A boundary with no edge

The line where a coiled shell’s whorls stop clearing one another is the most-drawn object in Raup’s space, and at no translation along the axis it is the hyperbola D = 1/W to the last bit of a double. Being exact is not the same as being an edge.

A line on a plot looks like a border between two territories. Whether it is one is a question about a continuous quantity crossing it, and the answer turns out to depend entirely on which continuous quantity is chosen. Two are available, both are properties of the same two circles, and they give opposite answers.

The two circles the whole thing is about #

A shell in this model is a circle swept round an axis, scaled by W every turn and held at a distance D from the axis. Two whorls that share an azimuth are two circles in the same half-plane, one turn apart, and everything about contact is decided by them. Circles at different azimuths lie in different half-planes and can only meet on the axis, which the model keeps them off.

So there is one pair to look at, and it can be measured in units of the earlier circle’s own outer radius. That whorl then spans the radial interval from D out to 1, and its cross-section is 1 − D thick. The model is self-similar, so this normalisation throws nothing away: the same pair recurs at every turn, only larger.

What linear overlap says #

The obvious measure is how far the two circles have pushed into each other: the sum of their radii, less the distance between their centres. With no translation it comes out as exactly 1 − W·D, in units of that outer radius.

That is a straight line in D with slope exactly W, and it reaches zero at D = 1/W with a corner rather than a curve. At an expansion of 2.5 the boundary sits at 0.4 and the overlap falls at two and a half units per unit of axis distance, all the way from the whole thickness of the whorl on the axis to nothing at the line.

By that measure the boundary is a kink #

Nothing about that picture is ambiguous. The quantity is positive on one side, zero on the other, and its slope does not change as the line is crossed. A quantity that goes to zero linearly picks out its zero unmistakably, and the boundary is exactly where a measurement of it would find one.

This is a real quantity and not an artefact of choosing badly. It is the depth of interpenetration — how far into the earlier whorl the later one reaches — and it is the number that would matter to anything asking how much material is in the wrong place.

The other measure is an area #

The second is the fraction of the earlier whorl’s cross-section that its successor covers. Call it the buried fraction: nought when the circles are clear of each other, one when the earlier one is entirely inside the later.

It is a different question about the same two circles. The first asks how far in the later whorl has come; the second asks how much of the earlier one has gone. On a boundary that behaved like a border they would agree, because both are zero on the line and positive inside it, and the shape of the approach would be a detail.

Three halves, fitted #

It is not a detail. The buried fraction leaves the boundary as the 1.499945 power of the distance below it — fitted over sixty points from a ten-millionth to a ten-thousandth below, worst residual 9.71 × 10⁻⁴ in the logarithm.

Three halves is not a number that needed fitting to be recognised. Two circles that cross by a depth δ enclose a lens whose depth is δ and whose width goes as the square root of δ, so its area goes as δ to the three halves. It is the standard shape of a tangency, and what the fit establishes is that the boundary is one.

What a three-halves power does to a reading #

An exponent above one means the quantity leaves zero flat. Halving the distance below the line divides the buried area by nearly three, and dividing the distance by a hundred divides the area by a thousand — so a shell just inside the boundary is buried by an amount that is not merely small but small in a compounding way.

At a hundredth below the line, 1.1824 per cent of the earlier whorl is covered. At a thousandth, 0.03847 per cent. At a ten-thousandth, 0.00122 per cent — roughly a hundred-thousandth of the whorl, which is less than the width of the line the boundary is drawn with.

The same quantity, at the other end of the same region #

Three tenths below the same boundary, 86.827 per cent of the earlier whorl is buried. That is the same shell, the same measure and the same boundary region: the axis distance has moved from 0.3999 to 0.1 and the contact has gone from invisible to nearly total.

A quantity that runs from a hundred-thousandth to four fifths across one boundary region is not describing a change of kind at the line. It is describing a change of kind spread across the whole interior, with the line marking where it starts rather than where it happens.

Which measure answers which question #

Both are correct and they are answers to different questions, which is the point rather than a complication. The distinction is the same one four packing criteria force: a claim that a boundary is sharp is not a claim about the boundary until the measure is named.

The question a reader actually has is whether two shells either side of the line are two kinds of shell. That question is about how much of one whorl is inside the next, because that is what a person or an animal would notice, and the area is the measure that answers it. The answer is no, and the linear measure cannot give it — falling to zero linearly is a property of a depth, and a depth of a ten-thousandth is not a visible thing either.

