Four finders, one option

The same corrugation turns up in four places — a beech leaf in its bud, an earwig’s hind wing, a folded gut lining, a deployable array — and none of the four learned it from the others. That is the ordinary shape of a convergence argument, and the ordinary conclusion drawn from it is that the corrugation must be a good solution to a problem the four share.

The conclusion may be right. What the observation supports is weaker than it looks, and the reason is arithmetic rather than biology: on the axis the four are supposed to be optimising, there is nothing to choose.

The axis with no freedom on it #

The folded footprint times the mean number of layers over it is the area of the sheet, exactly. It is a conservation law, it has no slack, and it has a consequence that is easy to state and easy to skip past.

A packing ratio is a layer count. A sheet folded into a thirtieth of its area has thirty layers on average — not approximately, not typically, but as an identity — and that is true of a corrugation, a twist, a crumple, a rolled tube and anything else a sheet can be made to do.

Measured on the patterns this collection folds, the identity holds to the resolution the folded states are sampled at — and it is the same identity a thick panel runs into as a pile: the Yoshimura shrinks sixty times and averages sixty layers; the waterbomb tessellation thirty-one and thirty-one; the preliminary base eight and eight.

So a lineage that needs to fit a leaf into a bud a thirtieth of its size is not choosing between designs with different layer counts. It is choosing thirty layers, and every arrangement that achieves the ratio achieves it that way.

What the four could have differed in, and did not have to #

If the ratio is fixed by the requirement, what is left for a lineage to get right?

Crease length is the obvious candidate. A pattern that reaches a given ratio with less creasing is cheaper to make, cheaper to maintain and less liable to fail at a fold — and unlike the ratio, it is genuinely free.

Measured across thirty-one patterns from four different constructions, the relation between how much a pattern shrinks and how much creasing it costs per unit of paper is weak: on logarithmic axes the correlation is 0.45, so knowing the ratio leaves most of the crease density undetermined. A corrugation that shrinks by three can cost anywhere from 4 to 25 units of crease per unit of paper.

That is the freedom a convergence argument ought to be about, and it is the axis nobody compares — because it is not what a folded organ is described by.

The three things a sheet can do, and how short the list is #

The safest reading above says the four converged on “the only kind of thing there is”. That deserves a count rather than an assertion, and the count is short.

A developable sheet that has to occupy less room than it does has three families of option, and every pattern in this collection belongs to one of them.

Corrugate. Parallel or nearly parallel creases, a fold angle, and a ratio equal to the layer count. The Miura, the Yoshimura, the accordion, the leaf pattern and every deployable array are here.

Roll. No creases at all: the sheet wraps a cylinder, and the ratio is set by the radius — though a corrugated sheet chooses its own cylinder when the two are combined. It is the one option that costs nothing in creasing and it gives up the flat folded state entirely.

Crumple. Creases at every angle, arriving by buckling rather than by design, with the pattern the sheet gives itself rather than one anybody chose.

Three families is not many, and it is the honest content of the “nothing else to arrive at” claim. What the list also shows is where the families differ: they differ in crease density and in whether they reopen, which are the free axes, and not in the ratio, which is the constrained one.

What convergence is evidence for, then #

Three readings of the same observation, in increasing order of what they require.

The weakest and safest. Four lineages needed to pack a sheet into a small footprint. Packing a sheet into a small footprint is corrugating it, because the conservation identity says the ratio is a layer count and layers come from folds. So the convergence is on the only kind of thing there is.

The middle one, which is measurable and mostly unmeasured, and which needs a population to be measured over before it means anything. The four converged on the same corrugation rather than on corrugation in general — the same crease directions, the same cell shape — and that is a claim about the free quantity rather than the constrained one. It would be evidence about optimality if the alternatives at the same ratio had been enumerated and shown to be worse, and they have not been.

The strongest, which is what the argument is usually taken to support. The corrugation is the best solution to the shared problem. Nothing in a convergence supports this without the middle step, and the middle step is a comparison across the free axis — which is exactly the comparison the identity leaves undone.

The free axis is not free inside a corrugation #

The correlation of 0.45 is measured across four constructions that share almost nothing, and it is worth asking what it looks like within the family the four organs actually belong to. It looks much worse for the argument.

Take the simplest corrugation: parallel creases across a sheet, folded to flat. The packing ratio is the layer count, the layer count is one more than the number of creases, and each crease runs the full width of the sheet. So the total crease length is the crease count times the width — which is the ratio times the width, whatever the spacings are.

Move the creases about however one likes. Bunch them at one end, space them geometrically, alternate wide and narrow cells: the ratio does not change, the crease count does not change, and the total length of crease does not change either. Crease density is fixed by the packing ratio inside the corrugation family, exactly as the layer count is.

So the axis this essay proposed as the free one, and on which a convergence argument might have been doing work, is not free where it matters. The 0.45 correlation is a fact about comparing a twist tessellation with a fold-and-cut pattern — objects no lineage was choosing between — and within the family the four converged on, the two quantities are the same quantity.

Which leaves a shorter list than the one above #

Squaring the correlation says the same thing about the wider comparison. At 0.45 the shrink accounts for a fifth of the variation in crease density and four fifths is something else — but the something else is which construction, and constructions are not alternatives available to a leaf.

What is genuinely left open, once the ratio has fixed the layers and the creases, is the second list this essay gives and nothing from the first: where the creases run, what shape the cells are, how the pattern behaves part-folded, and whether it opens by itself. Those are properties of the arrangement rather than of the amount, and none of them is visible in a ratio or in a total crease length.

