Nature’s Aesthetic: The Golden Ratio (Fibonacci Sequence)

glass spiralFrom art to technology, nature has always been a source of inspiration for humanity, often providing prime examples for any and all newfound ingenuity.

Whether attributed to a divine creator or the laws of physics, the design seen within the natural world motivated many to eventually discover a formula which would explain this apparent order.

Known as the golden ratio, it provides the mathematical foundation for designs found in nature and an aesthetic strived for in human creation.

Believed by many to express the inherent order and beauty within nature, the golden ratio has historically been imbued with near spiritual significance – leading some to refer to it as the “divine proportion.”

From seashells and flowers to hurricanes and galaxies, this pattern can be spotted throughout nature – and it has even been argued that the human eye is predisposed to interpret images which contain it.

shellRepresented by the Greek letter phi, Φ, the golden ratio is summarily estimated at 1.618, just like how pi is deduced to 3.14.

Being an irrational number like pi, it has the unique quality of being an infinite number without repetition past its decimal point.

National Geographic defines phi as being “a unique mathematical relationship. Two numbers are in the golden ratio if the ratio of the sum of the numbers (a b) divided by the larger number (a) is equal to the ratio of the larger number divided by the smaller number (a/b).”

Represented by the mathematical expression of “1: phi”, it can be visually represented as a so-called “golden rectangle”, with the ratio “of side “a” to side “b” the same as the ratio of the sides “a”-plus-“b” to “a.”

According to The Wire, given this visual representation one can subsequently “create a square on one side of the golden rectangle, and the remaining space will form another golden rectangle. Repeat that process in each new golden rectangle, subdividing in the same direction, and you’ll get a golden spiral, arguably the more popular and recognizable representation of the golden ratio.”

golden triangleThe most famous approximation of phi is through the well-known “Fibonacci sequence.”

An infinite series on numbers starting with 0 and 1, its pattern progresses by continually adding the last two numbers from the sequence.

For example,

0 1 1 (0 1) 2 (1 1) 3 (2 1) 5 (3 2) 8 (3 2) 13 (5 8)

As the Fibonacci numbers continue, their ratios (2/1, 3/2, 5/3, 8/5, etc.) approach evermore closer to 1.618 – phi.

2/1 = 2 3/2 = 1.5 5/3 = 1.666 . . .

Leonardo Da Vinci believed the Fibonacci sequence was inherently aesthetic – and was known to have utilized the mathematical pattern in many of his artworks.

fibonacciFirst described by the Greek mathematician Euclid in the 3rd century B.C. as “the division in extreme and mean ratio,” it appears in even earlier cultures.

Although Euclid and his contemporaries were aware of phi, it was only cited in the relation between two lines, not as the golden rectangle.

A commonly cited example of the ancient Greeks utilizing phi is The Parthenon, however, it is now known to have been constructed with the ratio [of] 4:9 – two whole numbers.

This architectural design is not surprising, for the Greeks of that era associated whole number ratios with Platonic concepts of perfection. Rather than coming out of Greece, some scholars now believe that the golden ratio first originated in Africa.

In Africa, it is common for designs to be rooted in the mathematical principle known as “recursion” – where a shape is divided into smaller versions of itself which remain entrenched within the whole, creating a “self-similar” fractal pattern.

Traditional fabrics from Ghana known as Kente cloth have been found with scaling patterns in sync with the Fibonacci sequence. The golden ratio is also found in ancient Egyptian sites such as the layout of the Temple of Karnak.

In his book The Elements of Typographic Style, Robert Bringhurst suggests Fibonacci had African influence, reminding readers that:

“If we look for a numerical approximation to this ratio, 1: phi, we will find it in something called the Fibonacci series, named for the thirteenth-century mathematician Leonardo Fibonacci. Though he died two centuries before Gutenberg, Fibonacci is important in the history of European typography as well as mathematics. He was born in Pisa but studied in North Africa.”

galaxyBeyond the spiraling pattern of galaxies, the golden ratio has now been identified to appear in individual stars. New data from the Kepler space telescope has revealed four stars which “pulsate at frequencies” at a ratio near the mathematical estimation of phi – 0.61803398875.

Their pulsating behavior is a result of their brightness fluctuating, due to their atmosphere expanding and contracting in response to pressure changes – thankfully for life on Earth, this behavior is not shared with the Sun, having a relatively consistent brightness.

In his 2002 book The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number, Astrophysicist Mario Livio claims that “The golden ratio has a long history in disciplines ranging from the physics of crystals to visual arts, and that “The golden ratio is special in that it is in some sense the most irrational of all irrational numbers.”

An irrational number is named as such due to its quality of not being able to be expressed as a ratio of whole numbers. While some irrational numbers can be calculated with their rational counterparts, some, like the golden ratio, are among the hardest to approximate.

According to Scientific American, the stars which appeared to pulsate in accordance with the golden ratio also exhibited “fractal behavior – never ending patterns that repeat on continuously smaller scales.”

Speaking to Scientific American, Livio concludes that the findings are important “from a dynamics perspective [for] it is quite intriguing to understand why systems would be attracted to this ratio.”

The mysterious allure of the golden ratio and the Fibonacci sequence is not limited to its aesthetic beauty.

These mathematical formulas have also been applied in the world of finance – with surprising results.

“Fibonacci retracement”, “is a tool that technical analysts use to guide their outlook about buying and selling behavior in markets.”

fibretracementFinancial investment researchers who adhere to this strategy look for mathematical hints of the famous formula while surveying stock market prices. Taking 1/phi, 0.618, or 0.382, – which is two places higher in the sequence – they translate these values into percentages – 61.8% and 38.2% – which they believe predict upcoming behavior.

Considered the “support level” for a stock, a dip of 38.2% – or more – in price two Fridays in a row is believed to be an indicator of a further dip downwards towards the second number, 61.8%.

Katie Stockton – founder and partner at the technical analysis firm Fairlead Strategies, LLC – argues that even if Fibonacci retracement is bogus, believing in it still has an impact.

Commenting to Smithsonian Magazine about past fluctuations in the price of gold, she states:

That was a big Fibonacci breakout that a lot of folks were watching, even to the extent that it became such a widely followed level that I think there becomes some self-fulfilling property to it.”

Similar strategies have even been applied financial assets as volatile as Bitcoin.

JC Parets, an “expert in technical analysis of markets” who runs a financial research site called All Star Charts, claims to have accurately predicted Bitcoin’s rise by applying the Fibonacci sequence.

In 2017 he correctly predicted that Bitcoin would surpass $6,500 before subsequently rising above $7,400. He continued to remarkably predict future peaks of the cryptocurrency as it continued to rise past $60,000.

Parets claimed that he could predict Bitcoin’s trajectory by segmenting price fluctuations before calculating the sum of two previous price points.

Whether it be to make money, create beauty, or explain the evolution of the natural world, the golden ratio expresses a magnetic resolution demanded by the mysterious desire of the Universe.

Author: Liam Penn (Dark Matters)

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