Four panels, and only the last two look different #

The ladder above is the argument in one picture. The four depths are a ten-thousandth, a hundredth, a tenth and three tenths below the line, and the buried fractions run 0.00122, 1.1824, 28.607 and 86.827 per cent.

The first two panels show what looks like tangency and what looks like a graze. The last two show shells anyone would describe differently from each other, let alone from the first. Nothing in that sequence happens at the boundary; the drama is spread over the interior, and the leftmost panel is nominally on the wrong side of a line that is supposed to separate two things.

A slider whose stops were taken from the boundary #

The figure below moves the opening’s distance from the axis through nine stops, from half the boundary’s own value to one and a half times it. The step is an eighth of the boundary, so the crossing is the fifth stop exactly rather than something that happens between two frames.

Why that matters more than it sounds #

A slider is an instrument, and its stops decide what can be seen. This collection already carries the counter-example: the shell section’s slider runs the expansion from 1.4 to 4.5 in nine stops of 0.3875, and at an axis distance of 0.42 the boundary is crossed at 2.380952 — which falls between the stops at 2.175 and 2.5625.

No frame of that slider is near the crossing. A reader dragging it sees whorls clearly free, then whorls clearly in contact, and never the state the figure exists to show. The transition happens in the gap, and the picture is honest about every frame it draws while being silent about the only thing anyone came for. Taking the stops from the boundary rather than writing them down beside it is a one-line difference and it is the difference between an instrument and a decoration.

The general form of that defect is a resolution nobody varied, and it recurs across this collection wherever a setting was chosen once and then read as though it were a measurement — a sample count fixed since the first commit is the same shape of error one subject over. A step chosen for convenience is a step that decides what the figure can report, and the only way to know which is to move it.

Where half the whorl is buried #

If the boundary is the level set of the buried fraction at zero, there is nothing special about zero. The level set at a half is a second curve inside the first, and it says where the contact becomes deep.

The share falls, and it has a limit #

At an expansion of 1.1 the half-buried level sits at 0.886530 of the boundary’s own axis distance. At 2.5 it is 0.592613. At 20 it is 0.508655, and the sequence is heading for a half.

So on a slowly expanding shell the deep contact is confined to the last eleven per cent of the interval between the boundary and the axis, and nearly all of the contact region is shallow. On a fast one, half of the interval is already deep burial. The line drawn on a morphospace is the same line in both cases and it hides two quite different interiors.

Read at fixed fractions of the way in #

The same fact stated the other way round is starker. Halfway from the boundary to the axis, an expansion of 1.5 has 74.17 per cent of its earlier whorl buried and an expansion of 4 has 56.42 per cent. A quarter of the way, they are 92.11 and 84.23 per cent. A tenth of the way, 98.14 and 95.98.

A shell a tenth of the way from the boundary to the axis is almost entirely swallowed whatever its expansion, which is the sense in which the interior is not a region so much as a short approach to a limit. What the expansion controls is how quickly that happens, and the range it controls it over is the last measurement anyone would read off a line.

The exponent is a property of the tangency #

Nothing in the three-halves law belongs to the expansion of 2.5 it was fitted at. The figure below redraws both measures at 1.5, where the boundary sits at 0.6666667, and it refuses to draw at all unless the fitted exponent is within a thousandth of three halves.

What changes with the expansion and what does not #

The slope of the linear measure is the expansion itself, so it changes by definition. The boundary moves, from 0.4 to 0.6666667. The half-buried share moves, from 0.592613 to 0.703242.

The exponent does not. That is what makes it a statement about the geometry of two nearly tangent circles rather than about any particular shell, and it is why the disagreement between the two measures is not something that could be settled by choosing a different expansion to argue at.

The expansion is the parameter with the widest range in the model and the easiest to recover from a drawn shell, since it is the growth factor of a logarithmic spiral. It sets where the boundary sits and how fast the interior deepens, and it sets neither the shape of the approach nor the fact that the approach is flat.

Nothing is ever swallowed whole, except on the axis #

One more quantity is worth stating because it bounds the whole interior. The margin against containment — how far the later disc’s centre is from swallowing the earlier one entirely — is exactly (W − 1)·D with no translation, checked to 10⁻¹² at eight settings.

That is zero only at D = 0. So at any distance from the axis at all, some part of every whorl stays outside its successor, however far inside the boundary the shell sits, and complete burial is reached at one point rather than over a region. A hundred per cent is the value on the axis and nowhere else.

The measure that gives way, and where #

The buried fraction has an exact formula, and it is the one thing in this geometry that genuinely fails. It subtracts a triangle from two nearly equal circular sectors, which is the classic way to lose every digit a double holds.

