The Fibonacci sequence explained, without the mysticism

The Fibonacci sequence is the easiest piece of mathematics to state and one of the most over-claimed. Each term is the sum of the two before it, which gives 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. That is the entire definition. Everything interesting follows from it, and so does everything dubious.

Where it comes from

The name is European and the sequence is older. Leonardo of Pisa, known later as Fibonacci, introduced it to Latin readers in the Liber Abaci of 1202 through a problem about rabbit populations: start with one breeding pair, assume each pair produces a new pair every month from its second month, and count pairs. The count follows the sequence. The rabbit model is biologically useless — nothing dies in it — but the arithmetic is the point.

Indian scholars had the sequence several centuries earlier, arising from a genuinely natural question in prosody: how many ways can a line of a given length be made from short and long syllables, where a long counts as two shorts? The answer is a Fibonacci number, and the reason is the same recursion — a line either ends in a short or in a long, so the count is the sum of the two shorter cases.

The ratio, and why it converges

Divide each term by its predecessor: 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619. The values alternate above and below and close in on 1.6180339887 — phi. This is not a numerical accident. If the ratios converge at all, the limit must satisfy the sequence’s own recursion, which gives x = 1 + 1/x, and therefore x² = x + 1. That is the golden ratio equation, and phi is its positive root.

The consequence worth carrying away is about approximation. Consecutive Fibonacci ratios are the best rational approximations to phi at their size — the near-misses. This is why they show up in plant growth: a plant placing organs at the golden angle produces visible spiral arms in Fibonacci counts, because those are the fractions that almost, but never quite, cause florets to align.

What the sequence does not explain

Fibonacci numbers appear in phyllotaxis and in a handful of combinatorial problems with the same recursive structure. Beyond that the record thins out quickly, and most popular examples fail one of three ways.

The first is selective counting. Anything with a countable number of parts will occasionally hit a Fibonacci number, because the small terms — 1, 2, 3, 5, 8 — are common counts. Five petals is not evidence.

The second is the shell. The nautilus is the most repeated example and it is wrong: the nautilus grows in a logarithmic spiral, but not one with a growth factor of phi. The measured factor is closer to 1.31, and the pictures that appear to match have had a golden spiral drawn over them at a convenient scale.

The third is markets. Fibonacci retracement levels in technical analysis are widely used and have no mechanism behind them, and a sequence that grows this slowly will land near almost any set of levels you care to nominate.

Two properties worth knowing

The sequence has a closed form. Binet’s formula gives the nth term directly from phi, without computing everything before it, and it works because the recursion has two roots — phi and its negative reciprocal — of which the second shrinks so fast that for practical purposes the nth Fibonacci number is phi to the nth power divided by the square root of five, rounded.

It also has a divisibility structure that is easy to check and surprising. Every third term is even, every fourth is divisible by three, every fifth by five. More generally, the greatest common divisor of two Fibonacci numbers is the Fibonacci number at the position given by the greatest common divisor of their positions — a tidy result and a reminder that the sequence is a piece of number theory, not a curiosity.

None of this diminishes the sequence. It makes the genuine case — placement at a fixed angle in a growing organism — considerably more interesting, because it is the one place where the recursion is doing real work rather than being pattern-matched onto a picture afterwards.