The cut along the axis

Raup’s three numbers are the expansion of the whorl per turn, W; the distance of the generating curve from the coiling axis, D; and the distance it travels along that axis, T. A median section — the cut across the axis that shows a nautilus as two nested spirals — carries the first two and is blind to the third, because every point on it lies at one height. The outline finds its own centre read W and D back off such a section with the centre free, and ended by asking for T: a second cut, along the axis, where each whorl appears at a height, and a round trip in all three numbers with the axis shared between the two cuts.

The second cut turns out to make the first unnecessary. One angle decides contact had already said as much in passing: a spire’s outline carries no W, a plan carries no T, and an axial section carries all three. Nobody had measured how well it carries them, whether its axis can be found from the cut the way the centre was found from the section, and what happens to each number when it cannot. That is the measurement here.

Seven circles and an axis #

A helicospiral shell grows its generating curve round the axis while scaling it by W every turn and moving it down the axis in proportion to its size. Taking the curve as a circle, the circle at azimuth θ has its centre at a distance $s(1 + D)/2$ from the axis, a radius of $s(1 - D)/2$ and a height of $T s$, where $s = A \, W^{\theta/2\pi}$. A plane through the axis meets that tube twice a turn, once on each side, and each meeting is the generating circle itself: whole, at its own distance and its own height. Three turns of shell therefore show seven circles, alternately right and left of the axis, each a factor of $\sqrt{W}$ larger than the last and further down.

Two shells are cut. One is a high spire, W = 2, D = 0.2 and T = 2, whose whorls stand clear of each other. The other is the shell the shells figures are drawn at, W = 2.4 and D = 0.42, given a low spire, T = 0.5, so that its W and D can be set against the median section’s readings of the same two numbers. The unit is the same as there: the innermost section’s inner edge sits one from the axis. Each circle is drawn at forty-eight points, and every point is moved by independent noise of a stated share of the rim, the outer edge of the widest section — the scale a person measuring the cut holds.

The recovery is told the points and which section each lies on, and nothing else. It fits seven numbers at once by least squares on each point’s distance from its circle: where the axis crosses the drawing and how it leans, the height of the apex on it, the scale, and W, D and T. It starts where a worker would, with the axis laid by eye along the column of the shell — a quarter of the innermost radius off and two degrees out — and the rest read from each section’s own circle about that axis by straight-line fits.

This fit needs no fence round it. The centre search for the median section had to be confined to a disc, because a centre run far outside the section sees every point at nearly one distance and the residual in $\ln r$ falls towards nothing. A circle has a radius and a centre whatever axis is assumed, and an axis run off to infinity redraws none of the seven, so the residual rises rather than falls as the axis wanders.

One cut does more than a section #

With no noise the round trip is exact. From the worker’s axis, over one, two and three turns and on both shells, the fit returns W, D, T and the axis to a hundred-millionth.

With noise it reads all three. Four hundred cuts of the high spire drawn to a thousandth of the rim give W to 0.023 per cent, D to 0.085 and T to 0.031; the low spire gives 0.036, 0.042 and 0.059. Ten times the noise gives about ten times each error — 0.239, 0.910 and 0.313 on the high spire — and the fit’s residual sits at the noise throughout, at 0.998 of it. That is the signature the median section also had: a reading limited by its drawing and by nothing in the method, with no floor and no error that grows faster than the noise.

The axis is found as the centre was. From a guess a quarter of a radius off and two degrees out, the fitted axis crosses the drawing within 0.0052 of the innermost radius of the true one and leans by 0.0072 degrees; on the low spire, 0.0029 and 0.0134. The median section’s centre over a turn came back to 0.0014 of the innermost radius, so the axis’s position is found a little less well and to the same order, and the axis’s lean, which a median section has no counterpart for, is found to a hundredth of a degree.

Across the axis or along it #

The obvious comparison is the shell both cuts can read. Held at the same noise, the median section and the axial cut of the W = 2.4, D = 0.42 shell give the following.

Over a turn the axial cut reads W to 0.058 per cent and the median section to 0.116: twice as well. Over two turns it is 0.040 against 0.055, and over three 0.036 against 0.060. D comes back to 0.045 per cent from both over a turn, and beyond that the two part: the axial cut holds at 0.042 to 0.045 over every span, while the median section’s D worsens to 0.093 over three turns, for the reason the outline finds its own centre gave — noise fixed against the rim is larger against the innermost wall the longer the span. And the axial cut reads T as well, to 0.139 per cent over a turn and 0.059 over three, where the median section reads nothing.

The reason is what each cut samples. A median section shows a continuous curve, but its reading of W is a slope in $\ln r$ against an azimuth, and both depend on where the centre is placed; its reading of D is a ratio of two walls at each azimuth, and the innermost stretch carries the most noise for its size. An axial cut shows only two sections a turn, but each is a whole circle, and forty-eight points on a circle fix its centre and radius far better than they fix a stretch of spiral. W is then the ratio of successive circles, read at seven places; D is inner edge over outer edge inside each circle, read seven times. The cut throws away the spiral between the sections and keeps what it needs from them.

