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A dodecahedron, a solid in which each side is a pentagon, is related to the Golden Ratio in that both the surface area and the volume of a dodecahedron of unit edge length are functions of the Golden Ratio. Plato described the dodecahedron as the solid ''which the gods used for embroidering the constellation on the whole heaven.''
A dodecahedron, a solid in which each side is a pentagon, is related to the Golden Ratio in that both the surface area and the volume of a dodecahedron of unit edge length are functions of the Golden Ratio. Plato described the dodecahedron as the solid ''which the gods used for embroidering the constellation on the whole heaven.''
==Fibonacci Numbers and the Golden Ratio==
Leonardo of Pisa (c1170 – 1250), commonly known as Fibonacci, was an Italian mathematician, who wrote an arithmetic treatise called [[Liber Abaci]] ( the book of the abacus) in 1202. In this book, Fibonacci describes a problem concerning rabbit popuations, the solution to which is the sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
The sequence can be continued by adding together the last two terms generated.
The numbers in the sequence are normally called Fibonacci numbers, and they have some remarkable properties, such as if the terms in the series are labelled with the natural numbers, then any Fibonacci number which occurs with a label which is prime, is itself prime. (for example, the 11th number in the sequence is 89. Both 11 and 89 are prime).
The relevance of this sequence here is that the ratio of the last two numbers generated at each stage in the Fibonacci sequence converges to the Golden Ratio. This can be checked:
* 1/1 = 1
* 2/2 = 2
* 3/2 = 1.5
* 5/3 = 1.6666..
* 8/5 = 1.6
* 13/8 = 1.625 etc.
It can be seen from this that successive ratios underestimate and overestimate the ratio, and the convergence is quite rapid.
For more on the sequnce, see: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
==The golden ratio in art==
==The golden ratio in art==