A limit erases a rhythm before it shows itself

What the growth lines carry treated a shell’s lines as a clock, and a good deal has been made of that since. A section’s spacings say what law its radius followed through time, provided that the section can resolve its finest lines. Whether a section can see its own limit showed that a section which cannot resolve them does not look blurred. It looks like an animal that changed its law. A limit is a length separated the two by a floor test, on the reasoning that a limit has a size and a short day has a share. And a limit that loosens along the whorl broke the floor test, and found something that still worked: the pile, the share of spacings in each run that bunch against that run’s shortest. A limit keeps one line per length, so what it leaves is piled at that length. An animal’s short days are a tail, so nothing bunches. Every limited section had a large pile and no unlimited section had one.

Each of those animals had independent days, and that is one thing a real animal is not. A nautilus lays down growth lines to a rhythm, and fortnightly cycles of a few bands are one of the things their septa and lines are supposed to record. The essay on the loosening limit named the obvious threat to the pile: a rhythm makes some days systematically short, and if those days are alike, they ought to pile too. A rhythm’s pile might reach a limit’s. It suggested the next test, that the position of the piled spacings might separate them, since a rhythm’s fall every fourteenth and a limit’s fall anywhere. And it asked whether the law a changing animal names can still be read from a section carrying both.

An animal with a fortnight in it #

The animals here are the same as before, a volume clock on a shell of four whorls with a growth factor of 3.2 per turn, four thousand lines laid at intervals drawn from a gamma distribution. Its days vary by 10 or 20 per cent. One interval in every fourteen is then multiplied by a stated factor, shortened by 30, 50, 70 or 90 per cent. A limit is added or not, at eight sizes from nothing to four parts in a thousand of the outer radius. Each combination is forty sections, read for the pile against the threshold already found, for the law the outer whorls name, and for two new measures of position. One is the correlation of each spacing with the fourteenth one after it, with each spacing taken against the median of its neighbours because the whorl widens. The other sorts the shortest tenth of the spacings into fourteen places in the sequence, and reports the fullest.

With no limit the picture is unambiguous. There is a level run of spacings with an ordinary scatter, and every fourteenth one plunges to half. No statistics are needed to see that. Turn the dial to half a part in a thousand, and the dips are harder to make out. By one part in a thousand they have gone, and what remains is a run of spacings with a lower edge to it and no regular structure at all. The rest of the essay is the numbers behind those three pictures.

A rhythm does not pile #

The threat does not materialise. With no limit, a rhythm’s pile is 0.0083 at a share of a half and the same at nine tenths, on animals whose days vary by a tenth. An animal with no rhythm at all reads 0.0124. At days varying by a fifth the corresponding figures are 0.0059 and 0.0070. All are far below the threshold, 0.036, that the essay on the loosening limit set from sections with no limit. So a rhythm’s pile is smaller than none at all: not one of the hundreds of sections read crosses the threshold, at any share of the rhythm.

It is simple to see why, once it has been measured. The pile asks what share of a run’s spacings lie within 5 per cent of its shortest. A rhythm has made many short spacings, but they are a share of an otherwise variable day: if days vary by a tenth, then the days shortened by half are spread over about a tenth of their size. Few lie within 5 per cent of the shortest of them, and a run’s shortest is now set further down by the shortened days, so the rest of the run is further away from it. A rhythm’s bunch lies low and has a real width. A limit’s bunch has none, because it was cut off at a length.

That carries a practical point worth stating. A high pile means a limit, and it can be read without any thought about rhythm. And the converse also has a use, since a rhythm brings the pile slightly down, to the low side of what independent days produce. The threat that a rhythm might be mistaken for a limit is closed.

A rhythm has its own signature #

The second half of the suggestion works as well, on sections with no limit. A section with no rhythm reads a correlation of $-0.02$ at a lag of fourteen lines. With a rhythm it reads 0.45 at a share of 30 per cent, 0.76 at a half and 0.97 at nine tenths, for days varying by a tenth. If days vary by a fifth, the same shares read 0.16, 0.44 and 0.89, so a stronger variation in the days takes away some but leaves plenty. The fold agrees. With no rhythm the fullest of the fourteen places holds 10 per cent of the shortest tenth of the spacings. With a rhythm of a half it holds 71 per cent, against a limit of 71.4 per cent that would mean every short spacing was a rhythm’s.

So it takes very little to establish a rhythm on a section that resolves its lines. A person with a set of spacings can read this directly, and the period would come out of the same scan, if it were not known in advance. Nothing here needs to be told fourteen. The figures are given at fourteen because that was the rhythm the animals had.

What a limit does to it #

The same figure, read from left to right, is the main result. Start from a rhythm of a half, reading 0.76. At a tenth of a part in a thousand the limit is too small to matter, and the correlation stays at 0.76. At two tenths it reads 0.75. At half a part in a thousand it is down to 0.28. At one part, 0.21, at two, 0.10, at three, 0.05, and at four, 0.02. A stronger rhythm does not hold out longer. At nine tenths the same limits read 0.83, 0.67, 0.06, 0.08, 0.05, $-0.01$ and $-0.05$, losing it faster.

That is perhaps backwards from what a person would guess, and the reason is the mechanism. The limit does not smooth the record. It removes lines: walking from the apex, it keeps a line and then discards everything closer to it than the limit allows. A line after a short interval is exactly what it discards. A strong rhythm’s short interval is so short that it is below the limit, and the line at its end is the one to go, so nothing short remains in the record to be regular about. A weak rhythm’s short days survive a little longer, since they are not quite below the limit.

Once a line has been lost, the index along the record is no longer tied to the animal’s days. A spacing fourteen lines on in what remains has no fixed relationship to a spacing fourteen days on. Whatever faint regularity survives is being measured against a sequence that has already slipped.