Three sections are enough #

Seven numbers need at least seven equations, and each circle supplies three — two coordinates of its centre and its radius. Two sections, half a turn of shell, supply six and cannot fix the seven; the reading refuses to start there. Three sections, one turn, supply nine.

Nine is enough. One turn of the high spire reads W to 0.051 per cent, D to 0.095 and T to 0.083. Four turns take W to 0.018 and T to 0.026, and D to 0.084 — barely moved. The numbers divide by where they are read. D is read inside each section, from its inner and outer edges, and a fourth turn adds sections without making any one of them more exact. W and T are read between sections, from how the circles grow and how far down each one sits, and a longer run of sections gives them a longer baseline, exactly as a longer span gave the median section’s W a longer arc.

T is read worse on the low spire, 0.139 per cent over a turn against the high spire’s 0.083, and the reason is scale rather than method. The low spire’s sections sit only half their own size apart in height, so the heights T is read from are small against noise fixed to the rim, and a relative error in a small quantity is a large one. By three turns the low spire’s T comes to 0.059 per cent, and by four to 0.056.

An axis left where it was laid #

A worker who lays the axis by eye along the column and reads the numbers about it, without fitting the axis, pays the price the joint fit avoids. On a clean cut of the high spire, that price is set by how the axis is wrong.

An offset goes into D. A quarter of the innermost radius off costs D 0.86 per cent, W 0.12 and T 0.31: D is the ratio of each section’s inner edge to its outer edge about the axis, and moving the axis sideways adds to one distance what it takes from the other.

A tilt goes into T, and on a high spire into D much more. Two degrees costs the high spire’s T 3.9 per cent and the low spire’s 4.5; that is the misplaced axis tilting each whorl’s height into its radius, which the planispiral round trip had expected. But it costs the high spire’s D 11.6 per cent and the low spire’s 1.3. The mechanism is the height. An axis leaning by $\psi$ is displaced sideways, at a section a height $h$ down, by $h \tan \psi$, and the high spire’s lowest section sits eighty units down with its inner edge eight units from the axis: two degrees moves the axis 2.8 units there, a third of the distance D is read from. The low spire’s lowest section sits sixteen units down with its inner edge fourteen out, and the same lean moves the axis 0.58 units there, a twenty-fourth of that distance.

The worker’s axis — a quarter of a radius off and two degrees out together — costs the high spire’s T 4.3 per cent and its D 13, and the low spire’s T 5.1. Five degrees costs the high spire’s D 39 per cent.

The laid axis leaves a mark #

Unlike the tilt of an oblique photograph in the outline finds its own centre, a laid axis is not a compensating failure: the residual gives it away. Held at the worker’s axis, cuts drawn to a thousandth of the rim leave a residual 25.9 times the noise on the high spire and 10.7 times on the low. On a clean cut it is 2.9 per cent of the rim on the high spire. A circle redrawn about a wrong axis has nowhere to go; it cannot shift to absorb the error the way a free centre shifted to absorb a squash, because the axis is the one thing it was not allowed to move.

So the axial cut’s own errors are of two kinds, and a worker can tell them apart. A reading with the axis held has a residual that shows the axis is wrong; a reading with the axis free has a residual at the noise and numbers limited by the noise. The first is always worth replacing with the second.

What trades against what #

The worry from the median section was that three numbers fitted jointly would trade against each other, so that a fit redraws the drawing well from numbers individually wrong. There the trade was between W and the centre, and the joint fit made it short. Here the axis is found so well that its trade is shorter still, and a different one appears.

Across four hundred cuts of the high spire the errors in W and T are correlated by −0.68, and across the low spire by −0.34. W and D are uncorrelated on both, at 0.07 and 0.02. The axis’s own error explains little of T’s: the fitted lean and T’s error are correlated by 0.35 and 0.44, so the lean accounts for about an eighth of T’s variance on the high spire and a fifth on the low.

The trade is built into how T is read. Each section’s height is $T s$ below the apex, and $s$ is the scale the fit assigns that section from W. A W read slightly high makes the lower sections’ scales slightly larger, and the same measured heights then need a slightly smaller T. The high spire, whose T carries most of each section’s position, shows the trade most. It is not a failure: it means an error in W should be read together with the error in T rather than beside it, and a worker comparing two shells’ W and T should compare them as a pair.

A saw that misses the axis #

Everything so far has had the right model. A real axial cut is sawn or ground, and the plane it leaves is parallel to the axis without passing through it. The model has no number for that, so the question is where its error goes and whether the residual says so — the same question a section seen from the wrong angle asked of an oblique photograph.

