## The Golden Section and The Divine Proportion

In 1509, mathematician Luca Pacioli (pictured left) wrote a dissertation entitled Divina Proportione, or the Divine Proportion, coining the term. The book, which was illustrated by Leonardo DaVinci, was also the first to refer to the Golden Ratio as the Golden Section. Pacioli stated 5 aspects of the ratio that made it "divine."

"As God, it is unique."

"As the Holy Trinity, it consists in the unity of three, being made of three terms."

"As God cannot be defined in words, it cannot be expressed as a rational number."

"As God, it is always equal to itself."

"As there are four elements which form all nature, so the holy proportion permits the construction of the dodecahedron, the expression of quintessence."

Many Renaissance artists referred to it as the Divine Proportion and it was used widely in art.

"As God, it is unique."

"As the Holy Trinity, it consists in the unity of three, being made of three terms."

"As God cannot be defined in words, it cannot be expressed as a rational number."

"As God, it is always equal to itself."

"As there are four elements which form all nature, so the holy proportion permits the construction of the dodecahedron, the expression of quintessence."

Many Renaissance artists referred to it as the Divine Proportion and it was used widely in art.

## Φ - Phi

The ratio 1.6180339887... was not referred to as Phi until the 1900's. The term was developed by American mathematician Mark Barr. Phi is he 21st letter of the Greek alphabet, and 21 is the 8th number in the Fibonacci sequence, from which the ratio is derived. The letter is equivalent to the letter F in the English alphabet, and Fibonacci begins with the letter F. Finally, it is the first letter of Phidias' name.

Phidias was a Greek sculptor, mathematician and architect. He studied Phi and used it when building the Parthenon in Greece. He also used Phi in the statue of Athena in Athens, Greece, and the statue of Zeus in Olympiad.

Phi with a capital 'P' is 1.6180339887... and phi with a lowercase 'p' is 0.6180339887..., the reciprocal of Phi.

Phidias was a Greek sculptor, mathematician and architect. He studied Phi and used it when building the Parthenon in Greece. He also used Phi in the statue of Athena in Athens, Greece, and the statue of Zeus in Olympiad.

Phi with a capital 'P' is 1.6180339887... and phi with a lowercase 'p' is 0.6180339887..., the reciprocal of Phi.

## Extreme and Mean Ratio

Euclid coined the term extreme and mean ratio, which was also used by Johannes Kepler in describing Phi. He also linked the number to the construction of pentagons.

## "A Precious Jewel"

Scientist Johannes Kepler used Phi in the development of the Kepler triangle, one of his many contributions to the world of math and science. Here's what he had to say about the Golden Ratio.

"Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio [the Golden Ratio]. The first we may compare to a measure of gold, the second we may name a precious jewel."

"Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio [the Golden Ratio]. The first we may compare to a measure of gold, the second we may name a precious jewel."