Phyllotaxis
the spirals are counted, not admired
Concept

Measurement sensitivity — where it appears

How much a reading moves when one of the conditions it was taken under moves, as against how much it scatters when nothing about them changes. A growth factor fitted to a nautilus section is sensitive to its assumed centre roughly as the inverse square of the span of arc, so one offset costs a half-turn fragment sixty times what it costs three turns.

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 The nautilus error budget with its three span-dependent entries priced at one span, against the gap the golden claim asks it to clear. The budget with the dividers set aside, but with its centre, span and oblique entries replaced by the worst joint error of a quarter-radius centre offset and a ten-degree tilt at one span of arc. 0.5 of a turn: 210.4%; 0.75 of a turn: 111.5%; one turn: 61.0%; 1.5 turns: 31.9%; 2 turns: 29.3%; 3 turns: 25.1%. The upper level line is the gap, 114.2%; the lower is the budget as it was priced, 54.4%, with its three coupled entries at 32.2% together. The budget clears the gap from 0.75 of a turn and not below it.

Three entries and one span

The error budget for a nautilus added seven ways a growth factor read off a section can be wrong, and asked whether they were independent. Three of them are not: the displaced centre, the span of arc and the oblique view are one error priced three ways, at two turns, over a turn and more, and at half a turn — 32.2 per cent together. A section has one span. Read together at one span, a quarter-radius centre and a ten-degree tilt come to 7.1 per cent at two turns and 188 at half a turn, and in their worst orientation they always add to more than their sum. So the budget refuses the golden spiral from three quarters of a turn of shell upward, and below that it cannot.

shells · Nautilus

Named alongside it

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

Claim testingGolden spiralGrowth factorLogarithmic spiralBranchError propagationFibonacciφ, the golden ratioHonest limitsMeasurement errorModel scopeφ⁴

All concepts