An opening that is not round

The cut along the axis found that one plane through a shell’s axis carries all three of Raup’s numbers. Each whorl crosses the plane twice a turn and shows there as its whole generating curve, and seven such sections over three turns, fitted together with the axis free, return the expansion W, the distance from the axis D and the translation T to a few hundredths of a per cent from a drawing accurate to a thousandth of its rim.

Every section in that measurement was a circle, because Raup’s generating curve is a circle. The essay ended on the obvious objection. A snail’s opening is not a circle: it is taller than it is wide, or wider than it is tall, and it is often turned against the axis. The fourth number divides the third had already shown what an ellipse does to Raup’s cube — its height divides the translation in the boundary where whorls touch — so the height of an opening and the translation of the spire are exactly the two quantities a cut would have to keep apart. The measurement here is the axial round trip on shells whose opening is an ellipse.

The cut, with the circles the fit wants #

The shells are the two that essay cut. One is a high spire, W = 2, D = 0.2, T = 2, its whorls standing clear of each other; the other is the shell the shells figures are drawn at, W = 2.4 and D = 0.42, with a low spire, T = 0.5. The generating curve at azimuth $\theta$ still has its centre at $s(1+D)/2$ from the axis and its height at $Ts$, with $s = A\,W^{\theta/2\pi}$. What changes is the curve itself: an ellipse with half-width $\rho = s(1-D)/2$ across the section and half-height $e\rho$ along the axis, turned by $\psi$ in the plane of the cut. At $e = 1$ and $\psi = 0$ it is Raup’s circle, and the inner edge of the opening still sits at $Ds$ from the axis when it is not turned.

The drawing is made exactly as before, each section at forty-eight points moved by independent noise of a stated share of the rim, and read two ways: with Raup’s seven numbers, by the recovery the axial essay built; and with nine, the same seven plus the aspect $e$ and the turn $\psi$, fitted together by least squares on each point’s miss from its ellipse and started where the circle reading ends.

W survives, D and T carry the misfit #

An opening a quarter taller than it is wide, turned 20°, drawn to three thousandths of the rim and read with circles, gives W = 1.9995 — right to the noise. It gives D = 0.1413 and T = 1.9032, where the shell was built at 0.2 and 2. The circles are visibly wrong: they bulge past the ellipses’ flanks and fall short of their tops, and the points miss them by 6.4 times the noise. Read with ellipses, the same drawing gives W = 1.9994, D = 0.2003, T = 2.0024, an aspect of 1.2524 and a turn of 19.65°, and misses by 0.93 times the noise. At an aspect of 1.5 the circles read D = 0.0859 and T = 1.8108 and miss by 12.4 times the noise; the ellipses read 0.2002 and 2.0028.

That W survives is not luck. Every section’s curve is the same ellipse scaled by $s$, so whatever circle the fit puts through a section is the same share larger or smaller than the ellipse in every section, and the ratio of successive sections — which is all W is — is untouched. The misfit has to go somewhere else, and there are only two places for it.

Where the misfit goes, in closed form #

A circle fitted to an ellipse’s points keeps the ellipse’s centre and takes something close to its mean half-axis, $\rho' = \rho(1+e)/2$. The centre stays at $s(1+D)/2$ from the axis and at height $Ts$, so the circle model reads the inner edge at the centre’s distance minus $\rho'$ and the size of the section from the outer edge at the centre’s distance plus $\rho'$. Written out,

$$D' = \frac{r_c - \rho'}{r_c + \rho'} , \qquad \frac{T'}{T} = \frac{r_c + \rho}{r_c + \rho'} ,$$

with $r_c = (1+D)/2$ and $\rho = (1-D)/2$ in units of $s$. On the high spire that gives D 28.6 per cent low at an aspect of 1.25, and the hundred cuts read 29.35. It gives T 4.76 per cent low, and the cuts read 4.89. On the low spire the form gives 11.8 and 3.50 per cent against 12.16 and 3.59 read. The small excess over the form is the circle’s own compromise, which leans a little toward the ellipse’s long axis rather than sitting exactly on the mean.

