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φ Phi and the Ancient Golden Rectangle (Part 2)

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By Patricia Kingswell

December 28, 2019
2:00 am
Los Angeles, CA

The “divine” Fibonacci sequence (1,2,3,5,8,13,21,34,55…) is connected to the Golden Mean or Golden Ratio (φ Phi), which shows up everywhere in nature, and in classical architecture and art. Around 300 BC, the Greek mathematician Euclid studied the “Golden Section” or “division according to the outer and middle proportion.” It describes a length that is divided into two parts, in such a way that the smaller part is to the larger part in the same proportion as the larger one is to the whole. This relationship can continue infinitely―both into the microcosmic and the macrocosmic. Around 1600, Johannes Kepler saw a relationship between the Fibonacci Sequence and the Golden Section. When applying the Fibonacci numbers as the length of the lines, it fit the sequence of the Golden Section. He

The “divine” Fibonacci sequence (1,2,3,5,8,13,21,34,55…) is connected to the Golden Mean or Golden Ratio (φ Phi), which shows up everywhere in nature, and in classical architecture and art.

Around 300 BC, the Greek mathematician Euclid studied the “Golden Section” or “division according to the outer and middle proportion.” It describes a length that is divided into two parts, in such a way that the smaller part is to the larger part in the same proportion as the larger one is to the whole.

This relationship can continue infinitely―both into the microcosmic and the macrocosmic.

Around 1600, Johannes Kepler saw a relationship between the Fibonacci Sequence and the Golden Section. When applying the Fibonacci numbers as the length of the lines, it fit the sequence of the Golden Section.

He observed that dividing a Fibonacci number with the proceeding number more and more closely approaches the irrational number φ Phi (1.61803398874989484820… ad infinitum), the longer the sequence is continued.

Therefore, φ Phi describes the Golden Section, where the smaller part is 1 and the larger part is φ Phi. Put simply: GOLDEN RATIO = 1 : 1.61 or GOLDEN RATIO = 1 : φ

“Since ancient times artists and architects have seen in the golden mean the most aesthetically satisfying geometric ratio,” says physicist Stephen Barr.

Leonardo Da Vinci illustrated the many “golden” geometric shapes involving φ Phi in the book “The Divine Proportion”, of which, the most common is the golden rectangle; a rectangle in which the proportions of its sides correspond to the golden ratio.

“Where the spirit does not work with the hand, there is no art,” said Da Vinci, who often used the ratio in laying out his compositions, especially when trying to imbue a sense of harmony, sacredness and beauty―like in his painting, “Salvator Mundi”.

“I found that Leonardo thought long and carefully about the Salvator Mundi, since he was trying to create a portrait of a divine being and used a number of devices to achieve the extraordinary presence and power of the image. I’m not surprised that he made extensive use of the golden ratio, the “divine proportion” and it is very interesting to see it mapped out in this way,” says Dianne Modestini, who restored the painting.

Knowledge of the golden ratio goes back to ancient civilisations. Ancient Mayans are known to have used it in their architecture, and according to some, so did the ancient Greeks and Egyptians.

Examples in architecture:

The GOLDEN RECTANGLE with a golden ratio of length to width:

The Parthenon, Athens (447 – 438 BC)
Notre Dame, Paris (1163 – 1250)
The Taj Mahal, India (completed in 1648)

The GOLDEN ISOSCELES/SUBLIME TRIANGLE in which two sides are in the golden ratio proportions:

The Great Pyramid of Giza
Pentagons and pentagrams (five-pointed star)

“The Golden Proportion, sometimes called the Divine Proportion, has come down to us from the beginning of creation. The harmony of this ancient proportion, built into the very structure of creation, can be unlocked… opening to us its marvelous beauty. Plato called it the most binding of all mathematical relations, and the key to the physics of the cosmos.”―Bonnie Gaunt