A fraction of nothing

A sentence of the form N per cent of Raup’s cube is geometrically impossible is easy to write and it appears wherever the model is introduced. It has a number in it, the number can be recomputed, and recomputing it is what this essay is.

The number is not a property of the geometry. It is a property of the rectangle somebody drew the axes on, and the same geometry answers anywhere from a twentieth to well over half depending on where the edges were put. Worse, it has a limit as the rectangle grows, and the limit is zero.

Six boxes and one hyperbola #

The geometry is fixed throughout and it is the exact one: with no translation along the axis, two whorls sharing an azimuth run into each other below D = 1/W and clear each other above it, located to the last bit a double holds. Only the rectangle changes.

The box the model’s own sampling grid uses — expansion from 1.1 to 6, axis distance from 0.02 to 0.9 — gives 37.07 per cent. Taking the axis distance over its whole range instead gives 34.62. Stopping the expansion at 3 gives 52.81. Starting it at 3 and running to 20 gives 11.16. Running from 1.01 to 100 gives 4.64.

A factor of 11.4 #

Those are five readings of one curve. The spread between the widest box and the tightest is a factor of 11.4, and nothing in the geometry moved between them: the same hyperbola, the same contact test, the same units.

The excluded areas themselves barely move between two of those boxes, which is the clearest way to see that the movement is in the denominator. Over the expansion range 1.1 to 6 the region under the curve has area 1.696449 against a rectangle of area 4.9. Trimming the axis distance to run from 0.02 to 0.9 takes twelve per cent off the rectangle’s height and leaves an excluded area of 1.598399 — a small loss from the numerator and a larger one from the denominator, so the share rises from 34.62 to 37.07 per cent without the geometry having been touched.

A quantity that moves by an order of magnitude under a choice nobody thinks of as a choice is a quantity whose value has to be quoted with the choice attached. That is the general form of the problem, and it is the same one a spread that triples when its window triples runs into a subject away: a statistic and its instrument setting arrive together, and detaching them is what makes the number quotable and wrong.

The tight box #

The tightest of the six runs the expansion from 1.1 only as far as 3. Over that interval the hyperbola is high — it starts at 0.909 and has fallen only to 0.333 — so the region under it takes up most of the rectangle, and the answer is 52.81 per cent.

The wide box #

The widest runs from 1.01 to 100, and the same curve has fallen to a hundredth well before the right-hand edge. Almost the whole of the added rectangle sits above the curve, so it dilutes the answer to 4.64 per cent.

Neither box is wrong #

That is the part worth being careful about. There is no correct box waiting to be identified, and picking one would not repair the statistic — it would only hide the choice inside a convention.

An expansion of 100 is a real shape: a bivalve is a coiled shell with an enormous expansion and less than one whorl of it. An expansion of 1.05 is a real shape too, a tight coil of many whorls. A box that admits both is defensible, a box that admits the middle is defensible, and they give 4.64 and 52.81.

Sampling is the other half of the choice #

Even with the edges fixed, the answer is not fixed. The widest box run from 1.01 to 100 gives 4.64 per cent when the expansion axis is sampled uniformly and 21.33 per cent when the same two decades are sampled geometrically — a factor of 4.60 from the weighting of one axis alone.

Why the geometric axis is not a trick #

It is the more natural of the two, which is what makes the factor uncomfortable. Expansion is a growth factor per turn: the step from 1.1 to 1.2 changes a shell far more than the step from 90 to 100, and every measurement of it is a fit to a logarithmic spiral’s one parameter and therefore a measurement of a logarithm.

So an axis that gives equal weight to equal ratios is the one a biologist would draw. It is also the one that reports four and a half times as much excluded ground over identical limits. Both weightings are honest and the number is not a measurement of the shells in either.

Where the fraction comes from arithmetically #

The reason all of this happens is one line. The region under the curve is the integral of 1/W, which is a logarithm. The box is a rectangle, and its area grows as a line.

A logarithm divided by a line has one limit and it is zero. Everything above is that statement made concrete: widening the box adds area to the denominator in proportion to the width and to the numerator in proportion to the logarithm of it, so every widening lowers the share, and no widening ever stops lowering it.

