Sir Isaac Newton

Isaac Newton Institute for Mathematical Sciences


Posters in the London Underground

January 2000: Maths Counts
(click on the image for larger version)

So, did you spot the pattern? To get the next number in the sequence you add together the previous two. We need two numbers to start the sequence off, and in the case of the Fibonacci sequence we start with 1 and 1.

Leonardo Fibonacci was born in Pisa in the 12th century. He was a merchant and customs officer of the time, travelling widely in North Africa. He was also one of the first Europeans to learn about the Arabic numbers 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 and to persuade other people to use them; before then everybody counted in 12's.

Leonardo was trying to find a way of modelling the population of rabbits. Let us suppose that any new pair of rabbits produces one pair in the next breeding season and one in the season after that, and then they die. This means that the total number of new pairs in a given season is equal to the number of new pairs born in the previous season, plus the number born in the season before that. So to find the next number in the sequence you add together the last number and the one before it. Starting with one pair of rabbits, you can easily generate the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... - the population of rabbits grows very quickly - actually exponentially fast!

The surprising thing about Fibonacci's sequence is that it turns out to occur in many different places in nature. The way in which the spiral patterns of sunflower seeds and pine cones grow is described by the sequence, and it is common for the number of petals on a flower to be a Fibonacci number. Four-leaved clovers are rarer than five-leaved ones because five is in Fibonacci's sequence and four isn't!

If you take the ratios of successive Fibonacci numbers - 1/2, 2/3, 3/5,5/8, 8/13, 13/21, 21/34, and so on - the fractions get closer and closer to the Golden Ratio (or Golden Section), , which is about 0.61803.... To be precise,

.

The Golden Ratio is a very special number, and has been known about since Greek times. Paintings with a height to width ratio of  have an especially aesthetically pleasing aspect. The Parthenon in Athens has the same ratio of lengths. The five-pointed stars on many flags of the world (for example, the European flag) are made by cutting the diagonals of a pentagon according to the Golden Ratio. The Ratio is sometimes called the "divine proportion", which is particularly apt as many religious paintings use it.

The beauty - and the usefulness - of mathematics depends in part on the way in which it can link together many apparently unrelated topics (for example, rabbits, sunflowers and painting) in a coherent whole. The worlds of nature and science are founded on a bedrock of mathematics.

Further reading:

  • The life and numbers of Fibonacci at the PASS Maths website
  • Fibonacci numbers and the Golden Section at the University of Surrey
  • Leonardo Fibonacci at the MacTutor History of Mathematics Archive
  • Fibonacci Number at Eric Weisstein's World of Mathematics
  • Home page of the Fibonacci Quarterly Journal

  • The Mathematics of Fibonacci's Sequence

    The Fibonacci sequence is defined by the property that each number in the sequence is the sum of the previous two numbers; to get started, the first two numbers must be specified, and these are usually taken to be 1 and 1. In mathematical notation, if the sequence is written $(x_0, x_1,x_2,...)$ then the defining relationship is

    \begin{displaymath}x_n=x_{n-1}+x_{n-2}\qquad (n=2,3,4...)\end{displaymath}

    with starting conditions $x_0=1,
x_1=1$. On dividing both sides of (1) by $x_{n-1}$, we obtain $1/R_n=1+R_{n-1}$ where $R_n=x_{n-1}/x_n$, the ratio of successive terms.

    As $n \to \infty, R_n \to R$ where $1/R= 1+R$, or $R^2+R-1=0.$ This quadratic equation has two roots; the one we need here is obviously between zero and one; it is

    \begin{displaymath}R={{\sqrt
5-1}\over 2}=0.61803...,\end{displaymath}
    the number known as the Golden Ratio.

    The number $R$ has some remarkable properties; for example, it is expressible as a "continued fraction'':

    \begin{displaymath}R={1\over\displaystyle 1+
{\strut 1\over\displaystyle 1+  {\strut 1\over\displaystyle 1+  {\strut ...}}}}\end{displaymath}

    In the theory of chaotic dynamical systems, is recognised as "the most irrational number" between 0 and 1!

    The spiral curve shown in the poster is a logarithmic spiral, a curve whose equation in polar coordinates is $r=ke^{a\theta}$where k and a are constants. The spiral patterns evident in the sunflower are of this form, and the numbers of spirals going in opposite senses are the conscutive Fibonaci numbers 34 and 55. The underlying reason for this may be found in many texts; see for example Conway JH and Guy RK The Book of Numbers, Springer-Verlag (1996), chapter 4.


    Credits:
  • Concept, graphic design, and background fractals: A D Burbanks
  • Sunflower photo: HK Moffatt
  • Poster text: Based on suggestions of B Krauskopf and A D Burbanks
  • Website text: RE Hunt, A D Burbanks and HK Moffatt

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