A law that never stopped changing

A shell that changed its law read a change that happened at a moment. The animal held one power of its radius constant for the first part of its life and a different one afterwards, and the sequence of whorl ratios came out flat, crossed and flat, with the crossing locating the moment to a fortieth of a turn.

A step is one shape. Nothing about an animal requires its growth to change that way, and the more likely thing — a juvenile becoming an adult over several seasons rather than overnight — is a change with no moment in it at all. That shell has no flat stretches for the reading to stand on. What it does have is the question here.

Writing down a change with no moment in it #

The definition the stepped shell used carries over without amendment. The instantaneous law is

$$p(\theta) \equiv \frac{d \ln (dt/d\theta)}{d \ln r}$$

which returns the power it came from on a shell with one law and integrates to $dt/d\theta = \exp\!\left(k\int_0^\theta p(u)\,du\right)$ with $k = \ln W / 2\pi$. A step makes $p$ piecewise constant; a gradual change makes it continuous. The simplest continuous choice is linear in the angle, running from one law at the apex to another at the aperture, and that is the shell drawn throughout here.

One thing does change, and it is worth stating because the rest of this subject’s arithmetic has been exact. With $p$ linear the integrand becomes $\exp(k(p_1\theta + s\theta^2/2))$, which is a Gaussian and has no elementary antiderivative. The whorl masses are computed by Simpson’s rule on a stated grid instead — and the grid is checked against the one case that does have a closed form, the case with no drift at all, where it reproduces the exponential integral’s shares to better than a part in a million million. An eightfold refinement then moves nothing, so the numbers below are the integral rather than the grid.

The one inexact thing in the whole thread, and how it is checked #

Everything about a shell with one law, and everything about a shell with two, is an exponential integral: the counts are closed forms, the ratios are closed forms, and a shell that changed its law never had to approximate anything. A law drifting continuously breaks that, and it is the first place in this thread where a number is computed rather than derived.

The check is the case with an answer. Setting the drift to zero turns the Gaussian back into an exponential, where the closed form is known, and the quadrature reproduces its whorl shares to better than a part in a million million at every growth factor and law read. Refining the grid eightfold then moves the drifting shares by a part in a hundred thousand million, which is a statement that the numbers are the integral and not the sampling.

That is worth doing rather than assuming, because a quadrature that is slightly wrong in a systematic direction would produce exactly the phenomenon this essay is about — an apparent power sitting a little outside the law the shell was keeping. The offsets reported below are between a twentieth and a sixth of a whorl, and a grid error a hundred million times smaller than that cannot be their cause.

The counts, and how unlike the stepped shell’s they are #

The stepped shell holds 43, 43, 43, 171, 1753, 17947. The first three counts are equal because an angular clock puts the same number of lines in every turn whatever the shell is doing, and the last two stand in a ratio of 10.24, which is $W^2$ at 3.2 per turn. Between them sits one count that belongs to neither regime.

The drifting shell holds 42, 63, 140, 465, 2303, 16987, and no two successive whorls anywhere stand in the same ratio as any other two. The two rows begin and end at the same laws and share no interior value. Twenty thousand lines is the same shell material in both cases; what differs is only when the animal was doing what.

That already settles the crude version of the question. A shell that changed gradually is not a shell that changed suddenly plus noise — the counts differ by factors, not by ones.

The sequence is a ramp #

Turning the counts into apparent powers — each ratio’s logarithm against $\log W$ — gives 0.3486, 0.6865, 1.0320, 1.3755, 1.7180. The successive differences are 0.3379, 0.3455, 0.3435 and 0.3425, which is as near a straight line as five whole-numbered counts allow.

A shell with one law gives a level sequence. A stepped shell gives two levels and a crossing. A drifting shell gives a ramp, and the three are different enough to name by looking. There is no fitting anywhere in that sentence: the sequence is what the counts are, and its shape is the measurement.

The same shell, stepped, for comparison #

Put beside it, the stepped shell’s sequence is 0.0000, 0.0000, 1.1868, 2.0010, 1.9998. Two of its five values are exactly the first law, two are exactly the second, and one is neither — and the one that is neither is not halfway between them but at 1.1868, which is where the closed form for a crossing on a whorl boundary puts it.