The fold says the same with less ambiguity. The fullest of the fourteen places goes from 71 per cent of the shortest spacings, with no limit, to 11 and 14 per cent at the largest limit, against a chance of about 7, and those largest-limit readings do not differ between animals with and without a rhythm. The selection of a maximum makes the chance level look higher than it is, and the no-rhythm animals read the same as the rhythmic ones.

In which order the two signs change #

The most useful thing about the two curves is how they relate. The pile crosses its threshold at the same step of the limit where the rhythm’s correlation collapses. At two tenths of a part in a thousand the pile is 0.009 to 0.014, far below the threshold, and the correlation stays at 0.67 to 0.75. At half a part in a thousand the pile has jumped to 0.048 to 0.056, over the threshold, and the correlation is 0.06 to 0.28.

So a section is in one of two states and there is no third. Below a certain limit a rhythm can be read and a pile is absent, and above it a pile is present and a rhythm cannot be read. The window between is at most a factor of about two and a half in the limit, where neither signal is clean. The two tests are measuring nearly the same event, the point at which a limit starts removing whole lines, from opposite sides. That suggests what a section can and cannot say, in a sharp form.

How few lines it takes to lose #

The limit’s size can be put in terms of what it does to the record, which makes that collapse look even sharper. The outer whorl’s ordinary spacing, on these sections, has a median of 0.72 parts in a thousand of the outer radius, with its shortest tenth under 0.48. So a limit of half a part in a thousand sits right at the shortest tenth of the ordinary days, and it removes 11 per cent of the lines, 3,456 kept out of 3,877. That is all it takes to bring a rhythm’s correlation from 0.75 down to 0.28.

At that limit almost nine lines in ten are intact, the mean spacing has hardly changed, and a person reading the section would not call it badly resolved. But the rhythm’s short days are the ones that are being lost, every one of them, and it needs only one line lost per fortnight or so to take apart any regularity that is counted in lines. The rhythm is more fragile than the record that carries it, by a good margin, and that is a reason not to expect it from anything but the best-resolved sections.

Three kinds of section #

Plotted on both axes at once, forty sections of each kind fall into groups that do not overlap. Sections with a rhythm and no limit have a pile under the threshold and a large correlation. Sections with no rhythm and no limit have both small. Sections behind a limit have a large pile and a correlation close to nothing, and the ones with a rhythm cannot be told from the ones without.

So the suggestion from the earlier essay turns out to be half right. Position does separate a rhythm from a limit. It does so, however, only because the pile already had, and position’s real contribution is to find a rhythm on a section which has none of a limit’s signature. The two tests together make three categories out of four possibilities, with the fourth, a rhythm behind a limit, inaccessible. It is the important one for anyone reading real sections, since it says what absence means. A section with a low pile and no correlation at a given lag has no rhythm at that lag. A section with a high pile and no correlation has told the reader nothing about rhythm whatever.

That has some consequence for any survey of rhythm in the record of real shells. If a survey’s sections vary in how well they were resolved, then the coarsest of them will all report no rhythm, however regular the animal, and any comparison across specimens or across time would be a comparison of resolutions. A pile read on the same spacings says which sections have that problem, and needs no extra measurement.

What it does to the law #

The third question has the least dramatic answer. A rhythm with no limit does not bother the reading of the law: the section names 3.00 at every strength, on a true 3, within what forty sections can resolve. A limit, as the essay on a section’s own limit found, moves the reading a long way, to 2.15 at two parts in a thousand and 1.65 at four. A rhythm under the same limit adds a little: 2.20 against 2.15 at two parts in a thousand for a share of nine tenths, 1.71 against 1.65 at four. So a rhythm shifts the reading by a few hundredths in the direction of the truth, because it spends some of the animal’s lines below the limit, where they would have been lost in any case, and the lines kept are slightly better spread for it.

A section can therefore be read for its law with a rhythm in it, if it can be read for its law at all. The rhythm is a small effect on a reading whose largest source of error is elsewhere. That is the same conclusion a shell that changed its law reached about a similar pair of effects, one large and one small.

What this does not establish #

The rhythm here is one short interval in every fourteen, perfectly regular. A real fortnightly rhythm is more likely to be a gentle rise and fall across several days, and slips in its timing would soften the correlation, so the numbers here are the easy case for detecting it. Nothing here is a claim about what real nautilus sections show or whether a fortnightly rhythm exists in them. The limit is fixed at one length along the outer whorl, and a limit that loosens might let a rhythm through in the outer part of a whorl where it has loosened most. That has not been tried, and it is the first thing that should be. All the sections are from one law, and the threshold was set on one set of animals.

How it could be wrong #

A rhythmic animal with no limit whose pile crossed the threshold would remove the claim that a pile always means a limit. A rhythm that kept a clear correlation behind a limit that had also raised the pile would mean there was a fourth category after all, and that coarse sections could be read for rhythm. A real section with a large pile and a clear fourteen-line signal would be the most interesting thing anyone could bring to this, since it would mean this account of what a limit does to lines was wrong.

Still open: a rhythm behind a limit that loosens #

A limit that coarsens along the whorl, or fines down along it, passes from one regime to the other inside one section. A correlation read by position along the whorl, in windows, should then show a rhythm appearing or vanishing at the place where the local limit crosses its threshold. The next measurement lays a rhythm under a limit with a gradient, and asks whether a rhythm can be read in the part of a section that resolves it while the other part shows a pile, whether the point of change marks the local limit to any useful precision, and so whether a section that loosens along its whorl carries a reading of its own resolution, taken from a rhythm that happens to be in it.