What is the Golden Ratio?

“It is the irrational number that is equal to (sqrt(5)+1)/2 = 1.618033989.

There
are just two numbers that remain the same when they are squared namely **0**
and **1**. Other numbers get bigger and some get smaller when we square
them: In fact, there are *two* numbers with this property, one is Phi and
another is closely related to it when we write out some of its decimal places.”

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#golden

Using your calculator to find Phi :

Enter the number 1.

Add 1. Take its
reciprocal.

Add 1. Take its
reciprocal.

Add 1. Take its
reciprocal.

Continue this.
You should be converging on the Golden Ratio

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#golden

What is the history of the Golden Ratio? Include information about the Fibonacci Sequence and its discoverer.

“Leonardo of Pisa, better known as Fibonacci, was
born in Pisa, Italy, about 1175 AD. He was known as the greatest mathematician
of the middle ages. Completed in 1202, Fibonacci wrote a book titled *Liber
abaci* on how to do arithmetic in the decimal system. Although it
was Fibonacci himself that discovered the sequence of numbers, it was French
mathematician, Edouard Lucas who gave the actual name of "Fibonacci
numbers" to the series of numbers that was first mentioned by Fibonacci in
his book. Since this discovery, it has been shown that Fibonacci numbers
can be seen in a variety of things today.”

“He began the sequence with 0,1, ... and then calculated each
successive number from the sum of
the previous two.

This sequence of numbers is called the Fibonacci Numbers or Fibonacci Sequence.
The Fibonacci numbers are interesting in that they occur throughout both nature
and art. Especially of interest is what occurs when we look at the ratios of
successive numbers.”

http://www.geom.umn.edu/~demo5337/s97b/fibonacci.html

How does the Fibonacci Sequence relate to the Golden Ratio?

“ By charting the
population of rabbits, Fibonacci discovered a number series from which one can
derive the Golden Mean. The beginning of the sequence: 0, 1, 1, 2, 3, 5,
8, 13, 21, 34, 55... Each number is the sum of the two preceeding
numbers. Dividing each number in the series by the one
which preceeds it produces a ratio which
stabilizes around 1.618034**”**

http://www.geom.umn.edu/~demo5337/s97b/fibonacci.html

What are other names for the Golden Ratio?

The
other names are Phi, and the (sectio aurea**)** meaning the golden section,
and the golden mean.

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#golden

This picture shows Fibonacci’s chart that he made while he was looking at rabbits, without knowing he was about to change math forever.

`A proportion is formed from ratios, and a ratio is a comparison of two`

`different sizes, quantities, qualities or ideas, and is expressed by the`

`formula a:b. A ratio then constitutes a measure of difference. The`

`perceived world is then made up of intricate woven patterns of...`

`differences... A proportion, however, is more complex, for it is a`

`relationship of equivalency between two ratios, that is to say, one`

`element is to a second element as a third element is to a fourth:`

`a is to b as c is to d, or a:b::c:d ...`