Mathematics of the Golden Ratio

This Golden Ratio truly is unique in its mathematical properties and pervasive in its appearance throughout nature. The “mathematically challenged” may be more interested in the appearances of Phi in nature, its application to art, architecture and design, and its potential for insights into the spiritual realm, but let’s begin with the purest of facts about Phi, which are found in mathematics.

Most everyone learned about the number Pi in school, but relatively few curriculums included Phi, perhaps for the very reason that grasping all its manifestations often takes one beyond the academic into the realm of the spiritual just by the simple fact that Phi unveils a constant of design that applies to so many aspects of life. Both Pi and Phi are irrational numbers with an infinite number of digits after the decimal point, as indicated by “…”, the ellipsis.

Where Pi or p (3.14…) is the ratio of the circumference of a circle to its diameter, Phi or Φ (1.618…) is the Golden Ratio that results when a line is divided in one very special and unique way. To illustrate, suppose you were asked to take a string and cut it. There’s any number of places that you could cut it, and each place would result in different ratios for the length of the small piece to the large piece, and of the large piece to the entire string. There is one unique point, however, at which the ratio of the large piece to the small piece is exactly the same as the ratio of the whole string to the large piece, and at this point this Golden Ratio of both is 1.618 to 1, or Phi.

What makes this so much more than an interesting exercise in mathematics is that this proportion appears throughout creation and extensively in the human face and body. It’s found in the proportions of many other animals, in plants, in the solar system and even in the price and timing movements of stock markets. Its appeal thus ranges from mathematicians to naturalists to artists to investors to mystics.

Part of the uniqueness of Phi is that it can be derived in many other ways than segmenting a line. Phi is the only number whose square is greater than itself by one, expressed mathematically as Φ² = Φ + 1 = 2.618. Phi is also the only number whose reciprocal is less than itself by one, expressed as 1/Φ = Φ - 1 = 0.618. Where 1.618 is represented in upper case as Phi or Φ, its near twin or reciprocal, 0.618, is often represented in lower case as phi or φ. The Fibonacci series, also a plot element in “The Da Vinci Code,” provides yet another way to derive Phi mathematically. The series is quite simple. Start with 0 and add 1 to get 1. Then repeat the process of adding each two numbers in the series to determine the next one: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on. The relationship to the Golden Ratio or Phi is found by dividing each number by the one before it. The further you go in the series, the closer the result gets to Phi. For example, while 5/3 = 1.666 and 13/8 = 1.625, go further into the series and you’ll find that 233/144 = 1.61805, a very close approximation of Phi, which to ten decimal places is 1.6180339887.

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