What the growth lines carry

A spiral with no clock left a negative result and a lead. The negative result is that a shell’s curve records nothing about how fast the animal grew: four deposition laws trace the identical locus and return the identical fitted growth factor. The lead is that the four put their marks in very different places, and a shell that laid down growth lines has kept those places.

So the question is whether the difference is a measurement rather than a picture. It is, it is exact, and it needs nothing a section does not already give.

The arithmetic is two lines #

Suppose the animal holds the pth power of its radius constant per unit time: p = 0 for a constant angular rate, 1 for a constant length added at the opening, 2 for a constant area, 3 for a constant volume. Then $r^{p}$ is linear in time, so the time spent crossing any stretch of shell is proportional to the change in $r^{p}$ across it.

Whorl $n$ runs from $r = W^{n-1}$ to $r = W^{n}$, so the change in $r^{p}$ across it is $W^{p(n-1)}(W^{p} - 1)$. That is a fixed factor $W^{p}$ larger than the whorl inside it, at every $n$ and every $W$. So

$$\frac{\text{lines in whorl } n+1}{\text{lines in whorl } n} = W^{p}$$

and at $p = 0$ the change in $r^{0}$ is undefined but the angle is linear in time directly, which gives a ratio of exactly one.

The counts, against what the formula says #

Nine hundred lines laid at equal intervals of time over three turns at 3.2 per turn give 300, 300, 301 under an angular clock; 63, 199, 639 under a length clock; 8, 80, 813 under an area clock; and 1, 27, 872 under a volume clock. Every one of those counts is within one line of the closed form’s prediction, at every growth factor and law read — and one line is the most that rounding a share of nine hundred to a whole number can move.

The ratios those counts give #

Dividing the outer two: 1.0033 against a predicted 1, 3.2111 against 3.2, 10.1625 against 10.24, and 32.3333 against 32.768. The departures are the rounding and nothing else; the closed form has no free constant in it and is not fitted to anything.

Why the ratio is the same between every pair #

The factor W^p does not depend on which whorl the pair is taken at, and that is what makes the reading usable. A shell whose outer whorls are broken away, or whose innermost is too worn to count, still gives the same answer from whatever pair survives — provided both whorls of that pair hold enough lines.

It also means the reading has a consistency check built into it. A shell of five whorls offers four ratios, and they should all be the same number. A sequence that drifts is a shell whose law changed, whose lines were not laid at equal intervals of time, or whose shape was not constant. Which of the three it is, the ratios cannot say; that they disagree at all is what a single averaged reading would have hidden.

Reading a ratio back to a law #

The growth factor W is already known — it is what the curve gives, and growth as a rule established that it comes back out of any drawn spiral to the last digit. So the power follows from one logarithm:

$$p = \frac{\log(\text{ratio})}{\log W}$$

Twenty readings — five growth factors by four laws — and seventeen of them are countable. Every countable one returns its own law’s power, the furthest 0.0121 from a whole number.

Why that is a round trip and not a fit #

Nothing is searched for and nothing is minimised. The count goes in, the growth factor goes in, and a logarithm comes out, which is then near a whole number or is not. So the reading has no residual and cannot report a poor fit — a property it shares with every exact inversion, and the reason the distance from a whole number is worth printing beside the answer. The residual is not the test is this collection’s account of what happens when a number meant to police a fit is asked to do work it cannot; an inversion with no residual at all is at least honest about having none.

That distance is the only warning the reading carries, and here it is never more than an eighth of the way to the next law. A ratio landing at 1.6 would not be a law at all, and would say the shell was not doing any of these things.

What a mis-measured growth factor does to the reading #

The power comes from dividing two logarithms, so an error in W enters the answer through the denominator. A growth factor read five per cent high turns a true power of 2 into 2·log W / log(1.05W), which at W = 3.2 is 1.92 — still nearer 2 than 1, and comfortably so.

