Fibonacci sequence

Medieval mathematician and businessman Fibonacci (Leonardo of Pisa) posed the following problem in his treatise Liber Abaci (pub. 1202):

How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?

It is easy to see that 1 pair will be produced the first month, and 1 pair also in the second month (since the new pair produced in the first month is not yet mature), and in the third month 2 pairs will be produced, one by the original pair and one by the pair which was produced in the first month. In the fourth month 3 pairs will be produced, and in the fifth month 5 pairs. After this things expand rapidly, and we get the following sequence of numbers:


This is an example of a recursive sequence, obeying the simple rule that to calculate the next term one simply sums the preceding two:

\[\begin{eqnarray*}F(1) & = & 1 \\ & & \\ F(2) & = & 1 \\ & & \\ F(n) & = & F(n-1)+F(n-2) \end{eqnarray*}\]

Thus \(1 + 1 = 2\), \(1+2=3\), \(2+3=5\), and so on.

This simple, seemingly unremarkable recursive sequence has fascinated mathematicians for centuries. Its properties illuminate an array of surprising topics, from the aesthetic doctrines of the ancient Greeks to the growth patterns of plants (not to mention populations of rabbits). Consider, for example, the following diagram:

Figure 1: Fibonacci spiral

Here we have taken squares with sides whose lengths correspond to the terms of the Fibonacci sequence, and arranged them in an “outwardly spiraling” pattern. Notice that the rectangles which result at each stage are all roughly the same shape, that is, that the ratio of length to width seems to settle down as we build the pattern outward. Notice also that the ratio of length to width is at every step the ratio of two successive terms of the Fibonacci sequence, that is, the ratio of the greater one to the lesser. These ratios may be thought of as forming a new sequence, the sequence of ratios of consecutive Fibonacci numbers:

\[\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, \frac{21}{13}, \ldots\]

This sequence converges, that is, there is a single real number which the terms of this sequence approach more and more closely, eventually arbitrarily closely. We may discover this number by exploiting the recursive definition of the Fibonacci sequence in the following way. Let us denote the \(n^{th}\) term of the sequence of ratios by \(x_n\), that is,

\[x_1=\frac{1}{1},\,\, x_2=\frac{2}{1},\,\ldots , \,\,x_n=\frac{F(n+1)}{F(n)},\, \ldots\]

Then using the recursive definition of \(F(n)\) given above, we have:

\[\begin{eqnarray*} x_n & = & \frac{F(n+1)}{F(n)} \\ & & \\ & & \\ & = & \frac{F(n)+F(n-1)}{F(n)} \\ & & \\ & & \\ & = & 1 + \frac{F(n-1)}{F(n)} \\ & & \\ & & \\ & = & 1 + \frac{1}{\frac{F(n)}{F(n-1)}} \\ & & \\ & & \\ & = & 1 = \frac{1}{x_{n-1}} \\ \end{eqnarray*} \]

Now supposing for the moment that the sequence converges to a real number \(x\) (a fact which requires proof, but we’ll leave that aside), we may observe that both \(x_n\) and \(x_{n - 1}\) have the same limit, that is,

\[\lim_{n \rightarrow \infty} x_n = \lim_{n \rightarrow \infty} x_{n-1} = x \]

Consequently, the real number \(x\) to which the sequence of ratios converges must satisfy the following equation:

\[x= 1+ \frac{1}{x} \]

This is a simple equation to solve for \(x\): it is really a quadratic equation, and its positive root is the value we are looking for:

\[\frac{1+\sqrt{5}}{2} \]

…or about 1.618. (It is not rational, so the actual decimal doesn’t repeat or terminate.)

This number was known to the ancient Greeks and was called by them the golden mean. It is usually denoted by the Greek letter \(\varphi\) (phi), and sometimes by \(\mu\) (mu). They believed that the proportion \(\varphi:1\) was the most pleasing, indeed the aesthetically perfect proportion, and much of their artwork, sculpture, and especially architecture made use of this proportion. A rectangle whose sides had this proportion was called a golden rectangle. (And that is the shape being more and more closely approximated by our “spiralling rectangles” above.)

Whether or not you agree with the Greeks’ aesthetic judgment, it’s a safe bet that Nature herself does:

Figure 2: Fibonacci Spiral in a Nautilus

The growth of this nautilus shell, like the growth of populations and many other kinds of natural “growing,” are governed by mathematical properties exhibited in the Fibonacci sequence. And not just the rate of growth, but the pattern of growth. Examine the crisscrossing spiral seed pattern in the head of a sunflower, for instance, and you will discover that the number of spirals in each direction are invariably two consecutive Fibonacci numbers.

Figure 3: Fibonacci Numbers in a Sunflower

The Fibonacci sequence makes its appearance in other ways within mathematics as well. For example, it appears as sums of oblique diagonals in Pascal’s triangle:

Figure 4: Pascal’s Triangle

The Fibonacci sequence, and recursive sequences generally, never fail to reward our study with surprising and intriguing connections between mathematics and the natural world. They are one of the great scenic byways in Platonic Realms.