The Golden Ratio is a proportion that has fascinated countless mathematicians. Although it was first studied in detail by Euclid around 300 B.C., the Golden Ratio was already known by the Egyptians, who used it in constructing the Pyramids at Gizeh.

To see the Golden Ratio, divide a line segment into two unequal parts so that the ratio of the larger part to the smaller part is equal to the ratio of the whole to the larger part -- a ratio that is approximately 1.618034 to 1. The value of this ratio, approximately 1.618034, is often represented by the Greek letter , pronounced Phi. Phi's decimal places continue forever in a non-repeating sequence, making it an irrational number. Although Phi looks complicated, it naturally arises in surprisingly simple situations, ranging from the breeding patterns of rabbits to the composition of priceless paintings.

The Golden Ratio is often found when comparing the side lengths of simple geometric figures, a famous example being the Golden Rectangle. In this figure, the ratio of the long side of the rectangle to the short side is . When such a rectangle is partitioned by a square whose sides are equal to the shorter side, the ratio of the sides of the resulting rectangle is also . This process can be repeated, so that every newly-created rectangle will have the same proportion.

The Golden Rectangle is frequently found in art and architecture, especially in ancient Greek temples. Although it is not known whether or not Greek architects intentionally applied the Golden Ratio to their designs, it does seem to appear frequently in any design derived from the square and compass. During the Renaissance in Italy, such artists and scientists as Leonardo da Vinci were fascinated by mathematics and their relationship to the physical and spiritual order of the universe, which is why they called the Golden Ratio the "divine proportion." Although Renaissance artists may not have directly applied the Golden Rectangle in making their paintings, their compositions often demonstrate its perfect mathematical proportions. This perfect proportion may be a part of the reason that these paintings still appear beautiful to us today.