While the proportion known as the Golden Mean has always existed in mathematics and in the physical universe, it is unknown exactly when it was first discovered and applied by mankind. It is reasonable to assume that it has perhaps been discovered and rediscovered throughout history, which explains why it goes under several names.
Uses in architecture potentially date to the ancient Egyptians and Greeks
It appears that the Egyptians may have used both pi and phi in the design of the Great Pyramids. The Greeks are thought by some to have based the design of the Parthenon on this proportion, but this is subject to some conjecture.
Phidias (500 BC – 432 BC), a Greek sculptor and mathematician, studied phi and applied it to the design of sculptures for the Parthenon.
Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos.
Euclid (365 BC – 300 BC), in “Elements,” referred to dividing a line at the 0.6180399… point as “dividing a line in the extreme and mean ratio.” This later gave rise to the use of the term mean in the golden mean. He also linked this number to the construction of a pentagram.
The Fibonacci Sequence was written of in India in about 200-300 BC and brought to the Western world around 1200 AD
What we now as the Fibonacci sequence is named after Leonardo Pisano Bonacci (aka Bigollo) of Pisa, an Italian born in 1175 AD, who later became known as Leonardo Fibonacci. His book Liber Abaci, published in 1202, introduced this sequence to Western European mathematics in the form of a math problem on the breeding of rabbits. He learned of it though while being educated in North Africa with an Arab master, where he was exposed to the much earlier knowledge of Indian mathematicians. Liber Abaci became a pivotal influence in adoption by the Europeans of the Arabic decimal system of counting over Roman numerals. (3)
The sequence itself though had been described as early as the 2nd or 3rd century BC in the works of Acharya Pingala, an Indian mathematician who enumerated the possible patterns of Sanskrit poetry that could be formed from syllables of two lengths.
The contributions of Pingala and Fibonacci are important, but it’s not apparent that anyone even realized its connection to the Golden Ratio until the 1600’s by Johannes Kepler and others.
It was first called the “Divine Proportion” in the 1500’s
Leonardo Da Vinci provided illustrations for a dissertation published by Luca Pacioli in 1509 entitled “De Divina Proportione” (1), perhaps the earliest reference in literature to another of its names, the “Divine Proportion.” This book contains drawings made by Leonardo da Vinci of the five Platonic solids.
The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty. Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his painting of “The Last Supper,” from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background.
Johannes Kepler (1571-1630), discoverer of the elliptical nature of the orbits of the planets around the sun, also made mention of the “Divine Proportion,” saying this about it:
“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.”
It is believed that Martin Ohm (1792–1872) was the first person to use the term “golden” to describe the golden ratio. to use the term. In 1815, he published “Die reine Elementar-Mathematik” (The Pure Elementary Mathematics). This book is famed for containing the first known usage of the term “goldener schnitt” (golden section).
The term “Phi” was not used until the 1900’s
It wasn’t until the 1900’s that American mathematician Mark Barr used the Greek letter phi (Φ) to designate this proportion. This appeared in the “The Curves of Life” (page 420) in 1914 by Theodore Andrea Cook . By this time this ubiquitous proportion was known as the golden mean, golden section and golden ratio as well as the Divine proportion. Phi is the first letter of Phidias (1), who used the golden ratio in his sculptures, as well as the Greek equivalent to the letter “F,” the first letter of Fibonacci. Phi is also the 21st letter of the Greek alphabet, and 21 is one of numbers in the Fibonacci series. The character for phi also has some interesting theological implications.
Recent appearances of Phi in math and physics
Phi continues to open new doors in our understanding of life and the universe. It appeared in Roger Penrose’s discovery in the 1970’s of “Penrose Tiles,” which first allowed surfaces to be tiled in five-fold symmetry. It appeared again in the 1980’s in quasi-crystals, a newly discovered form of matter.
Phi as a door to understanding life
The description of this proportion as Golden and Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for a single number to play, but then again this one number has played an incredible role in human history and in the universe at large.
(1) Page 25
(2) Page 157
(3) Page 158