A plane a distance $e$ from the axis meets each whorl a little past its meridian on one side and a little before it on the other, because the tube has to turn by an angle $\delta$ with $R \sin \delta = e$ before it reaches the plane. So every right-hand section is read slightly further along the spiral and every left-hand one slightly earlier, and each appears at $R \cos \delta$ from the axis rather than at $R$.

The free axis absorbs most of it. On the low spire a cut missing the axis by a fifth of the innermost radius moves W by 0.087 per cent, D by 0.021 and T by 0.148, and the fitted axis moves 0.033 of a radius to take up the rest. The errors grow in proportion to the miss, not as its square: the shift of one side forward along the spiral and the other back is first order in $e$, and the axis can only mimic part of it.

And on the low spire the residual does not say so. At that fifth of a radius it is 1.08 times the noise, while W is out by two and a half times its own noise error of 0.036 per cent. It stays within a fifth of the noise until the cut misses by nearly a third of a radius. On the high spire the same miss leaves a residual of 1.78 and W 0.16 per cent out, T 0.20: the lower sections of a high spire sit so far down that the fore-and-aft shift of a section moves it measurably off any circle the free axis can offer, and the fit visibly fails. The low spire is where a missed axis hides, and every nearly planispiral shell is a low spire.

Whorls in contact #

The sections here stand clear of each other and show all the way round. In most gastropods they do not: each whorl rests on the one before, and the part of its section against the previous whorl is shared wall rather than its own. A worker sees the outer part of each circle and not the inner.

D is what goes. With only the outer half of each section drawn, the high spire’s D is read to 0.355 per cent against 0.085 with the whole circle, and the low spire’s to 0.176 against 0.042 — four times worse on both. With three tenths drawn it is 1.22 and 0.61. W and T barely move: 0.025 and 0.039 per cent on the high spire with half of each circle, against 0.023 and 0.031. That is the division of labour again. D is the inner edge over the outer, and the inner edge is what contact hides; W and T are where the sections sit, and an arc of a circle fixes its centre nearly as well as the circle does.

This does not model the shared wall itself, which belongs to the previous whorl and could in principle be read as part of it. It prices only what losing the inner arcs costs. A worker after D in a shell with touching whorls should expect several times the error a clear-whorled shell gives, and should look for a section in which the umbilical side is exposed.

What changes for a worker with a sawn shell #

The planispiral round trip ended with advice about photography and residuals. The helicospiral one has four pieces of advice, each measured.

Cut along the axis and fit the axis. One cut then reads all three of Raup’s numbers, and on the shell the median section was read on it reads W twice as well over a turn and D as well, with T besides. What a spire buys found that T enters the contact boundary only as its square; a worker who wants to place a shell against that boundary can now read T from the same cut that gives the other two.

Never leave the axis where it was laid. Two degrees of lean costs T four per cent and a high spire’s D over ten, and the residual shows it at ten to twenty-five times the noise.

Read W and T as a pair. Their errors are correlated by −0.68 on a high spire, so a small difference in both between two shells, in opposite directions, is the signature of the fit rather than of the animals.

And distrust a low spire whose residual is at the noise. A missed axis moves W by several times its own noise error there and leaves the residual where the noise put it. The only protection is the saw: grind the cut down to the axis, where the innermost sections are smallest and the miss is largest against them.

What this reading assumes #

That the generating curve is a circle. The fourth number divides the third measured the contact boundary for eleven openings, and found that an ellipse’s height divides the translation; a fit that assumes a circle on a shell whose opening is an ellipse will read that ellipse’s shape into D and T, and nothing here says how. That the plane of the cut is parallel to the axis: a plane that leans to it is a different failure and is not read. That the whorls are clear of each other or, in the contact case, that only their inner arcs are lost. And that the noise on each drawn point is independent — a drawing traced by hand along a whole outline carries correlated errors, which the residual would understate.

Measurements that would overturn it #

A noiseless axial cut from which the fit, started at the worker’s axis, does not return all three numbers to rounding. Errors in W and D that are correlated across cuts. An axis held two degrees out whose residual stays at the noise. A missed axis on a high spire that the residual does not show. Each would mean the reading here is wrong rather than incomplete.

Still open: an opening that is not round #

Every section here is a circle, which is Raup’s own generating curve and the one the error budget for a nautilus and its successors used. A snail’s opening is not a circle; it is closer to an ellipse or a teardrop, and it is often turned against the axis. The next measurement is the same axial round trip on shells whose generating curve is an ellipse of stated height and turn, read two ways — with the circle model, to see where its misfit goes and whether the residual catches it, and with an elliptical model whose two extra numbers are fitted with the rest — asking whether an axial cut can separate the height of an opening from the translation of the spire, which the fourth number divides the third showed enter the contact boundary together.