golden ratio

The “Golden RatioMathworldPlanetmath”, or ϕ, has the value


This number gets its rather illustrious name from the fact that the Greeks thought that a rectangle with ratio of side lengths equal to ϕ was the most pleasing to the eye, and much of classical Greek architecture is based on this premiseMathworldPlanetmath. In , an aesthetically pleasing aspect of a rectangle with this ratio, from a mathematical viewpoint, is that if we embed and remove a w×w square in the below diagram, the remaining rectangle also has a width-to-length ratio of ϕ.

Above: The golden rectangle; l/w=ϕ.

ϕ has plenty of interesting mathematical , however. Its value is exactly


The value


is often called ϕ. ϕ and ϕ are the two roots of the recurrence relation given by the Fibonacci sequenceMathworldPlanetmath. The following hold for ϕ and ϕ :

  • 1ϕ=-ϕ

  • 1-ϕ=ϕ

  • 1ϕ=-ϕ

  • 1-ϕ=ϕ

and so on. These give us


which implies

Title golden ratio
Canonical name GoldenRatio
Date of creation 2013-03-22 11:56:02
Last modified on 2013-03-22 11:56:02
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 17
Author Mathprof (13753)
Entry type Definition
Classification msc 40A05
Classification msc 11B39
Synonym golden number
Related topic ProportionEquation
Related topic ConstructionOfCentralProportion
Related topic DerivationOfPlasticNumber