We are two students from Byram Hills High School who are researching the Golden Ratio, and it's connection to nature (biology) and architecture.  Please use the links on the left or above to explore the different areas. 

What is the Golden Ratio?

The golden ratio is generated when we say that the ratio that of the  segments is equivalent to the ratio of the large section over the whole segment.    In theory we are just saying the x is the mean proportional between the whole segment and the smaller segment.  By saying this we get the ratio

x+1/x=x/1

by applying the means extremes theorem of proportionality we get

x²=x+1

or

x²-x-1=0

by applying the quadratic formula we get

x=(1ħsqrt((-1)²+(-4)(1)(-1)))/2

x=(1+sqrt(5))/2

which we get the golden ratio moreover it is generated through the rabbit problem that Fibonnacci declared by Fibonnacci in the 1100's.  He said that if you had 2 rabbits and they produced  2 rabbits every month after the month that took them to mature.  Assuming the rabbits live forever how would the population grow?

From that problem he generated this sequence:

1,1,2,3,5,8,13,21...

By dividing any number in the sequence by it's preceding number you should get an irrational number similar to the one that we proved algebraically above.  Note that the higher in the sequence you go, the closer the quotient will be to 1.61...  This number is also expressed as the greek letter phi. Thus it is the Lim x->infiniti x/y where x is a number in the fibbonacci sequence and y is the number before it

Golden Rectangle

The Golden rectangle is a rectangle where one side is assigned the length 1 or x and the other length is proportional so that it's length would equal 1+sqrt(5) or x(1+sqrt(5)).    By looking at the graphic below, you can see what it looks like....

Golden spiral

The Golden spiral is generated when you start out with a rectangle with one side being a number in the Fibbonacci sequence the other side would be the number before it in the sequence.  Then you draw a square using the shorter side in the rectangle so that in the area excluded in the square another rectangle in the proportions of the Fibonacci sequence is produced.   Thus you continue this until the rectangles in the proportions of the Fibonacci sequence are squres.  The spiral is generated when you connected the opposite dots of the squares formed in hyperbola type shape beginning with the largest and going to the smallest you have the spiral.

Geometers Sketchpad

If you would like instructions on how to construct the Golden Ratio on Geometers Sketchpad, please click here.

[This Page was made by Jeff Shuster and Phil Harris]