So the direction of each error is set by one fact. A taller opening makes the circle larger than the section is wide, which moves the circle’s inner edge toward the axis — D falls — and makes every section look larger than it is, so a given height is a smaller share of its size — T falls. A squat opening does the reverse: at an aspect of 0.8 the high spire reads D 24.3 per cent high and T 4.05 per cent high. D moves six times as far as T because D is a difference of two nearly equal lengths on this shell, the inner edge sitting a fifth of the way out, and a difference magnifies a shared error.

The turn does nothing to either. At an aspect of 1.25, turning the opening by 20° leaves the circle reading’s D and T where they were at 0° to the third decimal place: a circle has no orientation, so the fit sees the turned ellipse’s centre and its mean half-axis, and neither moves when the ellipse turns. The circle model cannot see the turn at all, which also means the turn cannot hurt it.

A nearly round opening is the dangerous one #

The residual is the reading’s own warning, and the axial essay found it honest for the failures it tested. Here it is honest only for large departures.

At an aspect of 0.99 — an opening one per cent squatter than round, which no one would see by eye — the points miss their circles by 1.25 times the noise on the high spire. D is off by 13.9 of its own noise errors and T by 6.5. At 0.98 the miss is 1.82 times the noise and D is off by 28 noise errors. The miss passes twice the noise only at an aspect near 0.97 or 1.03, by which point D is forty noise errors wrong. On the low spire the same one per cent moves D by ten noise errors with a miss of 1.11.

That is the compensating failure the outline finds its own centre warned about, in a new form. There, a view five degrees off the section’s plane moved W by three noise errors with the residual at the noise. Here the model is wrong about the shape of every section and the fit absorbs most of the wrongness into D and T, because a circle of the right centre and a compromise radius is nearly as close to a slightly elliptical ring of points as the true ellipse is. What is left of the misfit grows in proportion to the departure, as D and T do, but the residual adds it to the noise in quadrature: a misfit half the size of the noise raises the residual by an eighth. A small misfit is therefore always cheap in residual and expensive in D, and a residual that looks like noise is a statement about the noise, not about the opening.

Fitting the ellipse costs little #

The repair is to stop assuming the circle. The cut shows each section whole, so an ellipse is no harder to fit than a circle; the question is what two more numbers cost the three that matter.

On a round opening the high spire’s W is read to 0.020 per cent either way, T to 0.031 with circles and 0.035 with ellipses, and D to 0.084 with circles and 0.141 with ellipses. D pays most, for the same reason it moved most: it is the number that depends on the section’s width, and the ellipse model now has a second width to spend. On the low spire the costs are W 0.033 either way, T 0.067 against 0.069, D 0.045 against 0.082.

On the opening that is actually elliptical, 1.25 tall and turned 20°, the ellipse model reads W to 0.020 per cent, D to 0.111 and T to 0.036 on the high spire, and the aspect to 0.058 per cent and the turn to 0.082°. On the low spire, W 0.033, D 0.062, T 0.077, the aspect 0.087 per cent and the turn 0.120°. Every one of those is a tenth of a per cent or better from a drawing accurate to a thousandth of its rim. On a round opening the turn is not read at all — its error runs to fifty degrees, since a circle turned is the same circle — and that is the right answer rather than a failure.

How little of a shell the ellipses need #

Two more numbers could have meant more of the shell. They do not. The axial essay found that a single turn — three sections — fixes the seven numbers of the circle model, and the same holds for nine.

Cut over one turn, the high spire with an opening 1.25 tall and turned 20° is read with ellipses to 0.048 per cent in W, 0.128 in D, 0.081 in T, 0.070 in the aspect and 0.086° in the turn, at a thousandth of the rim. Over two turns, 0.028, 0.122, 0.047, 0.059 and 0.086°; over three, the numbers above; over four, W and T no better than three. The low spire behaves the same way: 0.048, 0.069 and 0.136 per cent over one turn in W, D and T, 0.033, 0.062 and 0.077 over three. What more turns buy is W and T, which are read from how the sections change from one to the next and so gain from every extra section. The aspect, the turn and D, which are read from each section’s own shape, are nearly as good from three sections as from nine, because each section already carries forty-eight points of its own outline.

