A shell that changed its law

What the growth lines carry found a measurement that gets an animal’s deposition law out of a fossil: the lines in successive whorls of a shell stand in the ratio of its growth factor raised to the power of the radius the animal holds constant per unit time, so two counts and a growth factor the curve already gives name the law. Every reading there had one power for the whole shell. The ratio was then the same between every adjacent pair of whorls, and a shell of five whorls said four times over what a shell of two said once.

An animal is under no obligation to grow the same way from hatching to maturity. This is the same count on a shell that did not.

The curve is the same curve, and that is the difficulty #

Nothing in the picture above distinguishes the two shells as curves. That was the finding of a spiral with no clock and it is unchanged here: four deposition laws trace the identical locus, every mark any of them leaves lies on $r = W^{\theta/2\pi}$ exactly, and a fit to the curve returns the same growth factor whichever law put the points there. A shell that changed its law is still a logarithmic spiral. Its outline carries no trace of the change, and no amount of care taken over the outline will produce one.

So the whole of the evidence is in the marks, and the marks are a count per whorl. The question is what a sequence of counts does when the rule generating them changes part way along, and whether what it does is specific enough to be read backwards.

It turns out to be very specific, for a reason worth stating before any arithmetic. The ratio between two whorls’ counts depends only on the law in force across those two whorls, and not at all on the absolute rate the animal was working at. So every pair of whorls the change did not touch says exactly what it would have said on a shell that never changed, and the pairs it did touch say something that is neither law’s answer. The change is localised in the sequence, which is what makes it findable.

What “the law changed” has to mean, written down #

The generalisation needs care, because there are two ways to write it and they disagree about which whorl the change happened in.

Write the curve as $r(\theta) = e^{k\theta}$ with $k = \ln W / 2\pi$. A constant law holds $dt/d\theta \propto r^{p}$ — that is what “a constant amount of $r^p$ per unit time” means once the angle is the variable. Take that as the definition of the instantaneous law:

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

which returns the $p$ it came from on a shell with one law, and which integrates to $dt/d\theta = \exp\!\left(k \int_0^\theta p(u)\,du\right)$.

That form is continuous wherever $p$ is bounded, and the continuity is the point. The obvious alternative is to write $dt/d\theta = r^{p(\theta)}$ and let $p$ jump, which makes the animal’s angular speed jump by $r^{\Delta p}$ at the moment of the change — a factor of thirty-two at a radius of $3.2^3$ for a change from a length clock to a volume clock. That is not a modelling choice between two defensible things. It is a discontinuity in the animal, and it would put an enormous artificial spike into the count of the whorl the change fell in.

The constant that keeps the speed continuous cancels out of every ratio taken on one side of the change, which is why the two readings agree everywhere except across it, and why the crossed ratio is the one number that depends on which convention was used.

The counts, on a shell that changed and one that never stopped changing #

The two rows begin and end at the same place and differ everywhere between, which is the first sign that the counts carry more than the endpoints. On the stepped shell the first three whorls hold 43, 43 and 43 — equal, because the animal was advancing its aperture at a constant angular rate and an angular clock puts the same number of lines in every turn whatever the shell is doing. Then 171, 1753, 17947, each about a factor of ten larger than the last, which is $W^2$ at $W = 3.2$.

On the drifting shell the same twenty thousand lines fall 42, 63, 140, 465, 2303, 16987, and no two successive whorls stand in the same ratio anywhere.

Neither row needs any calibration to read. A count is a count, the growth factor comes off the curve by the recovery growth as a rule established, and the arithmetic below uses nothing else. In particular nothing here needs the animal’s calendar, which is the property that made the single-law version of this measurement worth having: a statement about how an extinct animal grew, from a fossil, with no living specimen to calibrate against.

The sequence, and the shape it has #

Dividing each whorl’s count by the one inside it and taking the logarithm against $\log W$ turns the counts into a sequence of apparent powers, and on a shell with one law that sequence is flat. Here it is flat, then crossed, then flat: 0.0000, 0.0000, 0.1410, 0.9074, 1.0001, 1.0000 on a seven-whorl shell of 18,466 lines whose law goes from a constant angular rate to a constant length added at three and a half whorls.

Two things in that sequence are worth separating. The flat stretches say what the laws were, to four decimal places, and they say it twice each, which matters below. The two values that are on neither level say where the change was — and the fact that there are two of them rather than one is itself information, because a change falling exactly on a whorl boundary leaves only one.

The rounding bars are the whole of the reading’s error. A count of $c$ lines is within one of its exact share, so the relative error on a ratio of two counts is at most the sum of their reciprocals, and the power divides that by $\log W$. At the outer end of this shell that is three ten-thousandths; at the inner end, where the whorls are emptier, five thousandths.

