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Phi: A Magical Number

By Daniel Taylor on June 19, 2011 04:06

Have you ever heard of phi? No, I am not talking about fee-fi-fo-fum from Jack and the Beanstalk. The phi (pronounced fi like "fly") I'm talking about the mathematical constant with properties just as magical as the beans in Jack and the Beanstalk. This constant is referred to by many names including the Golden Ratio, Golden Number, Golden Section, Golden Mean, and Devine Number, but is normally referred to as the Golden ratio because of how the number is derived mathematically.

Have you ever heard of phi? No, I am not talking about fee-fi-fo-fum from Jack and the Beanstalk. The phi (pronounced fi like "fly") I'm talking about the mathematical constant with properties just as magical as the beans in Jack and the Beanstalk. This constant is referred to by many names including the Golden Ratio, Golden Number, Golden Section, Golden Mean, and Devine Number, but is normally referred to as the Golden ratio because of how the number is derived mathematically.

Phi is an irrational number similar to pi. The golden ratio can be represented as a continued fraction or an infinitely recursive function.

Given a rectangle with sides of length phi and 1, phi can be defined as a number such that the partitioning of the given rectangle into a square with sides of length 1 and a new rectangle yields a new rectangle with the same ratio as the original rectangle. This may be better explained in a picture.

As it turns out, phi is exactly (1 + 51/2)/2 which is approximately equal to 1.61800339887499. Phi can also be represented trigonometrically as 2sin(54°).

You might be saying to yourself, "Well, that doesn't seem very magical to me. It's just a number." In reply to that, I must say that it isn't the number itself that is interesting. It is the fact that the number is everywhere, whether you realize it or not.

The Fibonacci sequence follows this magical ratio. The first 11 numbers of the Fibonacci sequence are as follows: 1, 1, 2, 3, 5, 8, 13, 21, 24, 55, 89. By dividing one number by the previous number, you get a number close to the golden ratio. As you progress farther to the right, you get closer to the golden ratio. The sequence converges on the ratio.

8 / 5 = 1.6000
13 / 8 = 1.6250
34 / 21 = 1.6190
89 / 55 = 1.6180

The Parthenon in Greece follows this ratio, Leonardo da Vinci's illustrations of Polyhedra and the Mona Lisa follow the ratio. The ratio of the size of most books is close to 1.6. Everything from postcards, playing cards, posters, wide-screen televisions, and photographs have a close resemblance to the golden ratio. The magnetic resonance of some cobalt isotopes is 1.6. Animal shells often resemble the golden spiral, which is derived from the golden ratio. The ratio of the first bone in your finger to the second bone is 1.6. The egyptian pyramids have a very close semblance to the golden ratio. The opening of Beethoven's Fifth resembles the golden ratio. The size of galaxies follow the golden ratio to some degree. Even your heartbeat has ties to the golden ratio. Phi truly is a magical number.