What a spire buys

A flat coil and a tower differ by one number. T is how far the opening travels along the coiling axis in a turn, it is zero for an ammonite and a nautilus and large for a whelk, and the line where the whorls come apart is exact only at zero.

Above zero the boundary leaves the hyperbola, and the interesting question is not that it does but how. It turns out to do three things, all of which can be stated as numbers: it enters as a square, it moves in one direction only, and it stops mattering above a threshold that has a closed form.

The displacement, written out #

Solving the contact quadratic at a stated translation and subtracting the flat answer gives how far under D = 1/W the whorls actually come apart:

1/W − D = 2T²(W − 1) / [ √((W + 1)² + 4W·T²) + (W + 1) ]

which for small translations is close to T²(W − 1)/(W + 1). The translation appears once and it appears squared, which is the whole of the first result. It is written with the radical in the denominator rather than as a difference of two nearly equal numbers, and there is a reason for that further down.

Three decades, and a ratio of a hundred #

The way to check a square law is to move the input by a decade and see whether the output moves by two. At an expansion of two the displacement reads 3.333259 × 10⁻⁵ at a translation of a hundredth, 3.333333 × 10⁻⁷ at a thousandth, and 3.333333 × 10⁻⁹ at a ten-thousandth.

The ratio between successive rows is 100.000. Not approximately a hundred, and not a hundred after a fit — the assertion this figure carries is that each ratio is within a hundredth of a hundred, and it holds.

Why a square and not a first power #

The reason is geometric and it is worth having, because a square law is what a boundary does at a tangency rather than at a crossing. On the flat coil the two generating circles touch; displacing one of them along the axis by T moves it perpendicular to the direction in which they were already in contact, and a perpendicular displacement of a tangency costs a distance proportional to its square.

That is the same reason a chord’s sagitta goes as the square of its half-length, and it is why the answer could have been guessed. Guessing it is not measuring it, and the ratio of a hundred is the measurement.

What that costs a shell that has just left the plane #

Almost nothing, which is the consequence worth stating in words. A shell translating by a hundredth of a turn has moved its boundary by three parts in a hundred thousand.

So the reading that a little translation helps a shell escape contact is wrong, not because the effect has the wrong sign but because it has the wrong order. A tenth of a turn buys 1.467 per cent of the contact region and a whole turn buys 70.261 per cent. What buys something is a spire, not a tilt, and the difference between those two sentences is the exponent.

The same law at a faster expansion #

The square is not an artefact of the expansion it was measured at. Drawn at an expansion of six, the curve has the same slope over the same four decades of translation; what changes is where it sits, because the leading coefficient carries the factor (W − 1)/(W + 1).

One expansion, across every translation located #

At an expansion of 2.4 the boundary starts at 0.416667 on the flat coil and walks down: 0.415638 at a twentieth of a turn, 0.412558 at a tenth, 0.400331 at a fifth, 0.380276 at three tenths, 0.318577 at a half, 0.207176 at three quarters and 0.066667 at a whole turn.

Read as gaps below the hyperbola those are 0.25 per cent, 0.99 per cent, 3.92 per cent, 8.73 per cent, 23.54 per cent, 50.28 per cent and 84.00 per cent. The first three are the flat part of a square law and the last three are the part that has arrived.

It never goes the other way #

The second result is a negative and it is stronger than the first, because it quantifies over everything located rather than over one expansion. Across all 3,200 boundaries — 481 expansions at each of eleven translations — the located curve sits at or below the hyperbola and in no case above it.

What one-way rules out #

It rules out translation ever putting a pair of whorls into contact that a flat coil leaves free. The ground a spire buys is exactly the strip between the moved boundary and the hyperbola, and it takes none back.

That is worth stating because the opposite is easy to believe. A shell walking along its axis is moving its opening towards the whorl two turns back as well as away from the whorl one turn back, and a plausible account has it trading one contact for another. It does not. The trade does not happen, and the reason it does not is the next result.

Nothing beyond the next whorl round ever decides it #

The contact test used throughout compares a whorl with its immediate successor and stops. That is an assumption and it was checked: 540,000 points, at four separations — two, three, four and six turns apart — and six translations including zero, looking for a shell that clears its immediate neighbour and strikes a turn further away.

Zero cases. The nearest pair always binds.

