What Is the Fibonacci Sequence?
By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.
The Fibonacci sequence is a list of numbers where each one is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21 and onwards. It genuinely appears in some parts of nature and is genuinely connected to the golden ratio, and it is also the subject of a great deal of exaggeration.
The rule, and where it came from
The rule is simply that each term equals the sum of the previous two. Starting from 1 and 1 gives 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on, growing steadily faster.
It reached European mathematics through Leonardo of Pisa, known as Fibonacci, in a book published in 1202, where it appeared as a puzzle about breeding rabbits. The sequence was already known in Indian mathematics centuries earlier, in the context of counting patterns of long and short syllables in poetry.
The connection to the golden ratio
Divide any Fibonacci number by the one before it and the answer gets closer and closer to about 1.618, the golden ratio. The further along the sequence you go, the more precise the approximation becomes.
This is a real mathematical property with a proof behind it, not a coincidence or a mystical claim. It follows directly from the structure of the recurrence relation that defines the sequence.
Where it genuinely appears in nature
Some of the natural examples hold up well:
- •The number of spirals in a sunflower head is usually a Fibonacci number
- •Pine cones and pineapples show Fibonacci spiral counts
- •The arrangement of leaves around a stem often follows Fibonacci fractions
- •Many flowers have petal counts of 3, 5, 8, 13 or 21
Why plants do this
The reason is not mystical. Leaves and seeds grow at a consistent angle from the previous one, and an angle related to the golden ratio packs them most efficiently without any two lining up and shading each other.
The Fibonacci numbers appear because they emerge naturally from that packing process. This has been modelled mathematically and reproduced in physical experiments, so the mechanism is understood rather than merely observed.
Where the claims go too far
A lot of popular material about Fibonacci is unreliable. The claim that nautilus shells follow a golden spiral is not supported by measurement, since their growth ratio is typically nowhere near 1.618.
Claims about the golden ratio in the Parthenon, the pyramids and Renaissance paintings usually rely on selectively chosen measurement points, and there is no documentary evidence the builders used it. Similar caution applies to claims about stock markets and human faces. The genuine mathematics is interesting enough without the additions.
Where it appears in computing
The sequence has practical uses beyond curiosity. Fibonacci numbers appear in the analysis of certain algorithms and data structures, including Fibonacci heaps, which are used in some shortest path implementations.
Fibonacci search is a technique for locating a value in a sorted array using only addition and subtraction, which was useful on hardware where division was expensive. The sequence also serves as the standard teaching example for recursion, since its definition is naturally recursive, and as the standard demonstration of why naive recursion without memoisation can be catastrophically slow.
The takeaway
The Fibonacci sequence adds each pair of terms to get the next, its ratios approach the golden ratio, and it appears in plant growth for a well understood packing reason, while many popular claims about it in art and architecture do not survive scrutiny.