spiral Pattern
http://members.tripo
d.com
Whorled Pattern:
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http://members.tripod.com
2 leaves at each node
n=2
Whorled Pattern:
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http://members.tripod.com
The number of leaves
may vary in the same
stem
n = vary
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