Fibonacci & the Golden Ratio
A pattern born from numbers
Summary
The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34… Each number is the sum of the two before it. This plain rule makes the ratio of consecutive numbers approach a constant called the golden ratio (φ ≈ 1.618). Squares drawn with the same numbers form a spiral, one that turns up all over nature.
In depth
In 1202 Leonardo of Pisa (Fibonacci) presented this sequence in his book "Liber Abaci" to solve a problem about breeding rabbits. Each number is the sum of the two before it: F(n) = F(n-1) + F(n-2), with F(1) = F(2) = 1. This simple recursion leads to a surprising constant: the ratio of consecutive numbers converges to the golden ratio φ = (1+√5)/2 ≈ 1.618. The structure also has a geometric side: squares whose sides are Fibonacci numbers fit together with the earlier ones into a spiralling pattern. This "golden spiral" is echoed in many places in nature: sunflower seeds, the curl of a seashell, the scales of a pine cone, and the arrangement of leaves on a plant (phyllotaxis). Mathematics explains part of this as a kind of packing optimisation: the golden angle (137.5°) spaces leaves and seeds so that they overlap as little as possible.
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