Why the nearest pair binds #

The reason is that the tolerance falls as the turns between two whorls rise. At an expansion of 1.2 with the opening a fifth of the way out, one turn tolerates a squared translation up to 19.00 and two turns up to 4.56.

So a shell that has translated far enough to clear its immediate neighbour has translated more than far enough to clear everything behind it, and the ordering is monotone in the separation. This is the same shape of argument as a window that turns out to be the neighbour elsewhere in this collection: a check over a larger neighbourhood is worth running once precisely so that the small one can be trusted afterwards.

The translations the boundary was located at #

Eleven, and they are not a grid: 0, a twentieth, a tenth, a fifth, three tenths, a half, three quarters, one, one and a half, two and three. They are spaced to be dense where the square law is still flat and sparse where it has finished, which is the same argument a ladder swept by a ratio rather than a step makes about its own axis.

The routine that draws a boundary refuses any translation not on that list rather than interpolating between two of them. A value interpolated between located translations would be the closed form wearing the bisection’s caption, and the whole force of the flat reading is that those two are independent.

Where the contact region empties first #

At three tenths of a turn, 70 of the 481 expansions have no contact region at all — no distance from the axis puts their whorls in touch. At a half turn it is 199, at a whole turn 328, and at three turns 430.

Those are exactly the expansions that have crossed their own threshold, and the count is the threshold curve read as a tally. It is also why the curves in the one-way figure end at different places: a curve stops where its region empties rather than where the drawing runs out.

What the flat hyperbola costs a spired shell #

Read as a residual against the located boundary, D = 1/W is wrong by 4.52 per cent of its own value at a twentieth of a turn, 18.09 per cent at a tenth, 72.25 per cent at a fifth and 99.32 per cent at three tenths.

So a figure that draws the textbook line beside a shell with any appreciable spire is drawing the wrong line, and by a margin that stops being a correction and becomes a different claim somewhere around a fifth of a turn. The hyperbola belongs to the flat coil and to nothing else.

The threshold, and it is a closed form #

Set the located boundary to zero and solve, and the translation at which the contact region disappears altogether is

T_free(W) = √W / (W − 1)

Above that, no distance from the axis whatever puts the whorls in touch. The region has not shrunk towards an edge; it has left the plane.

What the threshold says about a fast shell #

At an expansion of 1.1 the threshold is 10.4881 turns of translation, which is a shape no shell has. At 2 it is 1.4142, at 3 0.8660, at 6 0.4899 and at 20 0.2354.

So the faster a shell expands, the less spire it takes to come free — a twentyfold expansion needs about a quarter of a turn. Above roughly an expansion of one over the squared translation the contact region has gone entirely, which is the same statement read the other way round.

The threshold is a permission and not a reason #

This is the point at which a geometric result invites an explanation it cannot support. It is tempting to say that high-spired shells exist because the tower frees them from contact, and nothing here licenses that.

What the threshold says is that a spired shell is permitted to coil close to its own axis. Why any lineage does is a question about function, development and history, and this collection has no census to answer it with. A permission is not a cause, and the gap between those two is where most over-reading of a morphospace happens.

How much ground a spire actually buys #

Measured as area, in the box running from an expansion of 1.1 to 6 with the opening anywhere from the axis out to the whorl’s own radius: on the flat coil the whorls are in contact over 1.696449 of a box of area 4.9, which is 34.621 per cent of it. The figures call that share forbidden, and the word means one thing only — a circular opening there would run into the whorl behind it. It is where most gastropods live.

Translation eats that away. A twentieth of a turn frees 0.367 per cent of it, a tenth 1.467 per cent, a fifth 5.838 per cent, three tenths 13.031 per cent, a half 35.295 per cent, three quarters 58.091 per cent and a whole turn 70.261 per cent.

Slow, then sudden #

The shape of that sequence is the square law read as an area, and it is the most useful single picture of what a spire is worth. Half the contact region is gone at a translation of 0.640146 and all of it above 10.488088, which is the threshold at the box’s slowest expansion.

The curve is not a pure square over that range, and the reason is that the box is not one shell. Its slower expansions cross their own thresholds and leave the calculation altogether, so past a half turn the freed share is rising partly because the region is shrinking and partly because there is less of the box left to have a region in.

