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The Golden Ratio

An overview of its properties, appearances and applications

Based on the number 1.618, Phi, and also known as
the Divine Proportion, Golden Proportion, Golden Mean and Golden Section


“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.”  Johannes Kepler, 1571-1630


Golden ratio in Nautilus shellGolden ratio in body of an antGolden ratio in human faceGolden ratio in the Mona Lisa

Golden ratio in violin designGolden ratio in the Taj MahalGolden ratio in Toyota logo design

What makes a single number so interesting that ancient Greeks, Renaissance artists, a 17th century astronomer and a 21st century novelist all would write about it?  This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Proportion, Golden Mean, Golden Section and Divine Proportion.  It was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in “De Divina Proportione” in 1509, by Johannes Kepler around 1600 and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.” With the movie release of the “The Da Vinci Code”, the quest to know Phi was brought even more into the mainstream of pop culture. The allure of “The Da Vinci Code” was that it creatively integrated fiction with both fact and myth from art, history, theology and mathematics, leaving the reader never really knowing what was truth and what was not. This site studies this golden number Phi, and its mathematical cousin, the Fibonacci Series, both of which have roles in the plot of this murder mystery, and distinguishes between the myth and the math.

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 and foreign currency exchange. Its appeal thus ranges from mathematicians to doctors 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.  These can be stated as quadratic equations, the only positive solution of which is:

Phi, the Golden Ratio, expressed in its formula based on the square root of 5

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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