That tolerance is worth having, because a growth factor is exactly the quantity several earlier readings have shown to be harder to measure than it looks: what the centre costs puts a displaced centre at 4.56 per cent, and a spiral with no clock adds a sampling failure of four per cent on top. The reading survives both, because the laws are a factor of W apart and the errors are a few per cent.

Three readings that cannot be counted #

The other three of the twenty are not wrong. They are uncountable: the inner whorl of the pair holds fewer than twenty lines, so its count is not a quantity to divide by. At 4.5 per turn under a volume clock the three whorls hold 1, 9 and 891, so the pair to divide is 9 and 891. At 6.854 under an area clock they hold 1, 19 and 881, and under a volume clock 1, 2 and 898.

That is the same geometry from the other side. A steep clock on a fast shell puts nearly every line in the outer whorl — 99.0 per cent at 4.5 per turn — which is exactly why the ratio is large and exactly why the denominator vanishes.

So the reading has a window, and both edges are the growth factor #

At a low growth factor the four laws’ ratios are close together: at 1.6 per turn they are 1, 1.6, 2.56 and 4.10, so telling a length clock from an area clock means separating 1.6 from 2.56 on counts that carry their own rounding. At a high growth factor they are far apart and the inner whorls are empty.

The window between those is wide — every law is countable and named at 1.6, 2.2 and 3.2, and three of four at 4.5 — but it is a window, and which end a shell sits at is decided by the same number the measurement divides by.

Where the window is at its widest #

Between the two edges the four laws are both far apart and all countable, and at 2.2 per turn every one of them is: the counts are 300/300/301, 112/247/542, 31/149/721 and 8/76/817, and the predicted ratios 1, 2.2, 4.84 and 10.65.

A factor of 2.2 between adjacent predictions is a great deal of room for a count that is exact to a line. So the measurement is at its best on a moderately coiled shell, which is most of them.

More lines widen it in one direction only #

Nine hundred lines over three turns is a choice, and a shell with more lines would push the uncountable cases back: the inner whorl’s count is a fixed share of the total, so doubling the lines doubles it. At 4.5 per turn under a volume clock the middle whorl holds nine lines in nine hundred, so it would take about two thousand to reach twenty.

That is not an absurd number for a shell with daily lines and a few years of growth, which is worth knowing. What more lines do not fix is the other edge: the ratios at a low growth factor stay where they are however many lines are counted, because they are ratios.

Counting more whorls helps more #

The other lever is turns. The ratio is the same between any two adjacent whorls, so a shell of five whorls offers four ratios instead of two, and the inner ones are taken where the counts are smaller and the outer ones where they are larger. A reading that is uncountable between whorls one and two may be perfectly countable between four and five.

That is the practical advice the arithmetic supports: count the outermost pair of whorls that both hold enough lines, and report the pair along with the ratio. Nothing about the method needs the inner shell at all.

The counts at a fast shell, drawn #

The volume clock’s row is the whole difficulty in one picture. Two of its three whorls together hold ten lines of nine hundred, and a count of ten against a count of eight hundred and ninety-one is a ratio with a tenth of its own size in rounding. The reading is not wrong there; it is not a reading.

What this does to the septa argument #

What the septa count worked out what a nautilus’s septal count per whorl does to a growth factor read from one chamber to the next, and found that the count is an exponent: one septum miscounted at thirteen moves the factor by 9.14 per cent. That essay treated the count per whorl as a number to be got right.

This one says the count per whorl is also a measurement in its own right. If a nautilus’s septa were laid at equal intervals of time, their counts in successive whorls would stand in the ratio W^p, and at W = 3.2 the four candidate laws predict 1, 3.2, 10.2 and 32.8. That is a prediction about a quantity ammonoid and nautiloid workers already report, which is unusual for anything in Raup’s model. A septal count that is roughly constant from whorl to whorl is therefore a statement that the animal advanced its aperture at a constant angular rate — or that the septa are not laid at equal intervals of time, which is the more likely reading and is exactly the sort of thing the measurement can be made to say. Either way it is a statement about the animal rather than about its proportions, which is more than the nautilus question or anything else in this field has been able to get out of a section.

