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 7 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 A shell whose deposition law changes part way through, beside one that never does. Both panels are the same logarithmic spiral at 3.20 per turn, marked at equal intervals of time. On the left the animal holds a constant angular rate for the first 3.50 whorls and a constant length added after that; on the right it holds a constant length added throughout. The curves are identical to the last bit a double holds, because a curve records no clock at all. The counts per whorl are not: 56, 57, 56, 67, 191, 613, 1960 against the unchanged shell's, and the change is in where the marks crowd rather than in where the shell goes.

A shell that changed its law

An animal that grew as a juvenile under one deposition law and as an adult under another leaves a sequence of whorl ratios rather than one, and the sequence says where the change happened. The ratio across the change is a closed form that is neither law's — 6.72 between a length clock and an area clock at 3.2 per turn, exactly the average of 3.2 and 10.24 — and it is monotone in where inside its whorl the change sits, so it inverts. On a seven-whorl shell of 18,466 lines a change at 3.5 whorls comes back at 3.5001, in a band 0.027 whorls wide that holds the true position. The reading refuses a change in the outer three whorls or the inner three, because a plateau it will trust is two agreeing ratios and two ratios need three untouched whorls.

shells · Spiral A step and a drift between the same two laws, as sequences. Two shells, both starting at a constant angular rate and ending at a constant area added over 6 whorls. The stepped one changes at a single position and its sequence is flat, crossed, flat. The drifting one changes evenly and its sequence is a straight ramp. The largest difference between them is 0.624 in power, against a rounding of 0.0055 — so what separates a step from a drift is the shape of the sequence and never any one of its ratios.

A law that never stopped changing

A shell whose deposition law moved evenly from one end to the other gives a sequence of whorl ratios that is a straight ramp rather than a plateau, a crossing and a plateau, and the two are separated by more than the counts' own rounding on every shell holding three countable ratios. Each ratio on the ramp reads the law at the boundary it straddles — 0.3486, 0.6865, 1.0320, 1.3755, 1.7180 against 0.3333, 0.6667, 1.0000, 1.3333, 1.6667 — so the reading is local where a fit to the curve is global, and a fit handed the same shell returns the geometric mean of its ends with no warning. The reading that locates a single change refuses a drifting shell at every size, naming the number of ratios that agree with neither end.

shells · Spiral Every law but one is pulled towards a length clock. The power a shell's outermost countable pair names, against how coarsely its lines can be told apart. A length clock sits flat at one however much of its record is lost. Every other law bends towards it: an angular clock reads 0.993 where the shell had 0 and a volume clock 1.083 where it had 3. The reason is a closed form: where the limit binds completely the surviving count in a whorl is that whorl's arc over the limit, and a logarithmic spiral's whorl arcs stand in the ratio of the growth factor exactly. So a shell too worn to read reports the law of its own geometry.

A count that is not exact

Reading a deposition law off two whorls' growth-line counts divides one by the other, so a miscount that is the same in both divides out: four lines in five missed at random moves the answer by five thousandths and costs only scatter. What biases it is a miscount that varies along the shell, and there is one that always does. The arc between successive lines rises or falls with the radius according to whether the law is shallower or steeper than a length clock, so a section's resolution limit eats the inner whorls of a shallow shell and the outer whorls of a steep one, and eats evenly at exactly p = 1. Where the limit binds, a whorl's surviving count is its arc over the limit, and whorl arcs stand in the ratio W — so a shell too worn to read reports a length clock whatever law it had.

shells · Spiral How irregular successive growth-line spacings are in each whorl, for a section with no limit and two with one. For each whorl holding at least 20 resolved lines, the median over 40 sections of the spread of the logarithm of each spacing over the one before — how irregular successive spacings are, whatever the trend along the whorl. Length clock, days varying by 0.1, no limit: 0.140, 0.140, 0.143, 0.142, reading p = 1.00 and flagged on 0 of 40 sections. Volume clock, days varying by 0.1, limit 0.003: too few lines, too few lines, 0.146, 0.139, reading p = 1.87 and flagged on 0 of 40 sections. Volume clock, days varying by 0.3, limit 0.003: too few lines, too few lines, 0.424, 0.190, reading p = 1.87 and flagged on 40 of 40 sections. A test comparing the whorls flags a section when one whorl is much more regular than another, which a limit produces and one animal does not.

Whether a section can see its own limit

A shell section that cannot resolve growth lines closer than some distance reads every deposition law as nearer a length clock, and says nothing about it. Given an animal whose days vary, the section can often catch itself: a limit changes how irregular successive spacings are in the whorls where it binds, and one animal is not steady in one whorl and irregular in the next. On an angular clock a test comparing whorls flags every limited section, before the reading has even moved. On a volume clock it can miss a limit that has pulled the reading from 3 to 1.87 — when the animal's own days vary by a tenth, which is exactly as irregular as the limit leaves the whorl it binds.

shells · Spiral The floor test on one section: whether the shortest spacings follow the spacing around them. The outer whorl of one section cut into 16 runs of equal count; each run drawn at its median spacing, relative to the section's median run, and at its shortest spacing divided by the median of the 41 spacings around it. Filled: a volume clock read through a limit of 0.003 of the outer radius, a floor of 0.315 in length, reading p = 1.87; rank correlation -0.74, one-sided p 0.001. Open: an unlimited shell whose law drifted from 3 to 1.57, reading p = 1.87; rank correlation 0.12, p 0.675. An animal's short days are short in proportion to the spacing around them, so its divided floors do not move with the run's spacing; a limit's floor is one length, so where the spacing is wider the divided floor is lower. Both from an animal whose days vary by a tenth.

A limit is a length, and a short day is a share

A shell section that cannot resolve its finest growth lines names a steep law as a shallower one, and so does a shell whose animal really did change its law. Matched to name the same law from the same whorls, the two agree in their counts, and the test that compares how irregular each whorl is goes blind on both across the same band of limits. What separates them is what a limit is. Every spacing it leaves is at least one length, wherever on the whorl it sits, while an animal's short days are short in proportion to the spacing around them. Asked whether the shortest spacings along the outer whorl follow the local spacing, forty sections of a drifting or stepped law are flagged at most twice, and a limited volume clock in the blind band thirty-three to forty times.

shells · Spiral

Named alongside it

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

Claim testingGrowth factorHonest limitsWhorlResolutionSilent failureRefusalClosed formIdentifiabilityMeasurement errorSystematic errorLogarithmic spiral

All concepts