The golden ratio with a reasonable number of decimal places is as follows: 1.6180339887…
Phi is an irrational number and cannot be expressed as an integer fraction. Giving an almost qualitative definition of phi is impossible, as the number itself is a fundamental property of many available forms and shapes. A numerical definition is possible and is described as the fraction of two numbers, a larger number to a smaller number that is also equivalent to the sum of both numbers to the larger number.
Mathematically, it is expressed as such:
(b / a) = [(a + b) / b] = phi, where b is the larger number and a is the smaller number.
Phi has manifested itself since the early studies of geometry, a branch in mathematics that deals with shapes and forms. It is more than appropriate to have geometry describe the form of beauty mathematics partakes. Many geometrical instances have phi screaming itself to ancient Greek mathematicians. One example will be the mathematical representation of a star, the pentagram, a shape that is formed by five lines. These five lines are intersected in a very uncanny manner, in which each of the lines is intersected into sections that, when their lengths are represented in lengths, yield the golden ratio.
Since the Renaissance period, artists and architects had proportioned their designs to approximate the golden ratio. One such example is the construction of the golden rectangle, which is believed to be aesthetically pleasing. The Parthenon, a temple built for the Greek god Athena, protector of Athens, has its proportions heavily relied on the golden ratio. Many elements are evident in showing their structure proportioned to the golden ratio, and this indicates that the architects probably knew what the golden ratio is since their time, or alternatively, had a good sense of proportion that gave rise to this sort of proportionalising.
However much as humans want to assign a definitive quality to the golden ratio is still a question, and its significance can only be sought after by people seeking aesthetics rather than functionality. It is true that a more realistic quality should be embedded into the golden ratio. Nevertheless, this enigmatic figure will continue to dazzle eager minds with its abstractness.
Livio, 2002, The Golden Ratio. New York: Broadway Books.
Green, Thomas M. The Pentagram and the Golden Ratio. [Online] Available from: http://www.contracosta.cc.ca.us/math/pentagrm.htm. [Accessed on 11th August 2009]