The ratio across the change is a third number, in closed form #

With the change on a whorl boundary the mass either side is an exponential integral, the continuity constant cancels the position out entirely, and what is left has no free parameter in it at all:

$$R = W^{p_1}\,\frac{p_1}{p_2}\,\frac{W^{p_2} - 1}{W^{p_1} - 1}$$

with the $p = 0$ limit of $(W^p - 1)/p$ taken as $\ln W$. At 3.2 per turn a change from a constant angular rate to a constant length added crosses at 1.891, against flanking ratios of 1 and 3.2. From a length clock to an area clock it crosses at 6.72 — which is exactly $W(W+1)/2$, the average of 3.2 and 10.24, though that coincidence belongs to that one pair and not to the formula. From an area clock to a volume clock, 23.47 against 10.24 and 32.77.

The crossing ratio always lies between the two laws’ own, at every growth factor and pair of laws read. That is not obvious from the formula and it matters practically: it means a ratio landing between two laws is not an unreadable reading to be discarded, which is how the single-law measurement would have had to treat it. It is the reading that says a change happened there.

One measured ratio, inverted for one number #

When the change falls inside a whorl rather than on a boundary, that whorl is two exponential integrals joined by the continuity constant, and its count depends on where inside it the join is. The two laws are already known — they are the flat stretches — so the ratio of that whorl to the one inside it is a function of exactly one unknown. The function is monotone across the whole whorl, so it inverts by bisection, and the answer is a position.

Nothing is minimised here and nothing is fitted. One measured ratio goes in and one number comes out, which is the same shape as the reading it generalises: what the growth lines carry took one ratio and returned one power by a single logarithm, and this takes one ratio and returns one position by a single inversion. The consequence is the same too. There is no residual to interpret, so the distance from the answer to anything else has to be reported separately — which is exactly what the second crossed ratio is for.

The other crossed ratio was not spent, so it is a prediction #

Two ratios were crossed and only one was used. The other is therefore a number the reading predicts and the shell either meets or does not.

On the seven-whorl shell above the unused ratio is measured at 2.8732 and predicted at 2.8716, a difference of six ten-thousandths in the power against a rounding of three thousandths. That is a check the inversion did not buy itself, and it is the only thing standing between this reading and the failure that the residual is not the test describes — an instrument that answers every question put to it and reports nothing about whether the question was the right one.

The answer is an interval, and the interval is the rounding #

The counts are whole numbers, so a band of positions reproduces them equally well, and the width of that band is the resolution the counts allow. On the worked shell the change at 3.5 whorls comes back at 3.5001 with a band running from 3.4847 to 3.5121 — 0.027 whorls wide, or about one part in forty of a turn. The band is taken over every combination of moving each count by the one line rounding can move it, rather than over one count at a time, because a line shifted from one whorl into the next changes two counts together.

Reporting the middle of that band alone would be claiming a precision the counts do not carry. Reporting the band is the honest form, and it behaves as a band should: a hundredfold in lines narrows it from 0.027 whorls to 0.0003, and the true position lies inside it at every position the reading accepts.

That is the measurement’s answer to the question what the growth lines carry left open, and it is a better answer than that essay expected. It supposed the change would be located to the nearest whorl. It is located to a fortieth of one.

The comparison worth drawing is with the other instrument on this shell that was asked to notice a change. 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, and it returned 3.17490 with a residual it accepted — one number, no warning, and the change entirely invisible. The count of lines is the opposite instrument: it cannot be made to average a change away, because its answer is a sequence and not a number.

Where on a shell this can be done at all #

The reading needs to know the two laws before it can ask where they met, and it takes each from the whorls the change did not touch. One ratio agreeing with nothing is not evidence that it is a law rather than a crossing; two agreeing ratios are. And two agreeing ratios need three untouched whorls, so a change can be placed only when it sits at least three whorls from the apex and three from the aperture.

A shell of six whorls therefore has exactly one such position, at its third boundary. A shell of seven has a range one whorl wide, eight has two, nine has three. Below six there is none at all — the change is still perfectly visible in the counts, and where it happened is not.

The distinction between those two is worth keeping. A five-whorl shell whose sequence reads 0, 0, 0.14, 0.91, 1.00 has said that the animal changed from an angular clock to a length clock, and has said nothing trustworthy about when. Insisting otherwise is what a reading that trusted a plateau of one ratio would do, and such a reading accepts positions it then places wrongly, with nothing in the counts to say so.

The smallest change the lines can show #

Nothing above requires the two laws to be whole numbers. A change from a constant angular rate to a law holding $r^{0.2}$ constant is as well defined as a change to a length clock, and the sequence shows it in exactly the same way — only smaller.

How much smaller is decided by the counts. On a five-whorl shell of three thousand lines at 3.2 per turn the jump and the rounding cross at 0.02 in the power: a change of two hundredths is visible and anything under it is inside the arithmetic. Ten times as many lines halves that to 0.01, which is the test that says the threshold belongs to the counting and not to the laws.

Two hundredths is a fine resolution for a quantity nobody has ever measured on a fossil, and it is worth being clear about what it is a resolution in. It is not a rate. It is the exponent in a rate law, and a change of 0.02 in that exponent is a change in the kind of thing the animal was holding constant, not in how fast it was doing it.

