Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. - Johannes Kepler
Of the two treasures that Kepler mentions, the Pythagorean Theorem is certainly the most popular today. Anyone who has taken a middle school math class or a high school geometry class has been exposed to the theorem describing the relationship between the legs and hypotenuse of a right triangle. However, the "extreme and mean ratio" may be a little less familiar. Kepler's (actually Euclid's) "extreme and mean ratio" has gone by many names. It has been called the Golden Ratio, the Golden Section, the Golden Number, Phi, and the Divine Proportion. Philosophers, mathematicians, artists, and others have all found this ratio fascinating to study and explain. It has been contemplated by the world's brightest minds in various fields of study. What is so interesting about this number that it has captivated minds as far back as ancient Greece? As one considers the numerical properties of Phi and the variety of places in which this quantity appears, the answer becomes much clearer.