The same corrugation in four places

Four systems that share nothing arrive at nearly the same fold. A corrugated leaf packs into a bud. An insect’s hindwing stows under a case. A thin cylinder crushed along its axis buckles into a diamond pattern. A solar array folds for launch.

The temptation is to call this biomimicry, or convergent evolution, or a deep unity. All three are available, none of them is checkable, and this site is in an unusually good position to say something better: the requirements can be computed, and the set of patterns satisfying all of them is small.

Why convergence stories are usually worthless #

The standard form of the argument is a resemblance followed by an explanation, and the explanation is unfalsifiable.

Two things look alike; therefore the same pressures produced them. The problem is that no version of that sentence can be wrong, because any resemblance can be assigned a pressure after the fact and any difference can be assigned a constraint. The argument has no way of failing and therefore establishes nothing.

What would make it an argument is a prior statement of the requirements, with the check that the observed solutions satisfy them and that most alternatives do not. That is available here, because each requirement in the list below is something computed here, in a figure that refuses to draw when it fails.

The difference between the two versions is not rhetorical. A resemblance-based account is compatible with every possible observation and therefore cannot be used to predict anything: it cannot say what a newly examined system will do, and it cannot say what would be surprising. A requirement-based account rules things out, and ruling things out is the only way an explanation earns its keep.

It is also the reason this essay sits on a site about geometry rather than about biology. The claim being made is not about organisms — it is about how small the set of patterns satisfying a stated conjunction is — and that is a question with a computable answer, or at least a computable partial one. Where the argument touches organisms it is borrowing observations from elsewhere, and it says so wherever it does.

The requirements, each of them computed #

Four conditions, and every one has appeared in this field with a generator attached.

It has to pack. The packing fraction of each geometry is computed from its own parameters, and the spread across candidates is a factor of three. A fold that does not reduce the footprint substantially is not solving the problem.

It has to move under one input. One degree of freedom means one driver, and the freedom is counted from the kinematics rather than from the crease count. This is the sharpest of the four and the one that eliminates the most.

It has to open one way. The exposed span has to rise at every step of the motion, or something has to be available to push it backwards. For a leaf nothing is; for a wing and an array the requirement is weaker but the property is still worth having.

It has to be robust to being made slightly wrong. This is the one the site computes only indirectly, and it is why the tapered corrugation’s finding matters: the parameters that can vary without destroying flat-foldability are a small subset of the parameters one might want to vary, and a pattern that only works at exact proportions will not develop reliably or manufacture cheaply.

What survives all four #

The conjunction is restrictive in a way none of the individual conditions is.

Rolls pack best and fail the second and third. Fans pack moderately and fail at their apex, where equal sectors with alternating letters violate Maekawa outright. Patterns with many independent freedoms fail the second. Patterns that lock fail the third. Patterns whose angles must agree exactly fail the fourth.

What is left is corrugations and their two-directional relatives, which is a small family, and it is the family all four systems use.

The elimination is worth walking rather than asserting, because each step is a computation this field already ran. The roll’s failure is a kinematic one: its motion has no discrete parameter and no crease pattern, so there is nothing for a single input at a fixed point to act on. The fan’s failure is a theorem: at a vertex with equal sectors and alternating letters the mountain and valley counts are equal, and Maekawa forbids equality outright, which means the classic radiating fold does not close flat at its apex without something extra.

Patterns with many freedoms fail by counting, and the count is the one the census computes. Patterns that lock fail because a folded state can exist with no motion reaching it, which is a property established by the site’s own rigid-folding machinery rather than by inspection.

Four eliminations, four different kinds of argument, and none of them requires looking at an organism. That is what distinguishes a requirement list from a description: the list can be applied to a candidate nobody has ever built.

That is a different kind of statement from “these things look alike”. It says the resemblance is a consequence of the requirement list, and it comes with the list.

What each of the four is actually doing #

The systems are worth separating, because they satisfy the conditions for different reasons and one of them is not doing anything at all.