That makes the argument tighter rather than weaker. A convergence on the packing ratio is arithmetic. A convergence on the total creasing is the same arithmetic wearing different units. A convergence on the cell shape would be the first observation in the sequence that is about a choice, and it is the one the descriptions of these four organs come closest to supporting and never quite state — because a leaf’s corrugation and an earwig’s wing are described by their ratios, their fold counts and their outlines, and not by the quantity that would settle the question.

Where the option set really is small #

The argument above says the option set is not small in the way it looks. There is a sense in which it is genuinely small, and it is worth separating because it is the good version of the same claim.

A sheet that has to deploy — open by itself, from one input — is not choosing among all patterns. It is choosing among patterns with a single degree of freedom, and that is a very short list: a Miura has one, a general corrugation of parallel creases has one, and most patterns have none at all because they do not fold rigidly.

That is a real constraint, it is structural, and it does cut the option set down. It is also a different constraint from the packing ratio, and it is the one a convergence argument would have to invoke to be doing any work — a lineage converging on a Miura-like pattern because Miura-like patterns are the ones that open is a much stronger statement than a lineage converging on thirty layers because thirty layers is what a thirtieth requires.

What a lineage does not get to choose #

It is worth being concrete about which parts of a folded organ are decided by the requirement and which are left open, because the split is not where intuition puts it.

Decided by the ratio: the average number of layers, exactly. Also the total crease length to within the pattern’s family — a corrugation of n folds has n creases across the sheet, and the ratio is n.

Left open: where the creases run, what shape the cells are, how the pattern behaves part-folded, whether it opens by itself, how the folds are distributed across the sheet, and whether the layers are ordered in a way that lets the sheet reopen without tearing.

The second list is longer, and every item on it is invisible in a packing ratio. That asymmetry is what makes the ratio such an attractive number to report and such a poor one to compare designs by: it is the property that is easiest to measure on a folded organ and the only one with no freedom left in it.

An argument from convergence that wants to say something about optimality has to be about the second list. Four lineages arriving at the same cell shape would be a striking fact; four arriving at the same ratio is four lineages needing the same amount of room.

Which theorem was checked, and how #

The identity is verified rather than cited. The layers counted over each folded footprint, times the area of a sampling cell, must equal the total area of the paper — and they do, to within 1.6 per cent on every pattern, the residue being the sampling grid.

The correlation is computed over four populations at once, so that it is not a fact about one family of patterns: eight printed patterns, twelve twist tessellations, six quadrilateral meshes and five fold-and-cut patterns, thirty-one in all.

The freedom count is a count, made on the patterns rather than asserted from the literature.

And every number survives the biology being deleted. Nothing here computes a growth rate, an energy or a material property; the leaf and the wing enter as requirements — a ratio, a deployment — and every measurement is made on paper.

Where the model stops #

The identity is about flat folded states. A leaf in a bud is not folded flat, and its packing ratio is correspondingly worse than the pattern’s limit; the identity still holds at each stage of the fold, with a smaller ratio and a smaller mean layer count, so the argument survives with every number reduced.

Thirty-one patterns is thirty-one, drawn from four constructions that disagree about everything else. The correlation between shrink and crease density is measured over the patterns four constructions produce, and a different set would give a different number. What is not sensitive to the set is the identity, which is exact.

Nothing here is a claim about any lineage. No growth process is modelled, no developmental sequence is proposed, and no organism’s pattern is reconstructed. The essay is about what an observation of convergence licenses, given a conservation law about paper.

And the four are not four independent trials of anything. A leaf and a wing are both thin sheets folded by growth in a confined space, so their problems are similar in ways beyond the packing ratio, and treating them as independent draws from a space of solutions is a further assumption that the convergence argument makes silently.

What the picture cannot show #

None of these figures shows a lineage, a history or a selective pressure, and the argument is about what those can be inferred from. A table of shrink against crease density is a picture of what is available; whether anything chose among the available things is not in it and cannot be.

Nor can a figure show the option set that was not taken. The claim that a convergence is uninformative if the alternatives were all equivalent needs the alternatives drawn, and what is drawn instead is a scatter of the patterns this collection can build — which is a sample of the available rather than a census of it.

The idealisation, named #

Every sheet here has zero thickness, and the identity depends on it: layers lie on one another with nothing between them, so the mean layer count over the footprint is exactly the ratio.

A real leaf has thickness, which is why the surface-in-a-volume curve turns over — past some number of folds the layers themselves fill the box and the extra surface has nowhere to be. So the identity is the zero-thickness limit of a relation that bends, and the bending is the whole content of the rung one step down. What survives is the direction of the argument: at any thickness, the ratio and the layer count are the same quantity to within the thickness correction, and neither is free.

The generalisation #

A convergence is evidence in proportion to the size of the option set that was not taken. Four finders landing on one answer is striking when the answers were many and unremarkable when they were few, and how many there were is a question about the problem rather than about the finders.

The specific form here is the one worth carrying out of the subject. Where a conservation law relates the objective to the mechanism, the objective determines the mechanism and there is no design freedom on that axis at all. Anybody comparing solutions on such an axis is comparing rearrangements of an identity, and the interesting comparison is on the axes the law leaves free — which are usually the ones nobody has a number for.

The uncomfortable version, for this collection as much as for anybody: the numbers reported for folded organs are almost always packing ratios, because a packing ratio is what a folded thing is for. That makes it the number everybody has, and the identity makes it the number with nothing in it.

Where the ladder goes next #

The measurement this rung asks for and does not make is a census of the alternatives: for a stated packing ratio and a stated deployment requirement, every pattern that achieves both, with its crease density. Four lineages agreeing would then be a fact about a set of known size rather than about a set nobody has counted.

That is within reach for the corrugations — parallel creases, one degree of freedom, a ratio fixed — and out of reach in general, because enumerating patterns that deploy means deciding rigid foldability, which is the expensive half of the subject.