Golden Ratio: the fingerprints of God

The Golden Ratio is an interesting topic that pops up in many parts of our lives. Personally, I was told about this for the first time by advertisements of cosmetic and beauty product industries. As a promotion strategy, they commercially (ab)use this concept to entice many girls to aspire to seemingly dazzling idea of achieving “golden” and “perfect ratio.” The name itself demonstrates enough to give an impression of perfect beauty.

The Golden Ratio had been discovered and rediscovered over time historically, which is why it goes under many names: golden mean, golden proportion, divine proportion, etc. The earliest appearance of the Golden Ratio in history is found in design of Great Pyramids in Egypt. In ancient Greece, Euclid discussed it in Book 6, Proposition 30, in his work “Elements” in 300 BC. He showed how to divide a line at the ratio 0.6180399… : 1, which he referred to “dividing a line in the extreme and mean ratio.”


Image courtesy of, Gary B. Meisner.

He took a line that is one unit long and divided the line in two parts, such that the ratio of the shorter part of the line to the longer part is the same as the ratio of the longer part to the whole line; that gives a quadratic equation x/(1-x) = (1-x)/1, which yields two solutions x = (1 +/- √5) /2 . Since the ratio between positive quantities has to be a positive number, x= (1 + √5) /2 = 1.618… The term “mean” he used gave rise to the name golden mean, and it is considered the first written reference to the Golden Ratio. After Euclid, many mathematicians extensively studied the Golden Ratio and its unique properties, and since the early 1900s at Mark Barr’s suggestion, they started using the Greek letter phi (Φ) to symbolize the Golden Ratio.

In architecture and arts as well, the Golden Ratio has been a popular topic. The design of Parthenon in Athens, built by the ancient Greeks from 447 to 438 BC, appears to have features of the Golden Ratio. The scholars speculate that Phidias, a Greek sculptor and mathematician, designed the Parthenon based on his studies of Phi. The Golden Ratio can be found on the grid lines and the root support beam. Also the ratio of the structural beam on top of the columns to the height of the columns, and distance ratio of the width of the columns to the center line of the columns exhibits the ratio of 0.618… : 1. Some people question this speculation of the Golden Ratio being applied in the famous structure because first, there is no concrete evidence or written documentation that Phidias was aware of the Golden Ratio when designing the Parthenon, and second, time and history have damaged its original features and dimension, leaving scholars with no choice but to conjecture based on the remaining structure. Thus, it may not be accurate to conclude that Greeks have applied the Golden Ratio in the construction of the Parthenon.

During the period of Renaissance, Leonardo Da Vinci was known to have applied the Golden Ratio in his art works, which was called sectio aurea, meaning “golden section” in Latin. His illustrations of polyhedra in De Divina Proportione, “On the Divine Proportion,” by Luca Pacioli in 1509 and his other works including the Last Supper and Mona Lisa indicates that he incorporated the Golden Ratio to define the fundamental proportions of his drawings. Other Renaissance artists also used the Golden Ratio extensively in their paintings and sculptures to achieve balance and beauty, and it was also called as divine proportion after Da Vinci’s work in Pacioli’s book.


Image: Polyhedron drawn by Da Vinci, via Wikimedia Commons.

But the most interesting thing about the Golden Ratio, in my opinion, is found in mathematics. Mathematics sees the Golden Ratio as unique rather than beautiful. It has several properties that make it unique among all numbers.

1. If you square Phi, you get a number exactly 1 greater than itself.

Φ^2  = Φ + 1= 2.618…

2. If you take the reciprocal of Phi, you get a number exactly 1 less than itself. The reciprocal of Phi is often written as phi with lowercase p.

1/Φ  = Φ – 1= 0.618…

3. Using two properties above, Phi can be expressed in nested radicals and continued fractions.

As in #2, Φ = 1+ 1/Φ.  This can be expanded recursively. Given the initial approximation Φ(0)=1, in order to get the next approximation in the sequence Φ(n+1), we just add 1 to the reciprocal of the previous approximation Φ(n). This gives a formula Φ(n+1) = 1+ 1/Φ(n). Now we substitute the successive values of Φ(n) in the formula repeatedly to build up a sequence of continued fractions. The limit of these continued fractions as n goes to infinity is phi.

Screen shot 2014-10-26 at 8.40.07 PM

For nested radicals expression, take a look at the property in #1. The equation Φ^2  = 1 + Φ likewise produces the continued square root.


Image courtesy of, Gary B. Meisner.

Other than these amazing properties, Phi is connected to other mathematics. Most notable link is Fibonacci sequence, discovered by Leonardo Fibonacci, an Italian mathematician, in 1175 AD, but it is a mystery if Fibonacci was aware of the link between what he had discovered and Phi.

The Fibonacci sequence is defined recursively. It starts off with two ones, and each successive term is the sum of the two terms before it. So the Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34,… Johannes Kepler who showed that the Golden Ratio is the limit of the ratios of successive terms of the Fibonacci sequence. If we take the ratio of two successive numbers and continue on the fractions, we can see that ratios are getting very close to Phi.

2/1 = 2

3/2 = 1.5

5/3 = 1.666666…

8/5 = 1.6

13/8 = 1.625

233/144 = 1.618055556…

377/233 = 1.618025751…

The fractions of Fibonacci numbers give values that are alternatively higher and lower than Phi, and converge on Phi as the number increases. So the limit of F(n+1)/F(n) when n is approaching to infinity is equal to Phi.


Image: Wikimedia Commons.

It is not only mathematics that connects Phi and Fibonacci. The spiraling growth of leaves, flower petals, and seed pods follows Phi rotation. This minimizes the amount of overlap of any leaf by those leaves located higher on the stem, which allows each leaf to receive the maximum amount of sunlight for photosynthesis. So it is not a coincidence that Fibonacci numbers are commonly found in the plant’s spiral growth pattern of new cells because they essentially express the Phi ratio. Although it is not absolutely true for all plant species, both Phi and Fibonacci explain intricate and sophisticated patterns found in nature. Phi is amazingly connected to every parts of the world, which explains why its reference is so ubiquitous in a variety of areas such as arts, architectures, and mathematics. It has played a significant role in deeper understanding of life and universe, and I think it is why people often call it “the fingerprint of God.”


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