The errors scale with the drawing’s accuracy and nothing else. At three thousandths of the rim the high spire’s D is read to 0.33 per cent and T to 0.11; at a hundredth, a careful sketch, to 1.1 and 0.39, with the aspect to 0.75 per cent and the turn to 0.82°. None of the fitted numbers is biased beyond its own scatter at any of those accuracies, and the circle model’s bias does not shrink at all: D’s 29 per cent at an aspect of 1.25 is a property of the model, the same at a hundredth of the rim as at a thousandth. So the case for reading ellipses is strongest on the worst drawings, where a residual at twice a large noise hides the largest absolute misfit.

Height and translation come apart #

The question the fourth number left was whether a cut can keep an opening’s height apart from the spire’s translation, which enter the boundary where whorls touch together — the boundary one angle decides contact located for circles, and what a spire buys priced. Read with circles it cannot: a taller opening reads as a lower spire, 4.9 per cent lower at an aspect of 1.25, and the two readings of the shell are indistinguishable in T.

Read with ellipses, the two separate. The errors in the aspect and in T correlate at 0.33 across a hundred cuts of the high spire and at 0.15 on the low, and neither error is large — T to 0.036 per cent and the aspect to 0.058. The correlation rises with the length of the cut, to 0.59 over four turns of the high spire and 0.42 of the low, with neither error growing; why the longer cut ties them more closely was not traced, and at a correlation of 0.6 the two are still read separately, each to a few hundredths of a per cent. The axial cut sees the two quantities in different places: the aspect in the shape of each section, the translation in how the sections’ centres descend from one to the next. A weak positive correlation remains because both are read partly through each section’s vertical extent, but a cut accurate to a thousandth of its rim separates them to a few hundredths of a per cent.

So a shell’s position relative to the contact boundary, which depends on T over the opening’s height, can be read off one axial cut — provided the cut is read with the right curve. Read with circles, the high spire at an aspect of 1.25 reads T = 1.90 and an opening as tall as it is wide, and so a ratio of 1.90; the shell was built at a ratio of 2 over 1.25, or 1.60. The error in the quantity contact depends on is nineteen per cent, nearly four times the error in T alone, because the circle model gets both halves of the ratio wrong in the same direction.

The same arithmetic runs across the aspects the circles were read at. The circle model reports the ratio as its T over one, the true ratio is T over the aspect, and the first is the second times the aspect times one plus the bias in T. At an aspect of 0.8 the high spire’s ratio reads 16.8 per cent low; at 1.5, 35.7 per cent high; the low spire at 1.25 reads 20.5 per cent high. A tall opening read as a circle makes the spire look more open than it is, its whorls further from touching, and a squat one makes it look closer to contact. Any survey that places real gastropods on Raup’s cube from sections read with circles carries that error along the one direction in which the contact boundary is crossed.

What this does not establish #

That a real opening is an ellipse. A teardrop, a notched lip or a flared outer margin is none of the curves read here, and each adds shape numbers that an ellipse fit would absorb into its five exactly as the circle absorbed the ellipse into its three — which is the lesson of the nearly round opening, and the reason a residual at the noise should be read as permission to look at the shape rather than as confirmation of it. The cut is taken exactly through the axis and every section is whole; the axial essay priced a missed axis and a shared wall on circles, and neither has been repeated with ellipses. And a section seen from the wrong angle remains a separate failure: a photograph of the cut taken off its normal squashes every section by one factor, and a squash of a circle is an ellipse.

What would overturn it #

A cut of a shell with an elliptical opening, read with circles, whose W is biased beyond its noise. A cut read with ellipses whose aspect and T errors correlate strongly. A circle reading whose bias in D departs from the mean-half-axis form by more than the few per cent its compromise leans. Any of those would mean the account here of where the misfit goes is wrong.

Still open: a photograph of the cut #

The last caveat is the practical one. A worker does not measure a cut shell with calipers point by point; the cut face is photographed, and a photograph taken a few degrees off the face’s normal squashes every section by the cosine of the angle across one bearing — which turns a circular opening into an ellipse of aspect near one and turns every section’s centre by a shared amount. The next measurement reads photographed axial cuts two ways: with the opening’s shape free and the photograph assumed square, which will read the squash as a shell property, and with a single shared squash fitted beside the shell’s own five numbers, asking whether the shared factor across seven sections is enough to tell a squashed photograph of a round opening from a true elliptical opening photographed square.