# A word in your shell-like

In the mysteries and magic of the mathematical world, phi, the Golden Mean, Fibonacci's numbers and constant proportion are all enclosed in the gleaming silica of the nautilus shell. Shells, like sunflowers and the breeding patterns of rabbits, display arithmetic and geometric proportions which have fascinated mathematicians over the millennia.

"In the growth of a shell, we can conceive no simpler law than this, namely that it shall widen and lengthen in the same unvarying proportions... the shell, like the creature within it, grows in size but does not change its shape; and the existence of this constant relativity of growth, or constant similarity of form, is of the essence, and may be made the basis of the definition, of the equiangular spiral," wrote mathematician and biologist D'Arcy Thompson (1860-1948) in On Growth and Form (1917).

The nautilus grows in an equiangular spiral, defined as a monotonic curve that cuts all radii vectores at a constant angle. This mollusc's particular shape is governed by properties exhibited in the famous Fibonacci sequence (1,1,2,3,5,8,13,21...) in which each term is formed by adding the two efore. The ratios of consecutive terms can be proved to converge on one number, known as (Greek letter phi).

On each spiral the nautilus enlarges by . Phi is also known as the Golden Mean: Phi:1 is the proportion which the ancient Greeks considered to underlie all notions of beauty. Its decimal representation is 1.6180339887...

"Golden Rectangles" are those whose sides are related by , for example 13x8. If you increased the size of that Golden Rectangle by swinging the long side around one of its ends, so that you had a new long side of 21, the new short side would be 13: the new rectangle is also Golden. Creating new rectangles around each other in this way gives you - guess what? - the base pattern of a nautilus shell.

Not surprising, then, that the great 17th-century mathematician Bernoulli had "Though changed, I rise again" (Eadem mutata resurgo) engraved on his tombstone beneath an equiangular spiral, as perfectly displayed in the shell of the nautilus.

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