A spiral with no clock

Growth as a rule settled what a logarithmic spiral is: not a shape somebody admired, but what a thing grows into when it adds material at its opening without changing shape. Its one parameter is how much it grows per turn, and that parameter comes back out of any drawn curve to the last digit.

Every essay since has called that parameter a growth rate, and it is not one. It is a rate per turn of the shell’s own coiling, and a turn is not a unit of time. The curve is $r = W^{\theta/2\pi}$ and nothing in it says anything about $\theta(t)$ — which is the whole of what an animal’s growth rate means.

Four clocks a shell could keep #

An animal adding material at its aperture is doing something at a rate, and there are several plausible things the rate could be constant in. The aperture could sweep a constant angle per unit time. It could add a constant length of shell edge. It could lay down a constant area of new wall. It could deposit a constant volume of material.

Each of those is a statement that some power of the radius changes at a constant rate: the angle itself for the first, then the radius, its square and its cube. So the family is one number wide, and the four named laws are the whole powers in it.

Why the family is one number wide #

The four are not a list somebody chose; they are the whole of what a self-similar clock can be. A shell growing without changing shape has one length scale, its own radius, and every quantity a depositing animal could hold constant per unit time is some power of that scale — an angle is the zeroth power, an edge length the first, a wall area the second, a volume the third. A law holding r^1.5 constant is perfectly well defined and sits between two of the named ones; it is just not anything an animal is doing.

So the clock is one number, p, exactly as the shape is one number, W. Two parameters, and the curve carries one of them.

Every one of them draws the same curve #

Six hundred marks laid at equal intervals of time under each law, at each of five growth factors, and every mark lies on $r = W^{\theta/2\pi}$ — not to within a tolerance, but exactly, because the radius is computed from the angle by the same formula whatever set the angle.

That is worth saying plainly because it is the whole argument. The curve is a locus; a clock decides which points of the locus an animal occupied at which moment, and a fossil is the locus. That is the same distinction a measurement in steps had to make about dividers walked along a shell: where the instrument puts its points is a choice, and here the animal made it.

And the fitted factor is the same too #

A fit through each clock’s marks returns the same growth factor, to nine decimal places, wherever the fit works at all. A shell that took two years to reach maturity and one that took twenty are the same curve if their proportions are the same, and the measurement several earlier readings have refined is a measurement of proportion.

What that costs the argument about the nautilus #

The nautilus question is an argument about a number: a golden spiral grows by 6.854 per turn and measured nautilus sections give about 3.2, which is a factor of 2.14 and not a rounding error. Nothing here disturbs that, because both sides of it are statements about proportion.

What it does disturb is the language around it. A nautilus growing by 3.2 per turn is not growing more slowly than a golden spiral would; it is growing to different proportions. Two shells at 3.2 can differ arbitrarily in how long they took, and nothing in either section says which.

A shape parameter and a rate parameter are different kinds of thing #

The essays here have been careful about one kind of confusion and not about this one. Raup’s three numbers compresses nearly every coiled shell that has ever existed into three quantities, and every one of the three is a proportion: how fast the whorl expands, how far the opening sits from the axis, how far it travels along it. None of them is a speed, and the model has no time in it anywhere.

That is a virtue of the model rather than a gap in it. A morphospace of proportions is a space a fossil can be placed in; a space with a rate axis is not, because a fossil does not carry its own calendar. The mistake is only in reading a proportion as though it were a rate, and the word “growth” in “growth factor” invites it.

The marks are where the difference is #

Under an angular clock the three whorls of a three-turn shell hold near-equal numbers of marks — 300, 300 and 301 out of nine hundred. Under a length clock they hold 63, 199 and 639. Under an area clock, 8, 80 and 813. Under a volume clock, 1, 27 and 872.

Nearly the whole record is in the last whorl #

That is the practical consequence and it is severe. At 3.2 per turn the outermost of three whorls holds 33.4 per cent of the record under an angular clock, 71.0 under a length clock, 90.3 under an area clock and 97.0 under a volume clock. At 4.5 per turn the last three figures are 78.7, 95.1 and 99.0.

