The golden mean is a number that appears in a dazzling array of mathematical and natural structures. The golden mean has fascinated both mathematicians and amateurs since the time of Pythagoras, and perhaps even earlier, with the Egyptians. The great astronomer Johannes Kepler called the golden mean one of the "two great treasures" of geometry (the other was the Pythagorean theorem), likening it to a "precious jewel."

There are many ways to define the golden mean, which is usually written phi (). The earliest known appearance of phi among mathematicians is in the Brotherhood of Pythagoras, a group of scholars from the 6th century B.C. In order to recognize its members, the Brotherhood adopted a symbol called the pentagram, which is a pentagon with a five-pointed star inscribed in it. At the center of the star is a smaller pentagon. The golden mean is the ratio...