That also explains why the answer is so sensitive at the left-hand edge and so insensitive at the right. The curve is a reciprocal, so nearly all of the excluded area sits in the first stretch of the expansion axis, and moving the right-hand edge outwards adds almost nothing to the numerator while adding everything to the denominator. A box’s lower edge decides how much area there is to share out; its upper edge decides only what it is shared among.

Widening it, and watching #

Held at an axis distance running the whole unit interval and an expansion starting at 1.1, the share falls monotonically with the upper edge: 52.805 per cent out to 3, 34.621 out to 6, 24.801 out to 10, 15.346 out to 20, 7.805 out to 50, 4.560 out to 100.

The limit exists and it is zero #

Out to an expansion of 1000 the share is 0.682 per cent. Out to 10,000 it is 0.0912 per cent. There is no floor: continued far enough the geometry excludes as small a share of the box as anyone wants it to.

That settles the class of claim rather than one instance of it. A percentage of an unbounded space is not a small number or a large one — it is a number chosen by whoever bounded the space, and the fraction excluded is not a well-defined property of the model at all.

A logarithm against a line, drawn #

The two quantities can be put on one pair of axes, and the picture is the whole argument without any percentages in it.

Where the drawn box came from #

The box giving 37.07 per cent is not arbitrary either, and its history is instructive: it is the range the model’s sampling grid was written with, an expansion from 1.1 to 6 and an axis distance from 0.02 to 0.9, on a grid of 26 steps in each direction. Six hundred and seventy-six cells, and the range was never varied.

That is how a box becomes invisible. A grid is written once, with limits chosen so the resulting picture looks well filled, and every number read off it afterwards inherits the limits without anybody restating them. This collection has found the same thing one subject away, where a sample count had been a constant since the first line of the work.

What the grid costs as a measurement #

The grid is also a measuring instrument, and it can be scored against the located boundary. Reading it as a figure would — for each column of expansion, the last cell in contact and the first free cell bracket the boundary, and the midpoint is the best the grid can say — the 26-step grid misplaces the boundary by 0.014931 at worst and 0.008491 on average.

That is about half a grid step, which is the best a grid can do and is not a defect in it. The relative error is the more awkward number: 7.23 per cent at worst and 3.10 per cent on average, because the boundary is a reciprocal and a uniform grid in the axis distance is coarsest exactly where the boundary is smallest.

Refining it is the expensive way to be right #

Refining does work, and it works slowly. At 51 steps the worst error is 0.008717, at 101 it is 0.004317, at 201 it is 0.002160 and at 401 it is 0.001096 — the first power of the step, which is what a bracket-and-midpoint reading gives.

So going from 676 cells to 160,801 buys a factor of 13.6 in accuracy for 238 times the cells. Locating the boundary directly instead — 481 expansions at eleven translations, two hundred bisection steps each, 3,200 boundaries — takes 1.4 seconds and is exact to the last bit a double holds. The grid was never the cheap option; it was the option that existed.

And the third axis moves the answer too #

Everything above holds the shell in a plane. Allowing it a spire moves the number again, because translation along the axis is purely permissive: it frees ground that a flat coil holds in contact and never takes any back.

In the box from 1.1 to 6, the excluded share falls from 34.621 per cent with no translation to 34.114 at a tenth of a turn, 22.402 at a half, and 10.296 per cent at a full turn’s translation. So the same box gives a factor of more than three depending on a parameter the statement usually does not mention at all, and what a spire buys is a measurement in its own right.

One number that barely moves #

There is an exception, and it is the useful one. The translation at which half the contact region has been freed is 0.640146 in the box from 1.1 to 6, and 0.631856 in the box the model’s own grid draws — two boxes whose excluded fractions differ by seven percentage points agree on this to about one part in seventy.

That is what a quantity independent of the box looks like when one turns up. It is a threshold on the third parameter rather than a share of a rectangle, and the reason it travels is that it is defined by the geometry halving something rather than by a rectangle containing it.

Four choices, and all of them silent #

Counting them up: the two edges of the expansion axis, the two edges of the axis distance, the weighting of the expansion axis, and the translation the whole slice is taken at. Six decisions, none of which is usually stated, and between them they take one geometry from 4.64 per cent to 52.81 and from 52.81 down to 10.296.

