NOTE: The discussion that follows is adapted (in part) from Fascinating Fibonaccis: Magic and Mysteries in Numbers: by Trudi Hammel Garland (Palo Alto, California: Dale Seymour Publications, 1987)
The Golden Proportion, generated by the Golden Section (or Golden "Cut"), is, quite simply, a unique and special proportion deeply rooted in folklore, history and philosophy.
A "sacred ratio" used in the building of the pyramids of Egypt 2,600 years ago is mentioned in the Ahmes Papyrus (also called the Rhind Papyrus after the nineteenth-century Egyptologist of the same name). This is a mathematical handbook, of sorts; dated about 1650 B.C. Greek mathematician and astronomer Eudoxus of Cnidus (c. 370 B.C.) observed that his friends divided a stick into golden proportions when asked to find the most pleasing placement of a crossbar. Both Euclid and Pythagoras referred to the "rectangle of the Divine Section," believing it to be uniquely inspired by the will of God. Plato considered the Golden Proportion to be the "most binding of all mathematical relations, the key to the physics of the cosmos."
Down through the ages the Golden Proportion has influenced art, architecture, music, and poetry—to varying degrees, intentionally as well as unintentionally. Mathematicians have examined its many properties and written exhaustively on the subject. It occurs naturally and abundantly in the physical and biological sciences; and the Golden Proportion appears to operate today in other fields as diverse as psychology, computer technology, and investment analysis. For example, several psychology studies indicate that people tend to divide and categorize things—such as a person's positive and negative qualities—according to the Golden Proportion.
What is the Golden Proportion:
Simply stated, The Golden Proportion exists between a small and a large segment:
The proportion of the small segment to the large is the same as the proportion of the large segment to the sum of both.
If the small segment is 1 unit and the large unknown, the proportion can be stated algebraically:
1/x = x/(1+x)
Expressed either way, the Golden Proportion is the only number that is equal to its reciprocal ±1.
The Golden Proportion is the basis of the Golden Rectangle, whose sides are in the golden proportion to each other. The Golden Rectangle is considered to be the most visually pleasing of all rectangles. For this reason, as well as its practicality, it is used extensively in all kinds of design—art, architecture, advertising, packaging, and engineering— and can therefore be found readily in everyday objects.
One of the unique qualities that distinguishes the Golden Rectangle from other rectangles is the fact that, when a square is cut from it, the remaining rectangle is the shape of the original one. This is easy to see at a casual glance .