And the last whorl is the part that breaks #

A shell’s outer whorl is its thinnest, its most exposed and the first thing lost to abrasion or predation. So under any clock but the angular one, an animal keeps almost the whole record of its own life in the part of itself least likely to survive, and a fossil section’s inner whorls — the part that does survive — carry almost none of it.

That is not a statement about phyllotaxis or about shells in particular. It is what happens whenever a record is laid down at a constant rate in a quantity that grows geometrically.

Which whorl a measurement lands in #

Every measurement of a shell’s growth factor here uses points spread along the curve, and the essays have been careful about how far along — how far a centre must move turns on the span of arc a fit is given, and three points on a diameter on the fact that two diameters half a volution apart are exact where four hundred fitted points are not.

Those are all statements about where the points are in angle. The counts above are about where the shell’s own marks are in angle, and the two meet: a measurement taken on the outer whorl of a fast shell is taken where nearly all the growth lines are, and one taken on the inner whorls is taken where almost none are. If the growth lines are what a person measures between, the sampling is not theirs to choose.

The same picture at the golden factor #

A golden spiral expands by $\varphi^{4} = 6.854$ per turn, which is twice the nautilus figure, and the marks separate much further.

Under a volume clock, forty of the forty-one marks land in the outer whorl and one in the first, with none at all in the middle one. The curve is again identical in all four panels, and no measurement of its shape would distinguish them.

Where the instrument gives way #

There is a second consequence and it is a defect rather than a difficulty. The routine that recovers a growth factor from a drawn curve has to turn a list of points into a list of angles, and it does that by unwrapping: each point’s angle is taken to be within half a turn of the one before.

For a curve sampled at equal angles that is always true. For one sampled at equal times it need not be, because the early marks of a steep clock are very far apart.

Half a turn, and the fit stops working #

Of twenty readings — five growth factors by four clocks — five leave a gap past half a turn: the volume clock at 3.2, the area and volume clocks at 4.5, and the area and volume clocks at 6.854. Every one of the fifteen under half a turn returns the growth factor to within 10⁻¹⁵. Not one of the five over it does.

What the wrong answers look like #

They look like answers. A volume clock on a curve built at 3.20 returns 3.33 — four per cent out, with no warning, no residual worth noticing and no sign in the number itself. On a curve built at 6.854 the same clock returns 8.19.

Four per cent is larger than several of the effects measured on shells here. What the centre costs put a displaced centre at 4.56 per cent, and that was worth an essay; this is the same size and arrives from the sampling rather than from the geometry.

Which is an instrument fault and not a shell fault #

The curve is unchanged. What has changed is where the marks are, so the failure belongs to the routine and not to the animal — and it is exactly the kind of failure the residual is not the test warned about, where the number that is supposed to say whether the fit is trustworthy says nothing of the sort.

The repair is not difficult in principle: unwrap by following the curve rather than by assuming a bound on the step. It is worth naming rather than performing here, because the interesting part is that a measurement can be defeated by when a shell was marked rather than by anything about its shape.

A shell that changed how it grew is a different question #

One number for a shell that changes handed the fit a shell whose expansion rises from 2.8 to 3.6 a turn and got back 3.17490 — the geometric mean of the two ends, exactly, with a residual it accepted. That is a shell whose shape changed, and the fit hid it.

The clock is the other axis and it hides nothing, because it was never visible. A shell of constant proportion whose rate changed tenfold partway through is the identical curve to one that did not, with the same fitted factor and the same residual — and there is no arrangement of the fit that could distinguish them, since the two point sets are the same set. The first is a failure of an instrument and could be repaired; the second is not a failure at all.