The leaf is packing into a container and opening under growth. It needs all four conditions and its actuator is the weakest one in this field.

The wing is packing under a case and opening under a muscle at its base. It needs the first, second and fourth, and its version of the third is weaker because a muscle can pull both ways.

The array is packing into a fairing and opening under a single release. Same three, with the fourth replaced by manufacturing tolerance, which is a specification rather than a fact.

The buckled cylinder is doing none of it. It is not solving a problem; it is what a thin shell does when it is crushed, and the pattern is an outcome of a mechanical instability rather than an answer to a requirement anybody had. That it lands on the same geometry is the most interesting of the four cases and the one that most needs care, because it is the case where the language of solutions and pressures has nothing to attach to.

The fifth case, which is a sheet of paper #

There is a fifth arrival at the same family and it is the one this site started from, which makes it a useful control.

Traditional origami has corrugations and pleats in enormous quantity, and they were arrived at by people making things with their hands over a long period, with no requirement to pack into a container, no actuator to satisfy and no manufacturing tolerance. The requirements list above does not apply to a paper model at all.

That should be uncomfortable for the argument and it is worth facing rather than omitting. If a family of folds arrives through a route that does not satisfy the requirements, the requirements are not the whole explanation.

The available response is that the craft route has a requirement of its own that overlaps: a fold a person can make reliably, from a square, with a small number of reference points, and can undo and redo while learning it. That is a robustness condition of exactly the same kind as the fourth, arrived at through a completely different pressure, and it selects for the same tolerance to imprecision.

Whether that is a real explanation or a comfortable one is a fair question. It is stated here as the weakest link in the argument rather than smoothed over, because a convergence account that cannot name its own weak point is the kind this essay opened by dismissing.

The case that is not convergence #

The cylinder deserves its own paragraph because including it in a convergence story would be a mistake, and the mistake is instructive.

Convergence means independent arrival at the same solution to the same problem. The cylinder has no problem. It is minimising an energy under a load, and the pattern that comes out is the one that costs least, which has nothing to do with packing, deployment or degrees of freedom.

That the answer coincides is a fact about the geometry being doubly special: the same family that satisfies the deployment conditions also happens to be the one a shell finds cheapest. Whether there is a reason for that coincidence is a genuine question and this site cannot answer it — the energy calculation is not here and would not be this site’s to make.

What can be said is that the coincidence was noticed the wrong way round historically. The pattern was published as buckling analysis in 1955 and as a deployable in 1970, and the second was not derived from the first. The order of discovery is a matter of record and the direction of influence is not.

Distinguishing this from rediscovery #

There is a nearby argument in this site’s history field and the two are worth keeping apart.

The same vertex was found four times is about people: different investigators arriving at the same mathematical object, in different decades, in different languages, without knowing about each other. Its evidence is documentary — publication dates, citations, the absence of citations — and it is a claim about a scientific community.

This essay is about systems: a leaf, a wing and a shell arriving at the same geometry with no community involved at all. Its evidence is computational — the requirement list and what satisfies it — and it is a claim about a constraint.

They can look identical in a summary and they are established by completely different means. Confusing them produces the worst kind of convergence writing, in which a historical coincidence is offered as evidence of a natural law — or, just as often, a genuine constraint is offered as evidence that somebody was influenced by somebody else.

What would falsify this #

The requirement list makes the claim testable in a way the resemblance version is not, and it is worth saying how it fails.

If a system with all four requirements were found using a pattern well outside this family, the list would be incomplete or wrong. If a pattern in the family turned out to fail one of the conditions on computation, the list would be wrong. And if the family turned out to be large — if a great many patterns satisfy all four — then the shared solution would be a coincidence rather than a consequence.

The third is the one this essay is least able to close. Nothing here enumerates the patterns satisfying all four conditions; the argument is that each condition eliminates a great deal and that the ones examined survive. A proper version would count the survivors, and that is a piece of work this field has not done.

