Phyllotaxis
the spirals are counted, not admired
Concept

Growth clock — where it appears

The rule relating a shell's coiling to time: which power of its radius the animal adds at a constant rate. It is absent from the curve, since every such rule traces the same logarithmic spiral, and present in the growth lines, whose counts in successive whorls stand in the ratio of the growth factor raised to that power.

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

One spiral at 3.20× per turn, marked at equal intervals of time under four rate laws. The curve is identical in all four panels — every mark lies on r = W^(θ/2π) exactly, whichever clock put it there — and the marks are not. Under a constant angular rate the three whorls hold 14, 13, 14 marks; under a constant length added the three whorls hold 3, 9, 29 marks; under a constant area added the three whorls hold 1, 3, 37 marks; under a constant volume added the three whorls hold 1, 1, 39 marks. The growth factor is a rate per turn of the shell's own coiling, and a turn is not a unit of time; the whole of what an animal's growth rate means is in the spacing of these marks and none of it is in the curve.

A spiral with no clock

The growth factor a shell's curve gives up is a rate per turn of the shell's own coiling, and a turn is not a unit of time. Four clocks — the aperture advancing at a constant angular rate, adding a constant length, a constant area, a constant volume — trace the identical curve: every mark one of them leaves lies on r = W to the power theta over two pi exactly, so a fit through any of them returns the same factor. What differs is where the marks are, and the difference is enormous: at 3.2 per turn the outermost of three whorls holds 33.4, 71.0, 90.3 and 97.0 per cent of the record. It also breaks the instrument. The routine that recovers a growth factor unwraps the angle by assuming successive points advance less than half a turn, and five of the twenty readings here leave gaps past that — returning 3.33 where the curve was built at 3.20, and 8.19 where it was built at 6.85.

shells · Spiral The ratio of two whorls' line counts is the growth factor raised to the clock's own power. Each curve is W^p for one law: flat for a clock that advances the angle at a constant rate, W for one that adds a constant length at the opening, W² for a constant area and W³ for a constant volume. At 3.20× per turn those are 1.00, 3.20, 10.24, 32.77. So a count of lines in two successive whorls, divided, and read back through a growth factor the curve already gives, names the law — and the four are further apart the faster the shell expands.

What the growth lines carry

A shell's curve says nothing about how fast the animal grew, and its growth lines say all of it. Under a law that holds the pth power of the radius constant per unit time, the time spent crossing one whorl is proportional to the change in that power across it — so the lines in successive whorls stand in the ratio of the growth factor raised to p, and each whorl holds the count the closed form predicts to within the one line rounding can move. Dividing two counts and taking the logarithm against a growth factor the curve already gives returns p: 17 of 20 readings name their own law, the furthest 0.012 from a whole number. The other three are not wrong, they are uncountable — at 4.5 per turn under a volume clock the inner whorl of the pair holds one line.

shells · Spiral

Named alongside it

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

Claim testingGrowth factorHonest limitsLogarithmic spiralMeasurementWhorlClosed formIdentifiabilityInstrument settingModel scopeRefusalResolution

All concepts