How fast a shell has to expand for this to bite #

The gap between marks passes half a turn at about 3.2 per turn under a volume clock, about 4.5 under an area clock, and not at all up to 6.854 under a length clock, whose worst gap there is 0.223 of a turn. So the failure is not a corner case: 3.2 is the measured nautilus figure, and a nautilus that deposited a constant volume of shell per unit time would defeat the fit at its own growth factor.

Whether it does is unknown here and is the sort of thing a count of growth lines per whorl would settle. That the question is answerable from a sectioned shell, with no calendar and no living animal, is the one cheerful thing in this essay.

What a growth factor is worth once the clock is separated from it #

It is still worth a great deal, and the separation is what makes it defensible. A number that was doing two jobs badly — describing a shape and standing in for a rate — does one of them exactly. The proportion of one whorl to the next is measurable to the last digit from any drawn curve, comparable between two shells of different ages, and comparable between a fossil and a living animal without any assumption about how either grew. None of that was true of it as a rate.

What is lost is a set of sentences that were never supported: that one shell grew faster than another, that a tightly coiled shell is a slow grower, that a golden spiral would take longer to make than a nautilus’s. Each of those reads a per-turn factor as a per-year one, and each is exactly as wrong as reading a map’s scale as a speed.

What survives the clock #

Three things, and they are the things actually measured on these curves. The proportion of a whorl to the one inside it, which is the growth factor. Whether successive whorls touch, which is a statement about the same proportions. And the shape of the aperture, which nothing here moves.

What does not survive is any reading of the growth factor as a speed, and any comparison between two shells’ growth factors that is meant as a comparison between two animals’ rates of growth.

A slower shell keeps a fairer record #

One thing is worth noticing on the other side. At 1.6 per turn the four clocks put 33.4, 49.7, 64.9 and 76.8 per cent of the record in the outer whorl — spread across a range of forty-three points rather than the sixty-four at 3.2 or the sixty-six at 4.5.

So a tightly coiled shell is a better recorder than a loosely coiled one under every clock, and the shells that are easiest to measure for proportion are the worst at keeping time. That is not a coincidence; both follow from the same geometric growth.

What a shell would have to be for the four clocks to be the whole story #

Three assumptions, and each is a place a real animal could differ. The shell grows without changing shape, so it has one length scale — which one number for a shell that changes already shows is often false and often invisible. The deposition rate is constant in whatever it is constant in, rather than seasonal or episodic; a shell that grows in summer and stops in winter is an angular clock interrupted, and its marks are neither of those things. And the growing animal does not change law partway through, which a juvenile becoming an adult is a good reason to doubt.

None of the three rescues the growth factor. Each of them adds a way for a curve to be produced by a history it does not record, and the point stands in every case: the locus is what survives and the schedule is not in it.

What is claimed, in one line #

A shell’s growth factor is a rate per turn and not a rate per unit time, four rate laws trace the identical curve and return the identical fitted factor, and the only difference between them is where along the curve the marks fall — which is enough to put nearly the whole record in the outer whorl and, at steep clocks and fast shells, to defeat the fit entirely.

What it does not establish #

That any shell’s growth lines are laid at equal intervals of time. That is an empirical question with a literature of its own, and lines that are daily, tidal, seasonal or the record of a disturbance each give a different clock or none. Nor does anything here say which law a real animal follows — only that its curve cannot answer.

What would withdraw it #

A mark laid by one clock that does not lie on the curve. Two clocks’ marks returning different fitted factors where both leave gaps under half a turn. A reading with a gap under half a turn whose fit is wrong, or one over it whose fit is right. A share in the outer whorl that does not rise with the power the clock holds. Each is checked every time the measurement runs.

Still open: what the lines carry that the curve does not #

The four clocks are told apart by where their marks are, so a shell that has kept its marks has kept the difference — and the counts above are not scattered. 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, which is a fixed factor per whorl. The next measurement takes that seriously: whether the lines in two successive whorls really do stand in the ratio of the growth factor raised to a whole number, whether dividing two counts and reading the logarithm back through the growth factor names the law, and how many lines a whorl has to hold before the division means anything.