George Tom


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.”

Using your calculator to find Phi :


 Enter the number 1.

        blball.gif (967 bytes)Add 1. Take its reciprocal.

        blball.gif (967 bytes)Add 1. Take its reciprocal.

        blball.gif (967 bytes)Add 1. Take its reciprocal.

        blball.gif (967 bytes)Continue this. You should be converging on the Golden Ratio


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.”


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


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.


Fluffy bunnies family tree





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 ...