What the axis distance costs

Raup’s shell model has three numbers in it, and the line everybody draws across the space it spans is a relation between two of them: the whorls of a coiled shell run free of each other when the product of the whorl expansion and the distance from the axis exceeds one. That line is exact — located by bisecting the drawn circles, its residual is the last bit a double holds.

A great deal of effort has gone into the first of those two and none whatever into the second. Seven separate measurements price what can go wrong with the expansion, and the essay that carried them onto the boundary turned the line into a band by propagating all seven. It also said, in its last paragraph, that the band it drew was half a band — because nothing had priced the other number — and it recorded a guess about how that would come out: the distance from the axis is a length measured against another length, so it should be easier than the expansion rather than harder.

The guess is wrong about the reason and right about the conclusion, and the gap between those is the whole of what follows.

The measurement is a ratio on one line #

In the normalisation these shell measurements use, a whorl at azimuth θ occupies the radial interval from D·W^(θ/2π) out to W^(θ/2π). Divide the two and the growth cancels: along any ray from the coiling centre, the distance from the axis is the inner wall’s distance over the outer wall’s distance, and it is the same ratio at every azimuth.

That is a very different kind of measurement from the one that gives the expansion. Recovering W means fitting a rate to a curve — taking the logarithm of the radius against the unwrapped angle over as much arc as the section shows, and reading the slope. Recovering D means finding two points on one line and dividing. The three numbers and what each does is where that parameterisation is set out.

Three things follow from that shape, and they do not point the same way. A ratio along a line survives a projection. A ratio at one azimuth needs no arc to be read over. And a ratio of two distances measured from a guessed origin is ruined by a bad guess in a way a fitted slope is not.

Read from the true centre, every ray returns 0.42 — the shell’s own value, to the last bit a double holds, at all thirty-six azimuths tried. That is worth stating as a starting point rather than as a result: it says the reading is exact when the centre is right, so everything priced below is the centre’s cost and not the instrument’s.

What a bad photograph costs, which is nothing #

Take the good news first, because it is exact rather than small.

A photograph taken off the normal returns the coiling plane compressed along one direction by the cosine of the angle between the plane and the film. That is an affine map, and an affine map takes a line through a point to a line through the image of that point, scaling every length along it by one factor that depends on the line’s direction and not on position along it.

Both of the distances a reading of D uses lie on one ray. They are therefore scaled by the same factor, and the ratio comes through unchanged.

This is not a new fact about projections. It is exactly the mechanism the caliper measure already exploits, where three points on one line through the centre give a growth factor that an oblique view leaves alone; the same immunity, arriving at a different quantity for the same reason. What is worth recording is that the model’s second parameter has it for free, with no instrument designed around it.

Measured over every tilt to sixty degrees, every bearing and every azimuth, the largest relative departure is 2.7 × 10⁻¹⁵. That is rounding in the bisection that locates the wall crossings, not an error in the projection. An oblique view costs the distance from the axis nothing, and the entry in its budget is a zero rather than a small number.

The expansion, by contrast, pays 2.95 per cent for a camera ten degrees off the normal read over half a turn, and pays it because a fitted slope sees the projection as a periodic perturbation of its data that a partial span does not average away.

And what a guessed centre costs, which is a great deal #

Now the bad news, and it is the reason the guess quoted at the top of this essay was wrong.

Displace the assumed centre by a distance s along the ray being read. Both measured distances fall by s, so the reading becomes (D − s)/(1 − s) in units of the outer radius. Subtracting the same length from the top and the bottom of a fraction below one moves it toward one, and the move is first order in s.

The expansion’s centre error is not. A displaced centre perturbs the log-radius data with a term that is largely absorbed by the fit, and the measurement of that found it falling as the square of the span of arc. At a quarter of the innermost radius over the spans a real section shows, it is 6.88 per cent.

At the same quarter-radius displacement, the distance from the axis loses 79.3 per cent at the worst azimuth.

The slope on logarithmic axes settles the order rather than assuming it: halving the displacement from four hundredths of a radius to two hundredths halves the error, 8.71 per cent to 4.22, and halving again gives 2.11. The ratios are 2.065 and 2.013, and a second-order quantity would have given four.

One number in that figure needs its provenance stated, because two versions of it appear in this essay. The table above is swept over twenty-four displacement directions and sixteen azimuths and reads 70.4 per cent at a quarter radius; the headline 79.3 comes from a sweep three times finer. Both are worst cases, and a worst case found on a coarser grid is a lower bound — the finer sweep did not disagree with the coarser one, it found a worse ray the coarser one stepped over. The finer number is the one the budget carries.

The repair that costs nothing #

The first-order term has a property that rescues it: it is odd in the azimuth.

