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 5 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 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 A pair of dividers opened to 1 of the innermost radius, walked along two and a half turns and along four. A nautilus spiral growing 3.2 times a turn, stepped at a fixed opening of 1 of its innermost radius from the inner end outward. Over 2.5 turns the walk takes 17 steps against a floor of 22 and reads the factor +18.2% too high; Over 4 turns the walk takes 104 steps against a floor of 132 and reads the factor +4.2% too high. The opening at which every span reads exactly is the square root of 3.2 less one, 0.789 of the innermost radius: under it the first step never carries the angle past half a turn, whatever the span.

The dividers belong to the opening

The nautilus error budget's largest entry, dividers walked along the shell at 278 per cent, was set aside as the historical method. Priced at one span it turns out not to belong to the span at all. A person sets a pair of dividers to an opening and walks until the curve runs out, so the step count grows with the arc exactly as fast as the floor on it does: opened to less than the square root of the growth factor less one — 0.789 of the innermost radius for a nautilus — they read the factor exactly over every span from half a turn to six, and opened wider they read it too high over every span, least over the longest. The 278 per cent was nine steps along five turns, an opening of 37 innermost radii. With the dividers opened to anything up to five radii, the whole budget refuses the golden spiral from three quarters of a turn upward.

shells · Nautilus A pair of dividers opened to 1/50 of a shell's outer radius, walked over the outer 3 of its six turns. A nautilus spiral of six turns growing 3.2 times a turn. The section is taken to preserve only its outer 3 turns; the lost inner whorls are drawn faint. Dividers opened to 1/50 of the outer radius walk from the innermost preserved whorl to the rim in 48 steps; the right panel magnifies the first steps and the whorls inside them. In radii of the whorl the walk starts on the opening is 0.66, against the safe 0.789, so the walk reads the factor exactly. A walk at this opening is exact over any span shorter than 3.16 turns.

The rim sets the opening

A pair of dividers reads a nautilus's growth factor exactly when its opening is under 0.789 of the radius of the whorl it starts on — a quarter of a millimetre at the true centre of a real shell, which no hand can set. A real section starts where its whorls can be read, and a person sets the dividers against the shell in front of them. Measured that way, the rule becomes a span: dividers opened to a share f of the outer radius are exact over the last log((√k − 1)/f)/log k turns of any shell — 3.76 turns at a hundredth, 3.16 at a fiftieth — and the change-over falls exactly there at every opening tried. Held against the rim, their error grows with the span rather than falling, so a section with its centre broken away is read more exactly, not less; and the whole budget still refuses the golden spiral at every span from three quarters of a turn to six for any opening up to a fiftieth.

shells · Nautilus

Named alongside it

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

Growth factorClaim testingError propagationHonest limitsLogarithmic spiralMeasurement errorModel scopeMeasurement sensitivityWhorlBranchFibonacciφ, the golden ratio

All concepts