That distinction is the one a spiral with no clock established and it is worth holding on to here, because the sensitivity cuts the other way for everything else about a shell. The growth factor itself is the quantity every instrument in this subject struggles with: three points on a diameter has a whole account of two methods disagreeing over it at the third decimal place. The exponent is easier, because it is read from a ratio of two counts rather than from the positions of points, and a count has no centre in it.

What the reading costs in lines #

Every ratio in the sequence has to be countable, and the emptiest whorl of a shell under the steeper of its two laws holds a share of the total that falls as the growth factor to the power of that law times the whorl count. So the cost is geometric, and the difference between the shallow changes and the steep ones is not a difference of degree.

At 3.2 per turn a five-whorl shell changing from an angular clock to a length clock needs 348 lines. The same shell changing from an area clock to a volume clock needs 2,976,299. A six-whorl shell making that change needs fifty-four million, which is not a number of growth lines any animal lays down.

That is the practical shape of the whole measurement. It is available on shallow changes and on shells that kept most of their whorls; it is not available on the steep changes at all, and no care taken in counting will make it so. A shell with more whorls is worse for this reading and not better, which is the opposite of the advice every other instrument in this subject gives.

What a person would do with a real section #

Six steps, of which only the last two are new. Fit the curve for the growth factor over a span of arc wide enough to be stable, with an assumed centre whose error is stated — what the centre costs prices that error at 4.56 per cent for a displacement of a quarter of the innermost radius, and how far a centre must move says how far a centre would have to be wrong before the fit becomes something else entirely. Count the lines in every whorl that holds at least twenty of them. Divide each count by the one inside it and read each ratio back through the growth factor. Look at the sequence: if it is flat, the animal kept one law and the earlier reading applies unchanged. If it is flat, crossed and flat, take the two levels as the two laws and invert the inner crossed ratio for the position. Then report the second crossed ratio beside the prediction, and the band rather than its middle.

The one step in that list which is genuinely awkward is the third, and it is awkward on a real shell rather than here. Whether two adjacent whorls are countable is a judgement about preservation, and a whorl that is partly countable is worse than one that is not countable at all — it produces a count that is low by an unknown factor, and a count low by an unknown factor is exactly what this reading has no defence against, since it divides two counts and believes the answer.

Why it fails in the direction it does #

The reason is the same one that made what the growth lines carry refuse three of its twenty readings, and it is worth naming rather than rediscovering. A steep clock on an expanding shell puts nearly the whole record in the outermost whorl. Adding whorls to such a shell does not add readable material; it adds inner whorls holding one line each, and pushes the countable pairs further out where they are all telling the same story.

So the shells that can be read are moderately coiled ones with several whorls of countable lines and a shallow change — which, if a real animal’s growth changes in the way a juvenile’s does, is not an unlikely description of one.

What is claimed, in one line #

A shell whose deposition law changed once gives a sequence of whorl ratios that is flat, crossed and flat rather than constant; the flat stretches name the two laws exactly, the crossed ratio is a closed form that is neither of theirs and is monotone in where inside its whorl the change sits, so one measured ratio inverts to a position inside a band the counts’ own rounding sets — and a second crossed ratio is left over to check it.

What none of it says about a real animal #

That any shell’s growth lines are laid at equal intervals of time. That assumption is unchanged, untested here and untestable from a section, and a mis-specified clock is indistinguishable from a different law. That a real animal’s law is piecewise constant — a step is what “changed its law” usually means and it is not the only thing an animal could do. That a change found this way is a change in biology rather than in preservation: a run of whorls whose lines are harder to see would produce a step in the counts with no step in the animal.

Nor does anything here measure a shell. Every count is from a curve built to a stated pair of laws and then counted as a real section would be.

What would withdraw it #

A flat stretch whose ratios disagree by more than their own rounding. A crossed ratio outside the two flanking laws’ ratios. A located position whose band does not contain the position the shell was built at. The unused crossed ratio landing outside its own rounding of the prediction. A change accepted on a shell with fewer than three untouched whorls on either side. A band that fails to narrow as the shell is given more lines. Each is checked every time the measurement runs, and the last is the one that would say the band is something other than the counting’s arithmetic.

Still open: whether the shape of the change can be read as well as its place #

Everything here is a step, and a step is one shape. A shell whose law moved gradually would give a sequence with no flat stretches to take the two laws from, and the reading above would have nothing to stand on — but it would not give a constant sequence either, and whatever it does give is as much a measurement as this one. The next question is what the sequence looks like when the change is spread over the whole shell rather than concentrated at a point, whether a gradual change and a sudden one between the same two laws can be told apart at all, and what a reading that assumes a step returns when handed a shell that never had one.

Measured, the sequence is a straight ramp rather than a plateau and a crossing, each of its ratios reads the law at its own boundary rather than an average of the shell, the two shapes separate above 373 lines on a five-whorl shell, and the reading above refuses a drifting shell at every size — naming the number of ratios that agree with neither end.