Fibonacci goes Gold

The Golden Ratio

1.618033988749894842... The Golden Ratio is a "special" number that goes on forever, like Pi, without ever repeating. It is often symbolized as the greek letter Phi.

The Golden Rectangle

The golden Rectangle is believed to be the most aesthetically pleasing proportion. It is a rectangle in which the length to width ratio is the golden ratio. If the Length of the rectangle is equal to Width x1.62 (or Phi-the Golden Ratio) then it is a golden rectangle. Should I try that agian another way? Trudi Hammel Garland, in Fascinating Fibonaccis explains it this way:
If the small part is called S and the large part is called L, the proportions can be mathematically stated as follows:
S/L = L/S+L

That is about as far as my mathematics descriptions will take me, but if you need more, or a precise visual, then here's the place to go for some "Golden Geometry"

Fibonacci Squares

A Fibonacci square will work a bit differently, but with a similar result. If you begin with a square that is one by one (unit, cenitmeter, inch...no matter) and add another that is the same size you create a rectangle. If you continue adding squares whose sides are the length of the rectangle - that longer side will always be a Fibonacci number. If you keep adding squares, you will end up with a rectangle that gets closer and closer to the "Golden Rectangle"

This is my aproximation that bares only a symbolic, but nowhere near technically correct, representation of the above described process.

Golden Squares

Golden Triangles

In Fascinating Fibonaccis by Trudi Hammel Garland, Golden Triangles are explained thusly:

It is an isosceleles triangle with one short side in golden proportion to each of the two longer, equal sides. Fibonacci numbers can be used to construct such triangles... Among the interesting properties of the golden triangle is the fact that the bisector of a base angle (which is always 72o) cuts the side opposite it into the golden proportions. That bisector also cuts the triangle into two new isosceles triangles...whose areas are in golden proportion to each other. The process can be repeated endlessly.

Again, I have done my best to demonstrate this process, it is not a prefect representation:

Golden Triangle

The History Page

Natural Fibonacci

Playing with the Numbers (Previous Page)

Aesthetically Pleasing Fibonacci? (Next)

The Formulas for the Fibonacci Sequence.

Bibliography and some great Fibonacci links.