A ray pointing along the displacement has s positive and reads low. The ray opposite has s negative and reads high, by nearly the same amount. Pair them and the first order cancels, leaving whatever the spiral’s curvature and the second order contribute.

So a worker who reads the distance from the axis at one azimuth and a worker who reads it all the way round the whorl and averages are not doing the same measurement, and the difference between them is a factor of 4.91: 79.3 per cent at the worst ray becomes 16.15 averaged, at the same displacement with the same wrong centre.

Nothing about the second reading is harder. The section is already in front of the worker, the walls are already visible at every azimuth, and the arithmetic is a mean of numbers that were going to be measured one at a time anyway. It is the cheapest improvement in any of these error budgets, and it exists because of the shape of the error rather than because of any care taken.

What it does not do is remove the entry. Sixteen per cent is still two and a third times what the same displaced centre costs the expansion, and it is the price of not knowing where the coiling axis is on a shell whose innermost whorls are the hardest part to see.

The cancellation needs azimuths to cancel with #

An average over opposite rays needs opposite rays, and a section that shows a fragment of a whorl has none.

The residual runs from 17.1 per cent on a full turn to 30.9 on three quarters, 44.5 on a half, 52.9 on a quarter and 63.2 on a twentieth — swept over every direction the displacement could have and every placement the window could have, because neither is observable.

That curve is the third entry in the budget, and it is a conditional entry in exactly the sense the expansion’s dividers entry is conditional. A worker whose specimen is sawn through the whole whorl does not carry it. A worker with a photographed plate showing a sector does, and cannot tell from the plate how much.

The error has a sign, and it points at the boundary #

Everything above is an absolute size. The sign turns out to matter more.

The averaged reading is biased low, and it is biased low whichever way the assumed centre is displaced. On the shell drawn throughout this essay it comes out 9.33 per cent under the truth on average over the twenty-four directions the displacement was tried in, and not one of those directions produced a mean above the truth.

That is not what a measurement error usually does. An error with no preferred direction scatters a specimen about its true position, and a survey of many specimens recovers the truth by averaging. An error that always points one way moves the whole survey, and this one points toward smaller D — which is to say toward the contact boundary, from the evolute side.

A shell read from a mislocated centre therefore looks more involute than it is. Not sometimes: reliably.

The second thing that figure shows is worse, and it is the reason the recommendation at the end of this essay is hedged rather than clean. The error is not constant across the space. The first-order term carries a factor of the whorl’s own thickness, so the reading degrades as the whorls close up: the averaged error is 0.42 per cent at a product of 2.16, 1.03 at 1.92, 3.33 at 1.44, 5.81 at 1.20 and 9.33 at 1.008 — a factor of twenty-two from the outside of the figure to the inside.

The product 1.008 is the shell drawn throughout these figures, and it is a thousandth of the way outside the contact boundary. The distance from the axis is read worst precisely where the boundary is, which is the one place anybody wants to read it, and it is read best far away from the boundary where the answer was never in doubt.

And on the other side it cannot be read at all #

There is a harder limit under that one, and it emerged as a defect before it was understood as a result.

The reading pairs wall crossings by walking outward along a ray: a ray enters a whorl at its inner wall and leaves at its outer, so consecutive crossings in distance order are one whorl. That is true while the whorls run free. It is false the moment they touch — on an involute shell the next turn’s inner wall lies buried beneath the previous whorl, the crossings stop alternating, and the pairing quietly returns a “whorl” spanning two turns.

It did exactly that, and the first sweep across the parameter space reported a shell at D = 0.10 as being read with a 454 per cent error. The number was nonsense and the nonsense was informative: the reading had not become inaccurate, it had become a reading of something else.

The reading refuses those shells now, and the refusal is not a shortcoming of the method. A buried wall is buried in the specimen too. The inner length that the distance from the axis is a ratio of is not present on the section of an involute shell, and no care with the centre, the camera or the saw recovers it. What a worker can measure there is where the previous whorl’s outline crosses this one, which is a different quantity that happens to be visible.

This is the sharpest constraint in the essay and it lands on the criterion the band essay hoped for. A contact test that leans on D is unavailable on the whole involute half of the space — the half where most gastropods live — and is at its least accurate along the boundary itself. It is a test for evolute shells at a distance from the line, which is a real domain and a narrower one than “the cheaper number”. The alternative already on the shelf is the angle criterion, which reads contact from the half-angle a whorl subtends at the apex and carries the expansion’s error better above a certain expansion and worse below it.

Two budgets, and each is decided by one condition #

Set the two lists beside each other and the interesting thing is that they are different lists rather than the same list with different numbers.

The expansion carries seven entries and totals 332.6 per cent in the worst case, which is a budget larger than the 114 per cent claim it was assembled to decide. The essay that assembled it found that the total is not the point: one entry, the dividers at 278.2 per cent, decides the verdict on its own, and the dividers are a historical method a modern worker simply declines. Set them aside and the budget falls to 54.4 per cent.

