“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…”-Johannes Kepler. The Golden ratio, it is the divine proportion, the perfect number, the perfect shape. Scientists, artists, mathematicians have aspired to achieve this golden mean in their work as long as their occupations have been in existence. Its seemingly simple nature has fascinated many of the world’s greatest thinkers and inspired amazing projects and building feats. The golden rectangle is probably the shape most identifiable with the golden ratio. It is achieved by a rectangle within another; the smaller rectangle is the same as the larger rectangle it is contained in. The fraction a+b/a is taken from this geometric representation. a+b/a essentially is or equals the golden ratio. This ratio was supposedly discovered by the ancient Greek Pythagoras who believed it was sent from the gods as a divine solution to form. The ancient Greeks incorporated this ratio into almost all of their great building projects such as the Parthenon and the Oracle at Delphi. The Fibonacci sequence is a number pattern defined by relation and recurrence. This sequence is closely related to the golden ratio. The golden ratio, being 1.6180, is the limit of a term in the Fibonacci sequence divided by its predecessor. Almost every number in the Fibonacci sequence divided by its predecessor roughly equals the golden ratio as well. The twenty-first letter of the Greek alphabet, Phi or φ, often represents the golden ratio because the number cannot be completely and accurately represented in ratio form. When the ancient Greeks began incorporating the golden ratio into their buildings it was not the first time it had been used in building; However the Greeks where the first to understand exactly what the golden ratio is and how it directly applied to science and mathematics. The golden ratio was used as a basis for most ancient Greek temples, courts and markets. These were open stone buildings usually supported by columns and a central foundation. Temples were usually rectangular, wide buildings with one floor and a high ceiling supported by marble columns. The temples would commonly be completely surrounded by terraced steps four to eight levels high. An example of a simple, early solid-back, four pillar wide, temple in the temple of Athena Nike located on the Acropolis in Athens. It is also the earliest temple to be fully iconic of the Greek classical era. When measuring from above the base of the foundation to the roof of the temple over the width it has a ratio of 1.36, close to the golden mean, but not close enough to infer any real relationship. When measuring from below the front façade and steps at the base the foundation over the width of the roof a ratio of 1.615 is found. This matches almost exactly with the golden ratio and the golden triangle. The temples architect, Callicrates was an important Athenian architect known for his work on other temples such as the great Parthenon and his use of the golden ratio in much of his work.

As Europe receded into the dark ages after the collapse of the Roman Empire much of the study of mathematics and architecture was lost. It would resurface as the Catholic Church employed the world’s greatest engineers to create massive churches throughout the continent. One of the most famous of these new “Gothic” churches was the Cathedral of Our Lady of Paris, better known as Notre Dame. Completed in 1345 the cathedral incorporated many new Gothic-building techniques such as the flying buttress. It also incorporated an ancient building technique: the golden rectangle. Not including the bell towers the church’s height divided by its width creates a ratio of 1.55, extremely close to that of the golden mean. The exterior inlets above the