golden ratio
The “Golden Ratio”, or ϕ, has the value
1.61803398874989484820… |
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 premise. 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
1+√52 |
The value
1-√52 |
is often called ϕ′. ϕ and ϕ′ are the two roots of the recurrence relation given by the Fibonacci sequence. The following hold for ϕ and ϕ′ :
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•
1ϕ=-ϕ′
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•
1-ϕ=ϕ′
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•
1ϕ′=-ϕ
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•
1-ϕ′=ϕ
and so on. These give us
ϕ-1+ϕ0=ϕ1 |
which implies
ϕn-1+ϕn=ϕn+1 |
Title | golden ratio |
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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 |