None of the six is a scientific choice. They are the choices a person makes when deciding what will fit on a page, which is the same reason a share of a front turned out not to be the thing in a quite different experiment: a fraction carries its denominator, and a denominator chosen for convenience makes a fraction that measures the convenience.

What the fraction is not #

It is not the thing Raup is remembered for. His result was that large regions of the occupied space are empty — perfectly constructible coiling parameters that no animal appears at — and that is a census of real shells and not a computation.

The two get conflated because both are described as regions of the morphospace that are not there, and they are entirely different claims. One is measured by drawing a curve, the other by measuring shells, and this collection can do the first and cannot do the second. Quoting a geometric percentage as though it were an occupancy is the error the conflation produces.

What is actually under the curve #

The region below the hyperbola is where two whorls a turn apart would run into each other if the aperture were a circle. That is not a contradiction and it is not rare: involute shells, whose whorls wrap over one another, are the ordinary condition of most gastropods, and their apertures are indented where the previous whorl lies against them.

That indentation is outside the model rather than a violation of it. So even the correctly computed area is the area a circular generating curve excludes, and every percentage above would move with a differently shaped aperture by an amount nothing here measures.

The region above the curve is no emptier. An evolute shell, whose whorls are free of one another, is a planispiral ammonite or a ram’s horn, and there are a great many of those too. Both sides of the line hold real animals, which is the fact that makes a share of either side uninterpretable even before the box is argued about.

The honest statement is a curve #

What survives all of this is the boundary itself, which needs no box: with no translation the whorls of a shell touch when the product of its expansion and its axis distance is one, exactly, at every expansion from 1.02 to 20 and to the last bit a double holds.

A curve is a stronger statement than a percentage and it is shorter. It also composes: the translation moves it in a stated way, and the closed form for where it moves to is a single expression rather than a table of shares.

Or a shell’s own distance from it #

The second honest form is per shell. The section this collection draws at an expansion of 2.4 and an axis distance of 0.42 sits 0.80 per cent outside the boundary. The same axis distance at an expansion of 1.6 sits 32.80 per cent inside it, with 51.36 per cent of its earlier whorl buried under the next.

Those numbers are about shells and they do not move when somebody redraws a rectangle. They are also the ones a reader can check against a specimen, whereas what a summary throws away here is precisely the identity of the shells being summarised.

And a fraction is quotable with its box attached #

None of this makes the fraction meaningless, only unquotable on its own. Thirty-seven per cent of the box from 1.1 to 6 in expansion and 0.02 to 0.9 in axis distance, sampled uniformly, at no translation is a true sentence and a useful one, because it says what the grid a reader is looking at contains.

The failure is entirely in the dropping of the second half. A number that requires six qualifiers and is usually given none is a number whose usual form is not a weaker version of the truth but a different claim.

What this does not show #

It does not show that the region under the curve is small or unimportant. It shows that its share has no value independent of a box, which is a statement about the ratio and not about the region.

It also does not show that anybody was careless. Quoting a fraction is the natural thing to do with a two-dimensional picture, and the picture is genuinely informative — the boundary itself is exact and worth drawing. What the arithmetic here establishes is only that the ratio drawn from it is a property of the frame.

What would refute it #

A demonstration that the excluded share converges to something above zero as the box widens. That is the whole of the claim and it is checkable in one direction: the sequence out to expansions of 3, 6, 10, 20, 50, 100, 1000 and 10,000 falls monotonically to 0.0912 per cent and would have to stop falling somewhere.

It would also be refuted by the six boxes agreeing. If the spread across them were a few percentage points rather than a factor of 11.4, the fraction would be nearly a property of the geometry and quoting it without a box would be a rounding rather than an error. The measured spread is what makes the case, and it is the one number here that could have come out the other way.

Why the percentage is tempting anyway #

Because it is the one form of the result that fits in a sentence, and because a morphospace invites it: once a space is drawn as a rectangle, the eye reads areas and asks what share of the picture is which. That reading is correct about the picture and says nothing about the model.

The same temptation has a milder form throughout this collection, which is why round numbers are not a sample had to be established rather than assumed. A figure is an instrument with settings, and the quickest way to turn one into a claim about nature is to quote a quantity that the settings decide.