**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**