by Nikhat Parveen, UGA

According to legend, the Greek Philosopher Pythagoras discovered the concept of harmony when he began his studies of proportion while listening to the different sounds given off when the blacksmith’s hammers hit their anvils. The weights of the hammers and of the anvils all gave off different sounds. From here he moved to the study of stringed instruments and the different sounds they produced. He started with a single string and produced a monochord in the ratio of 1:1 called the Unison. By varying the string, he produced other chords: a ratio of 2:1 produced notes an octave apart. (Modern music theory calls a 5:4 ratio a "major third" and an 8:5 ratio a "major sixth".) In further studies of nature, he observed certain patterns and numbers reoccurring. Pythagoras believed that beauty was associated with the ratio of small integers.

Astonished by this discovery and awed by it, the Pythagoreans endeavored to keep this a secret; declaring that anybody that broached the secret would get the death penalty. With this discovery, the Pythagoreans saw the essence of the cosmos as numbers and numbers took on special meaning and significance.

The symbol of the Pythagorean brotherhood was the pentagram, in itself embodying several Golden Means.

The Greeks, who called it the Golden Section, based the entire design of the Parthenon on this proportion.

**
The Greeks knew**

**it as the**

__Golden Section__

**
and used it for beauty
and balance in the
design of architecture**

Phidias (500 BC - 432 BC), a Greek sculptor and mathematician, studied phi and applied it to the design of sculptures for the Parthenon.

*
Porch of Maidens,
Acropolis, Athens*

Euclid
proved that the diagonals of the regular pentagon cut each other in "extreme and
mean ratio", now more commonly known as the golden ratio. Here we represent the
golden ratio by phi. F_{n} is the nth Fibonacci number.

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