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The Divine Proportion, matrices and Fibonacci numbers

Published online by Cambridge University Press:  01 August 2016

H. Brian Griffiths
H. Brian Griffiths*
Affiliation:
35 Glen Eyre Road, Bassett, Southampton, SO16 3GA
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Extract

A friendly engineer recently sent me a version of Figure 1 below, and asked me to explain the connection between it and the well-known Fibonacci Problem (FP) about calculating a population of rabbits. He knew that Figure 1 was related to the ‘Divine Proportion’ or ‘Golden Ratio’ ϕ ( = ½(√5 - l), which also occurs in the solution to FP, and wondered how such different problems could be related by such a number. (He unfortunately regarded ϕ as 0.618 exactly, thus missing a lot of stuff to arouse curiosity.) I knew of various references that I could recommend, but none covered all the things my engineer mentioned, so I constructed the following mathematical development, leaving the relations with biology and architecture to be explained in the books referenced later, see for example [1]. A matrix approach is used here, and may be new to those readers of the Gazette who may be quite familiar with the other material.

Type
Articles
Information
The Mathematical Gazette , Volume 92 , Issue 523 , March 2008 , pp. 14 - 21
Copyright
Copyright © The Mathematical Association 2008

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References

1. Thompson, D’Arcy, On growth and form (abridged edition, edited by Bonner, J.T.), Cambridge University Press (1961).Google Scholar
2. Griffiths, H.B. and Oldknow, A., Mathematics of models. Ellis Horwood, Chichester (1993).Google Scholar
3. Coxeter, H.S.M., Introduction to geometry, Wiley, London (1963).Google Scholar