The distance from the axis carries three entries and totals 60.6 per cent averaged. And it has the same structure: one entry decides it.

The partial section carries 73.4 per cent of the total and the assumed centre the remaining 26.6. Decline the first — saw further, or photograph the whole whorl — and what is left is 16.15 per cent, which is the irreducible price of not knowing where the axis is.

That both budgets are dominated by a condition rather than by a fact is the most useful thing here, and it is why summing either one is the wrong operation. A budget’s total answers how badly a reading could go, which is rarely the question. Its decomposition answers which part of the method to change, which is always the question, and in both cases the answer is one sentence long.

Where the two numbers cross #

The two error curves have different shapes, and different shapes cross.

The expansion is ruinous on a fragment and improves without limit: at the same displaced centre it costs 91.2 per cent over half a turn, 24.4 over one turn, 9.5 over one and a quarter, and 1.08 over four. The distance from the axis costs 28.1 per cent over half a turn, 17.0 over one — and 17.0 over four, because a whorl holds every azimuth there is and a second whorl holds no new ones.

They cross at 1.12 turns of arc, located between the two sampled spans that straddle it and to no finer a grid than those.

So the recommendation that was said to hang on which number is cheaper does change, and it changes conditionally rather than outright. A section showing more than about one and an eighth turns should be read for its expansion, and the contact criterion should be the one that leans on W. A section showing less — a fragment, a worn specimen, a published plate cropped to the aperture — should be read for its distance from the axis, and the criterion should lean on D. At half a turn the difference is a factor of 3.2 in favour of a parameter nobody had bothered to price.

The same rule states the case for a tilted view with no arithmetic at all. A photograph whose obliquity is unknown and unrecorded destroys nothing in D and perturbs W by an amount that depends on a span the reader cannot check. If the only thing known about a plate is that its camera angle is not known, the distance from the axis is the number to read from it.

What this does not establish #

That a worker makes errors of these sizes. Every figure here is a worst case under a stated condition, in the same sense as every entry in the expansion’s budget, and a careful measurement of a well-prepared section avoids most of them. A quarter of the innermost radius is the displacement treated throughout as a gross error, chosen so that the two parameters are priced at one condition rather than because anybody’s centre is that bad.

That these are all the ways the reading can go wrong. They are three ways it has been measured. A shell whose generating curve is not a circle, an aperture damaged at the lip, a section cut off the coiling plane rather than merely photographed off it: none is in the sum, and a shell that walks along its axis enters the boundary through a third number this essay holds at zero throughout, and the aperture shape in particular is known to move boundaries in this model by amounts that were measured separately.

And nothing here touches whether the two errors are independent. The centre and the window plainly are not — the partial-section entry is the centre entry with fewer azimuths to cancel over — so the total above is the sum of two quantities that share a cause, which is exactly the objection the expansion’s own budget records against itself. Whether the boundary is sharp at all is a separate question that two continuous measures across it answer differently.

What would withdraw it #

A reading from the true centre that did not return the stated distance from the axis at every azimuth. It returns it to 2.2 × 10⁻¹⁶, which is the last bit a double holds, and the same figure computed against the drawn outline rather than against the model agrees to 1.8 × 10⁻⁷, which is the polyline’s own discretisation.

A tilt that moved the reading by more than rounding. Seven tilts, eight bearings and twelve azimuths give 2.7 × 10⁻¹⁵.

A displacement sweep whose error did not halve when the displacement halved. It halves twice, at ratios of 2.065 and 2.013.

Or an averaged reading that improved past one whorl, which would mean the azimuths were not what the cancellation was using. It is identical at one, two and four turns to the last bit.

Still open: reading all three numbers back off one section #

Both of Raup’s first two numbers now have a price, and the natural next step is not a third price but a round trip.

Everything measured here reads one parameter from a section whose other parameters are known, which is the comfortable case and not the one a specimen presents. A worker with a sawn shell has an outline and nothing else, and has to produce all three of W, D and T from it — with the centre unknown, the coiling plane unknown, and no way to check any of the three against the others except that together they must redraw the outline.

That is a different question from either budget, because the three are not independent once the centre is free. A centre guessed too far out reads the expansion one way and the distance from the axis another, and the third number decides how much of what is on the plate is a projection of something out of plane. The measurement is the whole recovery: build a shell at stated numbers, hand the drawn outline to a recovery that is told nothing, and ask how close the three come back and whether the residual says which of them is wrong when one of them is.

The interesting failure to watch for is a compensating one. Three parameters fitted jointly to one outline can trade against each other, so a fit may redraw the section beautifully from numbers that are individually well out — which is exactly the shape of failure a per-parameter budget cannot see, and the reason the round trip is the test rather than the sum of the three entries.