Saying so is not a formality. An argument of this shape stands or falls on the size of the surviving set, and the site has the machinery to make a start — the assignment enumerator, the rigid-folding solver, the packing arithmetic — while the enumeration itself is over a space of patterns rather than of assignments, which is a much larger and less well-defined object. A count over a restricted family, say all corrugation-like patterns on a grid up to some size, would be an honest partial answer and is the obvious next rung.

Until that exists, this essay is a well-specified conjecture rather than a result, and it is better labelled that way than dressed up. The part that is established is the requirement list and what each condition eliminates; the part that is not is the claim that little else survives.

A partial count, from work already done #

The missing enumeration is not entirely missing. Two restricted versions of it have been run elsewhere on this site, and both give the same answer, which is worth putting here because a partial count is what the argument is short of.

Restrict to repeating rules on a quadrilateral-grid corrugation. A rule is six bits — a letter per row parity, a letter per combination of column and row parity — so sixty-four candidates. Sixteen satisfy the vertex conditions. Those sixteen are four objects once the reversal of every letter and the shift of the starting cell are quotiented out, and all four fold to the same corrugation.

One object out of sixty-four.

Restrict instead to repeating rules on a waterbomb tessellation, a different grid with two kinds of vertex. Nine bits, five hundred and twelve candidates, thirty-two survivors — and the thirty-two produce a single folded object, panel by panel.

One object out of five hundred and twelve.

Two families, two independent enumerations, and in both the surviving set is a single member. That is a considerably stronger statement than “each condition eliminates a great deal”, and it is the shape the essay’s third falsifier asks for.

And the geometry direction collapses too #

The obvious objection is that a count over rules is not a count over patterns, since a rule is one letter table and a pattern also has widths, heights and angles in it. That direction has been measured as well, and it collapses.

The grid corrugation’s rule table was swept over six deliberately awkward geometries — even columns, a one-sided ramp, a taper of twenty to one, taller rows, a steeper zigzag — and returned the same sixteen every time, with the same forty-eight failures and the same failure modes. The reason is that the vertex condition at a corrugation is a linear equation over the two-element field, and a linear equation has no room in it for a length.

So the continuum of geometries does not multiply the count. It is one rule applied to a family of shapes, not a family of rules. The survivors over patterns are therefore no more numerous than the survivors over rules, which is the step that lets the two enumerations above stand as partial answers rather than as answers to an easier question.

What is still owed is the range of the restriction. Both counts are over patterns already known to be corrugation-like, so they establish that little survives inside the family and not that little survives outside it. A count that started from a wider space is the rung this argument still needs — but “one of sixty-four and one of five hundred and twelve” is a great deal better than a conjecture with nothing under it.

The idealisation, named #

Every geometry in the comparison is an idealised one: zero thickness, ideal hinges, rigid panels. All four systems violate all three, and the corrections point the same way for all of them.

The list of requirements is also idealised in a specific way. It contains no requirement about what the deployed surface has to do — carry a load, catch light, exchange gas, hold a shape against wind — and those requirements differ enormously across the four systems. That the geometric requirements coincide does not mean the systems face the same problem; it means they share a sub-problem.

And the fourth condition, robustness, is stated qualitatively here. Making it quantitative would mean measuring how far each parameter can move before flat-foldability fails, which is computable and is not computed anywhere on this site.

Where this goes next #

This is a one-rung anchor and the rung it would grow into is the counting one named above: how large is the family that satisfies all four conditions.

The field has one essay left, and it is the one that says what the whole field does not know. The organism is not the model collects every computation in these fifteen essays beside the claim it does not establish, which is the rule the field was built to and which this essay leans on more heavily than any other — because a convergence argument is exactly where a figure-first site would be most tempted to let a computed geometry stand in for an observed one.

The habit worth carrying out of this rung is smaller than the argument and more reliable. When a resemblance is offered as an explanation, ask what the explanation forbids. If the answer is nothing, the resemblance is a description with an explanatory tone of voice, and no amount of additional examples will convert it into an argument. If the answer is a list, the list is the actual claim and the examples are its evidence.