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The Divine Proportion, matrices and Fibonacci numbers

  • H. Brian Griffiths (a1)

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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.

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References

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

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The Divine Proportion, matrices and Fibonacci numbers

  • H. Brian Griffiths (a1)

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