The percentage belongs to the box #

Every one of those area figures is a fraction of a box, and the box was chosen. The same geometry in the range this collection’s own morphospace figure draws gives a forbidden area of 1.598399, half gone at 0.631856, and a forbidden share falling from 37.069 per cent at no translation to 11.012 per cent at a whole turn.

Those are different numbers for the same shells, and how much they can be made to differ is a separate question with its own answer. Here the point is narrow: the freed share is a ratio of two areas in the same box and is far more stable than either, which is why it is the quantity reported.

The subtraction that had to be avoided #

The displacement is the difference between two numbers that agree to a dozen places at small translations, and taken as a subtraction it throws away up to 13 of the 16 digits a double holds. At a translation of a ten-thousandth the answer is around 3 × 10⁻⁹ and the two operands are around 0.5.

Computing it through the radical’s conjugate — the form written at the top, with a sum in the denominator instead of a difference in the numerator — removes the cancellation entirely. That is why the three decade probes read 3.333333 to seven figures rather than degrading as the translation falls, and it is the reason the square law is visible at a ten-thousandth at all.

What the older contact test got wrong #

The collection carried a contact test that decided the axial half as a separate inequality, as though the plane and the axis could be checked one at a time. They cannot: the true condition is T²(W − 1)² < (1 − W·D)(W − D), and it does not factor into a plane part and an axial part.

Over 1,640,000 points the two disagree on 9.236 per cent of the box, and the disagreement is lopsided — 141 to 1 towards calling a shell free whose whorls in fact run into one another. At the worst point it allows a translation below 0.909 where the geometry allows 10.488.

Why that error survived #

Because almost every figure here is drawn on the plane, and on the plane the separable test is exact. Its first half, W·D < 1, is the boundary itself.

That is the ordinary way a defect in a parameterised routine survives: the parameter it is wrong in is the one nothing varies. It is the same failure as an option nobody moved, and the remedy is the same one — sweep the parameter rather than reason about it.

The test has since been replaced by the inequality above, and the sweep that found the disagreement is kept rather than retired with it. It compares the routine against a bisection of the two drawn circles, so what it now reports — zero disagreements in 1,640,000 points — is an agreement between algebra and geometry rather than between algebra and itself. A check that could only ever have agreed would have been worth nothing either before or after.

What cannot be drawn from here #

One more limit, and it is about pictures rather than numbers. The routine that turns a shell’s parameters into an outline used to read each ring’s radial extent and discard its height, so a spired shell and a flat one at the same expansion and axis distance produced the identical outline. It carries the height now, and returns the axial section beside the plan.

That repairs the routine and does not change what a plan view can show. An outline in the coiling plane is a projection, and a projection of a translating shell is the flat shell — correctly, since that is what looking down the axis returns. So the figures in this essay are curves in the parameter plane and not sections of towers, which is the honest thing to draw and is why no shell appears in it. What the repair buys is that a figure wanting to show a spire now has the height to draw it with, rather than a plan captioned as an elevation.

What the result does not say about shells #

It does not say that whorls in contact are a problem. Contact is the ordinary condition of most gastropods, an involute shell wraps each whorl over the one before, and the region the geometry calls forbidden is forbidden only to a circular generating curve.

A real involute shell has an aperture indented where the previous whorl lies against it, which is outside the model rather than a violation of it. Every threshold above would move with a differently shaped aperture by an amount nothing here measures, and none of them is a statement about which shells an animal can grow.

What would refute it #

The square law fails if a decade of translation ever buys other than two decades of displacement, and three decades were tried at two expansions. The one-way result fails at a single located boundary above the hyperbola, and 3,200 were located. The nearest-pair assumption fails at a single shell that clears its neighbour and strikes a turn further back, and 540,000 points were searched.

The threshold is the one that could not be refuted by more sampling, because it is derived rather than measured — it fails only if the contact quadratic is wrong, and that is checked against bisected circles at every located boundary.

What the third number turns out to be #

A parameter that does nothing at all until it does everything. Below a tenth of a turn it is invisible; by a whole turn it has freed seven tenths of the contact region; above its threshold there is no contact region left to free.

That is a strange shape for a biological parameter and it is worth carrying into any reading of where shells sit in the space. The distance from the boundary is not a linear currency, and what happens to a shell approaching the boundary from the other side turns out to be stranger still.