The Golden Ratio
If a line of length x is divided into sections y and z such that x/y = y/z, the ratio of x to y or of y to z is called the golden section or golden ratio, which we will call Φ.
x
——————|———
y z
The Pythagoreans had as their symbol a regular pentagon with a line drawn between vertices to produce a star. This produced 9 triangles with another regular pentagon in the center. Each line between vertices is divided into 3 sections, two long and one short. The ratio of a long line to a short line is the golden ratio. The golden ratio can be computed from the formula:
Φ = (√5 + 1)/2
An interesting property of Φ is that Φ2 = Φ + 1. Another interesting property is that 1/Φ = Φ - 1. In other words, the reciprocal of Φ is the same as the portion of Φ to the right of the decimal point. There is no other number with this property. The number sequence 0,1,1,2,3,5,8,13,21,34... is called the Fibonacci sequence after Leonardo Pisano (Leonardo of Pisa), who is sometimes called Leonardo Fibonacci. Each number in the sequence is the sum of the previous two numbers. The last number in the sequence divided by the 2nd to the last number in the sequence approaches Φ as the sequence approaches infinity. This relationship can be stated as a formula: Φn ≤ Φn-1. This information has been used in the analysis of algorithms.
More on the Fibonacci numbers in part 4. See Roger Pink's thread on The Golden Ratio in his blog on formulas.
Have any of you used this info to analyze algorithms?
S
Number Theory & Mysticism Part 3
Re: Number Theory & Mysticism Part 3
