Phyllotaxis
the spirals are counted, not admired
Concept

Golden spiral — where it appears

A logarithmic spiral whose radius grows by the fourth power of the golden ratio — 6.854 — in one turn. It is the curve popular accounts say a nautilus follows, and measured sections give about 3.2, so the name is a claim about a number rather than a description of a shell.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A golden spiral and a nautilus spiral over 2.5 turns, from the same start. After 2.5 turns the golden curve is 7× larger. The growth factors are 6.85 and 3.2, a factor of 2.14 apart.

The nautilus question

A golden spiral grows by 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14 — not a rounding error, not an artefact of where the centre is assumed to be, and not close.

shells · Nautilus Every way a growth factor read off a section can be wrong, against the gap it has to clear. Each bar is a worst case computed by the library that measured it. dividers 278.2%, span 22.4%, septum 9.14%, ontogeny 8.99%, centre 6.88%, clock 4.06%, oblique 2.95%. The line is the gap the golden claim asks the measurement to resolve: 6.854 against 3.2 is 114.2 per cent. Added without cancellation the 7 sources come to 332.6%, which is outside that gap — so the claim is NOT refused by a section read carelessly.

The error budget for a nautilus

Every way a growth factor read off a shell section can be wrong has been priced here, one essay at a time. Added up they come to 332.6 per cent in the worst case and 279.6 in quadrature, against a golden-spiral claim that is 114.2 per cent away — so the budget does not refuse the claim at all. One entry decides it: the dividers, at 278.2 per cent on their own, and the dividers are the historical method and the only route measured that pushes a nautilus towards a golden spiral. Set them aside and the budget falls to 54.4 per cent and the claim is refused twice over. What the same budget cannot settle is anything smaller than half: 3.2 against 3.4 is inside it, and stays inside it until six of the seven sources are controlled.

shells · Nautilus

Named alongside it

The objects these essays reach for when they reach for this one.

Claim testingGrowth factorBranchError propagationFibonacciφ, the golden ratioHonest limitsIdentifiabilityLogarithmic spiralMeasurement errorMeasurement sensitivityModel scope

All concepts