The contrast is sharper than the two shells’ counts suggested. The drifting shell has no value repeated anywhere; the stepped shell repeats each of its laws twice and never takes an intermediate value more than once. Those are different sequences in the way a staircase and a slope are different, and the rounding bars are small enough that the difference is not a matter of interpretation at any of the five points.

It is also worth noticing what the stepped shell’s sequence does not do. It does not rise at the ends. A reading that mistook a drift for a step would have to explain two ratios agreeing at 0.0000 and two more agreeing at 2.0000, and a drift produces neither.

Each pair reads the law where it sits #

The line drawn under those points is the law the animal is actually keeping at the boundary between the two whorls each pair straddles: 0.3333, 0.6667, 1.0000, 1.3333, 1.6667. The readings sit on it, a little outside.

How far outside is worth a number, because it is a bias rather than a rounding. Solving each reading back for the angle at which the shell was keeping that law puts it at 1.046, 2.060, 3.096, 4.127 and 5.154 turns, against boundaries at 1, 2, 3, 4 and 5. So each pair reads the law at a point between a twentieth and a sixth of a whorl outside its own boundary, drifting further out the faster the law is moving. At the inner end that offset is smaller than the counts’ own rounding and at the outer end it is a hundred times larger, which is what makes it a bias to be named rather than absorbed.

The instrument is local, and the other instrument on the same shell is not #

This is the property that separates the two things a section can be measured for, and it runs opposite to intuition about which is the more robust.

One number for a shell that changes handed a growth-factor fit a shell whose expansion rose steadily from 2.8 to 3.6 a turn. It returned 3.17490 — the geometric mean of the two ends, exactly — with a residual it accepted, and a change of sixty-four per cent passed as one logarithmic spiral. That is what a fit does: it is a statement about the whole curve, and the whole curve is what it averages.

A count of lines cannot be made to do that. The outermost countable pair on a shell drifting from an angular clock to a length clock reads 0.8148 where the local law is 0.80 and the average of the ends is 0.50. On one drifting the other way it reads 0.1966 where the local law is 0.20 and the average is again 0.50. The reading lands on the local law every time and on the average never, because the ratio of two adjacent whorls has nothing in it from the rest of the shell.

So a section yields a global number and a local one from the same material, which is worth more than either. The practical version of that is a recommendation about which shells are worth the work, and it points the other way from every earlier instrument in this thread. Three points on a diameter needs two diameters half a volution apart and nothing else; what the centre costs needs a span of arc wide enough for the fit to be stable. Both want a clean piece of shell and are indifferent to how much of the rest survives. The count of lines wants whorls — as many as hold twenty countable lines each — and is indifferent to how clean any one of them is. What the growth lines carry already noted that the two readings fail independently — a displaced centre biases the growth factor and leaves the counts alone. This is the stronger statement: they are measuring different things, and a disagreement between them is information rather than a contradiction.

What the step reading does when it is handed a drifting shell #

An instrument that answers every question put to it is the failure the residual is not the test is about, and the step reading was built with that in mind: it takes its two laws from ratios that agree with each other, and a ratio agreeing with nothing is not one.

Handed a drifting shell it refuses at every size, and on the shells long enough to say why it names the right reason. Six whorls: three ratios agree with neither end. Seven: four. Eight: five. Whatever happened on such a shell was not one step, and the reading says so instead of choosing a place. On the shorter shells it refuses for the other reason — too few countable ratios to hold a plateau on each side — which is a weaker refusal and would have been given to a genuine step as well.

The control matters as much as the refusals. The same reading, on a seven-whorl shell that really did step returns 3.5001 for a change made at 3.5, so the refusal is a statement about the shell and not about the instrument being broken.

How far apart the two shapes are #

The difference between the two sequences is a property of the two laws and barely moves with the shell’s size: 0.22, 0.26, 0.30, 0.34, 0.32 and 0.37 in apparent power from three whorls to eight. The rounding rises steeply, because the same three thousand lines spread over more whorls empties the inner ones: three times the rounding goes 0.009, 0.017, 0.030, 0.053, 0.094, 0.167.

