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Golden Ratio
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Golden Ratio

The golden ratio, so named because it has been believed to possess inherent beauty, is the ratio between two values a and b, such that .

Basic Description

Geometric

From a geometric sense, the golden ratio can be represented using a line segment divided into two sections, of lengths a and b, respectively. If a and b are appropriately chosen, the ratio of b to a is the same as the ratio of a + b to a.

The value of this ratio, denoted , turns out not to depend on the particular values of a and b, as long as they satisfy the proportion above.

Algebraic

We may algebraically solve for the ratio by observing that ratio satisfies the property

This ratio is denoted by the Greek letter phi, .

By using the expression , we may substitute in and simplify the expression to one containing only and constants.

Using the quadratic formula An equation, , which produces the solutions of equations of form , we are able to find the positive solution, . This algebra shows that if two numbers a and b satisfy the proportion described earlier, then this is the only value the ratio can have.

Applications

Triangles

Phi is used to construct the golden triangle, an isoceles triangle that has legs of length and base length of 1. Similarly, the golden gnomon has base and legs of length 1.

These triangles can be used to from a pentagram, which has several golden ratio proportions.

These triangles can be used to form fractals and are one of the only ways to tile a plane using pentagonal symmetry.

History

The Greeks were aware of this ratio, but did not consider it particularly significant with respect to aesthetics. It was not called the "divine" proportion until the 15th century, and was not called "golden" ratio until the 18th century.

Since then, myths have suggested that the golden ratio is the most aesthetically pleasing ratio, and subtly incorporated into architecture and art throughout history. Among the most common are the Parthenon and Leonardo Da Vinci's Mona Lisa's incorporation of the ratio. Even more esoteric myths propose that the golden ratio can be found in the human facial structure, the behavior of the stock market, and the pyramids.

Mathematicians have solidly debunked these ideas as wishful thinking or sloppy math analysis. Additionally, there is no solid evidence that supports the claim that the golden rectangle is the most aesthetically pleasing

However, some artists have used it explicitly, such as Salvador Dalí in his painting The Sacrament of the Last Supper.

A More Mathematical Explanation

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Properties

Infinite Fraction

The golden ratio can also be written as a infinite fraction by using recursion.

If we evaluate truncations of the continued fraction by evaluating only part of the continued fraction and replacing by 1, we produce the ratios between consecutive terms in the Fibonacci sequence.

Thus we discover that the golden ratio is approximated in the Fibonacci sequence.

As the Fibonacci sequence approaches infinite length, the ratio between the consecutive terms approaches the golden ratio.

Proof of the Golden Ratio's Irrationality

We will use the method of contradiction to prove that the golden ratio is irrational.

We start with the equation that defines the ratio of . We will assume variables a and b are integers.

(1)

If a and b are integers, their sum c must be an integer as well.

We then substitute this expression for a into our original expression.

(2)

Here we arrive at our contradiction. If b and a are both integers, then it follows that is a rational number. However if b and c are both integers, it must be possible to arrive at a fraction of lowest terms. When we simplify the fraction to lowest terms, say , expression (1) still holds.

If a fraction is in lowest terms it cannot have any equivalent fractions with a smaller denominator. Thus we have shown that for to be rational would be impossible, and therefore it must be irrational.

For More Information

Markowsky. “Misconceptions about the Golden Ratio.” College Mathematics Journal. Vol 23, No 1 (1992). pp 2-19.

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