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Golden Ratio Project

Geometry project by Laura, Shannon, and Adam
by

Laura W

on 28 May 2010

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Transcript of Golden Ratio Project

The Golden Ratio:
by Laura,Shannon,and Adam
Nature
Objectives
What is the golden ratio?
What is the Fibonacci Sequence?
Where is the golden ratio found in nature?
Who discovered it? When?
Who discovered it? When?
Ratio of following number in F sequence to previous number in F sequence.
Honeybees
Family Tree
Golden Ratio displayed: 8/5, etc.
Leaf Spacing
First leaf on stem: #0
Next leaf same position: leaf #8
Number of rotations to get to leaf 8: 5
Golden Ratio displayed: 8/5
Nautilus Shell
1
1
5
3
8
13
21
Golden Ratio Displayed: 8/5
Constructed: by drawing:
2 "1 unit" squares
a "2 unit" square on edge of first two squares
a "3 unit square on edge of a one- and two- unit square
a "5 unit" square... etc.
Then: using two sides of
each square as radii, several
'quarter circles' drawn, centers the corner closest to center of spiral.
Rose Petals
8 petals in center
5 petals in next layer
Golden Ratio displayed: 8/5
(Eigth leaf takes 5 rotations.)
Fibonacci Sequence:
a sequence of numbers in which each number equals the sum of the two preceding numbers
Discovered by:
Leonardo of Pisa (Fibonacci-"son of Bonaccio")
in 1202.
Golden Ratio:
approximately 1.618,which equals the sum of the number’s reciprocal and 1
a+ b/a= a/b= ~1.618
Discovered by: Pythagoras
sometime during his life, 582 - 500 BC.
number of bees in previous generation:present generation
(between successive rectangle lengths.)
leaf #: rotation # from 0
inner layer: outer layer
1
1
2
1
2
3
1
2
3
4
5
Assume you have a male and a female rabbit.
The first month they're alive, they can't mate, and, when they do, they produce a male and female rabbit. Also assume they don't die. Every time, the end result of the pairs of rabbits is a fibonnaci number
Month
Number of rabbit pairs
4
1
2
3
Rabbits
Jesus Flower (Passflora)
In the Passion flower shown (left), there are two examples siding towards the Fibonacci sequence. There are five green petals and five pink petals (bottom right), as well as there being three light green carpels (below).
Sneezewort
We can see that each axil consists of a certain amount of branches.
Spiral Galaxies
The golden spiral is seen in spiral galaxies. One example of a spiral galaxy is the Milky Way, which is about 100,000 light years in diameter. It is amazing how the golden spiral can be seen on such a large scale.
Plant Growth-spacing of leaves:
http://library.thinkquest.org/trio/TTQ05063/phibeauty2.htm
Number of petals/rose pic:
http://www.goldennumber.net/plants.htm
ancestors of male honey bees: female honey bees: http://mikegothard.files.wordpress.com/2009/09/european-honey-bee.jpg
Jesus flower:
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#petals
sneezewort:
http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm
rabbits:
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits
Female Rabbit:
http://tailsmagazines.files.wordpress.com/2008/12/rabbit.jpg
Male Rabbit:
http://www.creationsbydawn.net/pi/tutorials/rabbit.jpg
http://www.life-tree.net/spiralcity/goldenSpiralTemplate.jpg
This is the shape of the shell of an animal called the nautilus.
Pictures from Adam S. and his Mom
HoneyBee
http://www.dvorak.org/blog/wp-content/uploads/2009/08/honeybee2.jpg
male bee
1 Parent
2 grandparents
3 gt gp's
5 gt,gt gp's
...8 gt, gt, gt gp's, etc.
Do these numbers look familiar? That is because the number of branches and leaves in each axil are Fibonacci numbers.
Here is an example of a sneezewort plant.
If we draw horizontal lines through each axil:
We can see that each axil consists of a certain amount of leaves.
2
Reflection
In general, we knew that the Fibonacci sequence was found in nature in many ways, and we knew the first few numbers. We didn't know much about the Golden Ratio or how it was found.
The most surprising thing for us was how many places you could find examples of these numbers. For example, in space and in dragonfly wings. Also, it was surprising how the Fibonacci sequence is so constant within the golden ratio.
Knowing this information will spark an interest in geometry seen in nature. When we notice many things around the world, whether it pertains to nature, the human body, architecture, and art, we will always think of the Golden Ratio.
We constructed our own!
http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
http://marybethbrown.files.wordpress.com/2009/10/milkyway_garlick.jpg
Full transcript