Three digits above a ten-millionth, and a negative area below #

Measured against the asymptotic lens area, which has no subtraction in it, the exact formula holds three digits only while the overlap is deeper than 10⁻⁷ of the whorl’s radius. At 10⁻⁹ it is wrong by a factor of nearly five, at 10⁻¹⁰ by a factor of four hundred, and at 10⁻¹⁴ it returns −3.644 × 10⁻¹⁰ — a negative area, which is a value the quantity does not have.

This is why the power-law fit above has a floor as well as a ceiling. The window from a ten-millionth to a ten-thousandth is bounded below by arithmetic and above by the point where the asymptotic form stops being asymptotic, and a fit run outside it would be fitting the failure rather than the geometry.

The shells this collection already draws #

Three of them sit close enough to the line to be worth reading against it. The section drawn at an expansion of 2.4 and an axis distance of 0.42 has a product of 1.0080 — it is 0.80 per cent outside the boundary, with nothing buried at all and whorls that clear each other by less than one part in a hundred.

The same axis distance at an expansion of 1.6 is 32.80 per cent inside, and 51.36 per cent of its earlier whorl is buried. The same expansion of 2.4 with the opening at 0.2 is 52.00 per cent inside, at 64.55 per cent buried. Those three sections illustrate the three numbers and they turn out to be a shallow miss, a half burial and a deep one.

What a sharp boundary would have looked like #

Stating the negative outcome is worth a paragraph, because this result is a shape and not a threshold, and a shape is easy to assert. A sharp boundary would show the buried fraction leaving the line with a finite slope — an exponent of one, like the linear measure — so that a shell a ten-thousandth inside would be buried by a ten-thousandth of something rather than by a hundred-thousandth of everything.

The fit would have returned 1.0 and it returns 1.499945. It would also have shown a jump, and there is none: both measures are continuous at the line, and the disagreement is entirely about the derivative. Neither measure supports the idea that anything discrete happens at the crossing, which is the reading a line invites and the one the numbers refuse.

The distinction between something that steps and something that slopes is worth the trouble of measuring, because a word that names a threshold is worth much less if the threshold turns out to be a level crossing on a smooth curve. Elsewhere in this collection a cost that could have declined smoothly falls in one step, and the answer decided what a word in the collection was allowed to mean. Here the answer goes the other way, and it decides the same thing.

What the model is not being asked to say #

The circles here are the model’s circles. Real apertures are not circles, they are ovals and slots and, in ammonites, elaborately lobed, and a real involute shell has its aperture indented where the previous whorl lies against it. That indentation is outside the model rather than a violation of it, and it means every burial figure above would move with a differently shaped aperture by an amount nothing here measures.

The same caution applies to the line, which is why the exactness of the hyperbola is a statement about a circle and not about a shell. It also applies to the third parameter: with any translation along the axis at all the boundary moves, and what a spire buys is measured separately.

What none of this says about which shells exist #

Whorls in contact are the ordinary condition of most gastropods. An involute shell is one whose whorls wrap over each other and an evolute shell is one whose whorls are free, and both regions of the space hold a great many real animals. Nothing here is a statement about which shells an animal can build.

That distinction cost this collection a retraction once already, when the region below the line was drawn in the colour reserved for failed claims and called impossible. The correction is on the record and the discipline it requires is the same one the census this site cannot do demands: the geometry can be computed and the occupancy has to be measured, and a computed boundary attached to an uncomputed meaning is the specific mistake available here.

What would refute this #

The exponent is the claim, so a fit returning something other than three halves over a window inside the arithmetic’s range would refute it — and the two failures either side of that window are known, measured and drawn above, which is what makes the window a choice rather than a convenience.

The rest is checkable by construction. The buried fraction and the contact predicate are computed independently, by drawn circles and by closed form, and they agree on every one of the located boundary values; the linear overlap is exactly 1 − W·D and departs from it by less than 10⁻¹² over a hundred and twenty axis distances. A disagreement between the drawn circles and the formula at any single setting would take the whole of this with it, which is a stronger test than a measure that only ever agreed with itself could offer.

The line is exact and it is not an edge #

Both halves have to be held at once, and they are not in tension. The hyperbola is located to the last bit a double holds and it is the right curve to draw, because it is where a quantity that is genuinely there changes sign.

What it is not is a border between two populations. The quantity that decides whether two shells look like two kinds of shell leaves that line flat, at the three-halves power of a tangency, and reaches its interesting values a long way inside. A line on a morphospace is worth drawing and it is worth knowing what a reader crossing it would actually see, which is: for a considerable distance, nothing at all.