So the two lines converge, and on a long enough shell of fixed line count they would meet. That is not what stops the reading first. What stops it first is countability — at three whorls only two ratios are countable, and two ratios have no shape at all. A shape is a statement about three numbers, and a comparison of two values is a comparison of laws rather than of shapes.

A floor in lines, not in whorls #

Sweeping the line count instead puts the floor where it belongs. On a five-whorl shell at 3.2 per turn the two shapes separate at 373 lines: below that they sit inside the counts’ own rounding, and a shell that changed suddenly cannot be told from one that changed all along. Above it the margin widens without limit.

Three hundred and seventy-three is a small number for a shell with daily lines, and it is worth being clear about what makes it small. The two shapes differ by about three tenths of a power, which is a large difference; the counts only have to be good enough to see three tenths. The step’s position, by contrast, needed 18,466 lines on a seven-whorl shell. Telling the two kinds of change apart is much cheaper than placing one of them, and a palaeontologist with a worn section can often do the first and not the second.

What a drifting shell can be asked for instead of a position #

A step has a position; a drift does not, and asking for one is the mistake the refusal prevents. What a drift has instead is a rate — how fast the power was moving, in powers per whorl — and the ramp gives it directly as its own slope. The four successive differences above average 0.3423 per whorl against a shell built at 0.3333, which is the same reading as the offsets already described and carries the same bias.

That is a smaller claim than a position and it is the honest one. A sequence that rises steadily says the animal’s law moved steadily; its slope says how fast; and neither statement needs anything the flat-crossed-flat reading needed, because there is nothing to locate.

It is also a claim with a unit in it that deserves care. A rate of 0.34 powers per whorl is not a rate per unit time, and the distinction is the one a spiral with no clock was written about: a whorl is not a unit of time either, and the number of whorls an animal takes to change its law says nothing about the number of seasons. What the sequence measures is how the law moved against the shell’s own coiling, which is the only clock a fossil has. Turning that into seasons needs the very thing the whole reading is trying to establish, and the circularity is not removable.

Where this leaves a section with a sequence on it #

Three shapes and three answers. A level sequence says one law, and what the growth lines carry reads it as a single power. A flat, crossed, flat sequence says one change at one place, and a shell that changed its law locates it. A rising sequence says a law that moved throughout, and its slope is the rate.

What no sequence of five ratios can do is separate a drift from two steps close together, or from a step plus a drift, and nothing here pretends otherwise. The shapes named are the ones a five-ratio sequence can hold, which is a statement about the shell’s length as much as about the animal.

What is claimed, in one line #

A shell whose deposition law moved evenly gives a rising sequence of apparent powers rather than a level one or a crossed one, each of its ratios reads the law at a point between a twentieth and a sixth of a whorl outside its own boundary, and the reading built to locate a single change refuses such a shell at every size rather than placing a change that never happened at a place.

What none of this settles about a real animal #

That growth lines are laid at equal intervals of time, which is the assumption under every reading in this thread and is not testable from a section. That a law that changes gradually changes linearly in the angle — linear is the simplest continuous thing and an animal is under no obligation to be simple. That the three shapes are the only ones, or that a real sequence must be one of them.

And nothing here measures a shell. Every count is from a curve built to a stated law and then counted as a real section would be, which is what makes the refusals meaningful and the numbers provisional.

What would withdraw it #

A drifting shell whose apparent powers are level, or whose successive differences disagree by more than the counts’ rounding. A reading on a drifting shell that lands further from the local law than from the average of the shell’s ends. The step reading returning a position for a drifting shell at any size. The step reading refusing a shell that did step, at a size where its plateaux are countable. The separation between the two shapes failing to widen as the shell is given more lines. Each is checked every time the measurement runs.

Still open: whether the lines were laid at equal intervals of time at all #

Every reading in this thread rests on one assumption and none of them tests it, because the clock is the thing being measured and a mis-specified clock is indistinguishable from a different law. That is true of a single shell. It may not be true of a population: lines laid daily and lines laid tidally differ by a known factor, and lines laid at equal intervals of arc rather than time give a ratio of exactly the growth factor whatever the animal is doing — which is one specific reading that should always be distrusted. The next question is whether a set of shells of different sizes, all read this way, constrains the clock itself, and in particular whether a spread of apparent laws that tracks specimen size rather than scattering at random would say the lines are not a calendar.