Three whorls is the smallest useful shell #

Two whorls give one ratio and no consistency check, so a shell of two is a reading that cannot be wrong in any way it would report. Three give two ratios, which is enough to notice a drift and not enough to say where it starts. Five give four, and four numbers that should all be equal is a real test.

That is a statement about which fossils are worth the work rather than about the arithmetic, and it points the other way from most of the measurements above: three points on a diameter needs only two diameters half a volution apart, and how far a centre must move found that a short span of arc is where a fit becomes unstable. For the growth factor, less shell is a difficulty. For the clock, less shell is fewer ratios, and fewer ratios is a weaker claim rather than a worse number.

Which assumption the whole thing rests on #

One: that the lines are laid at equal intervals of time. Nothing here tests it and nothing here can, because a clock is the thing being measured and a mis-specified clock is indistinguishable from a different law.

What can be said is what the arithmetic would look like if the assumption failed in a stated way. Lines laid at equal intervals of arc would give a ratio of W at every law, since arc length on a logarithmic spiral is proportional to radius — which is the length clock’s answer, arrived at with no animal in it at all. So a measured ratio of W is the one answer that should be distrusted, and the other three are informative.

What a person would actually do with a section #

Four steps, none of them new except the last. Fit the curve for W, over a span of arc wide enough to be stable and taken with an assumed centre whose error is stated. Choose the outermost pair of whorls that both hold at least twenty countable lines. Count the lines in each. Divide, take the logarithm against log W, and report the power with its distance from the nearest whole number beside it.

What that yields is a statement about the animal’s deposition law, from a fossil, with no calendar and no living specimen. It is a small statement and it is not a proportion, which makes it different in kind from everything else this field measures.

Why this is not the same measurement as the growth factor #

Both readings come off the same section and they are independent in a way worth stating. The growth factor is a property of where the curve goes; the power is a property of where the marks are along it. A shell could be measured for one and not the other — a worn section with countable lines on two whorls and no reliable centre gives a ratio and no W, and a clean section with no visible lines gives a W and no ratio.

They are also independent in their failure modes. A displaced centre biases W and leaves the counts alone; a lost outer whorl leaves W alone and removes the pair with the most lines in it. So a shell that yields both is a shell with two separate things measured on it, and the pair is worth more than either.

What is claimed, in one line #

The growth lines in successive whorls of a shell grown at a constant rate in some power of its radius stand in the ratio of its growth factor raised to that power, exactly; so two counts and a growth factor the curve already gives name the deposition law, wherever both whorls hold enough lines to divide.

What it does not establish #

That any shell’s lines are laid at equal intervals of time, that a real animal holds a whole power of its radius constant, or that the four named laws are the only candidates — they are the whole powers of one family, and a law at p = 1.5 is well defined and unlisted. Nor does anything here measure a shell. Every count is from a curve built to a stated law and then counted as a real section would be.

What would withdraw it #

A whorl holding a count more than one line from the closed form. A ratio of two countable whorls that is not the growth factor raised to a whole number. A countable reading naming a law other than the one it was built at. A power landing more than a fifth of the way to the next whole number. A reading declared countable whose inner whorl holds fewer than twenty lines. Each is checked every time the measurement runs.

Still open: a shell that changed its law #

Every reading here has one p for the whole shell, and the ratio is then the same between every adjacent pair of whorls. An animal that grew as a juvenile under one law and as an adult under another would give a sequence of ratios rather than one — and the sequence would say where the change happened and what it changed to, at the resolution of one whorl. The next measurement builds shells whose power steps partway through, counts the ratios whorl by whorl, and asks how sharp the change looks and how small a change in p can be seen against the rounding that a whole number of lines forces.