Four materials, four optima

The census returns one. At a stated packing ratio with parallel creases there is a single corrugation, its crease count is fixed by its ratio, and its total creasing is fixed by the same identity — so four lineages arriving at the same corrugation had nothing else to arrive at.

That settles the shape and leaves the number. A corrugation is not one object; it is a family indexed by how many folds are in it, and the four lineages do not have the same answer to that question. They cannot have, because the answer depends on the material and their materials are not alike.

The cost side, and it is linear in the fold count #

Nothing in a body folds on a line prices the correction that this rung trades against. A crease in an organism is a compliant region rather than a line: a patch of thinner or differently-oriented material that bends, with a radius below which it will not go.

Bending a sheet through half a turn on a hinge of radius ρ consumes (π − 2)ρ of surface — the arc round the outside of the turn is longer than the fold it replaces, and the difference is spent rather than folded. That is the same arithmetic the crease radius essay derives and it does not depend on anything biological.

The consequence is a subtraction. A sheet of extent S folded k times has k hinges, so k(π − 2)ρ of its surface is hinge and S − k(π − 2)ρ is panel. The cost is exactly proportional to the fold count, with no threshold and no saturation: the tenth fold costs what the first did.

The benefit side, and it is also linear #

The benefit is the packing ratio, and the conservation identity has already fixed it. A uniform corrugation of k folds packs to k, because its ratio is its mean layer count and the layers are the panels.

So the benefit is linear in k too, and a linear benefit against a linear cost sounds as though it should have no optimum at all. It has one, and the reason is that the two are not being added. The benefit is a ratio — paper over footprint — and the cost comes out of the numerator.

Write it out. The paper left to fold is S − k(π − 2)ρ. The footprint that paper occupies at k layers is (S − k(π − 2)ρ)⁄k. The ratio delivered is therefore

D(k) = k · (1 − k(π − 2)ρ ⁄ S)

which is k times something falling linearly in k: a downward parabola through the origin.

The optimum has a closed form, and it is a reciprocal #

A downward parabola through the origin peaks halfway to its other root. The other root of D is where the sheet is entirely hinge, at k = S⁄(π − 2)ρ, so

k* = S ⁄ 2(π − 2)ρ

and the ratio it delivers is exactly half of that, k*⁄2 = S⁄4(π − 2)ρ.

Two things follow immediately and both are worth stating plainly.

The best fold count is inversely proportional to the hinge radius. A material whose hinges are twice as fat should be folded half as finely. Not slightly less finely, and not less finely in some regime — half, exactly, at every sheet size and every ratio.

The best packing available is also inversely proportional to it. The material does not merely change where the optimum sits; it changes how good the optimum is, by the same factor. A material with a tenth of the hinge radius reaches ten times the packing, and no cleverness with the pattern recovers any of that, because the pattern has already been fixed by the census.

The figure computes the peak by searching over whole fold counts and then checks it against the closed form, which is the arrangement that makes the curve evidence: a sign error in the cost would move the numerical peak and the check would fire.

Four materials, and they are not close #

The radii that matter here span a wide range and the essays that established them are elsewhere in this collection.

A compliant hinge in a leaf is a fold in a lamina a few tens of microns thick, and its radius is set by how much the tissue can be thinned before it tears. A wing hinge in an insect is a membrane between two sclerotised panels, engineered by the animal to be as narrow as it can be, and narrower relative to the panel than a leaf’s. A fold in a gut lining is a fold in a tissue several cells thick with a blood supply running through it, and its radius is much larger relative to the sheet. A hinge in a deployable array is a mechanical joint, whose radius is whatever the engineering allows, which does not scale with the panel at all, and which brings a thickness problem of its own.

Put four radii spanning a factor of ten into the closed form and the optima span a factor of ten. That is the finding, and it is not a subtle one: two lineages that agree completely about the geometry and differ by a factor of ten in their hinge radius should differ by a factor of ten in the number of creases they use.

The one number all four share #

Four optima that differ by a factor of ten look like four unrelated answers. They are not. There is a quantity that is the same at every one of them, and it is the useful half of this rung.

At k* the hinge share is k*(π − 2)ρ⁄S, and substituting the closed form gives exactly one half. At its own optimum, every material has spent half its sheet on hinges. Not approximately, not typically: a half, for every radius, every sheet size and every ratio.

That is not a coincidence about paper. It is what happens whenever a benefit proportional to a count meets a cost proportional to the same count taken out of a fixed budget, and it is the pattern the surface-in-a-volume ladder named after meeting it twice — a quantity that looked unbounded in the idealised model turning over as soon as the sheet was given a thickness, with the maximum where half the competed-for resource had gone.

So the four lineages do share something. They do not share a fold count, a packing ratio or a crease density. They share the position in their own trade-off that the optimum sits at, and that is a fact about the shape of the arithmetic rather than about any of them.

It is also the most testable prediction in this field, and the one nobody appears to have checked. Measure the hinge width and the panel width on a folded organ and the two should be about equal at the optimum. Not because anything in the organism knows about parabolas, but because a lineage that has been pushed towards the top of a curve of this shape ends up at the point where those two are the same. A structure with hinges much narrower than its panels is under-folded; one with hinges comparable to its panels has gone past.

How wrong the fold count can be, which is less wrong than it looks #

A peak is flat, and flatness is the standard reason not to take an optimum seriously. If a structure at some distance from its best fold count is nearly as good, then four lineages at four different optima might all be doing fine at any one of them, and the spread of the optima would say nothing.

The arithmetic settles that, and it settles it against the reprieve.

Write u for the fold count as a multiple of that material’s own optimum. Substituting k = u·k* into D and dividing by D(k*) gives

D(u·k*) ⁄ D(k*) = 2u − u² = 1 − (u − 1)²

which is a parabola in u with its peak at one and its roots at zero and two. So the quality of a fold count is one minus the square of how far off it is, measured in units of the optimum itself.

That has a forgiving half and an unforgiving one, and they are the same half. Nine tenths of the best ratio is available anywhere between 0.68 and 1.32 times the optimum, which is a comfortable window: a structure need not hit the number to within a fold. But the parabola reaches zero at exactly twice the optimum — that is where the sheet has become entirely hinge — and goes negative beyond, which means the structure has more hinge than paper and packs worse than a sheet with no creases in it at all.

Now put the four materials into it. A material whose optimum is ten times another’s is, at that other’s fold count, at u = 0.1 or u = 10. At u = 0.1 it delivers 19% of what it could. At u = 10 it delivers nothing whatever — it is past the root by a factor of five, and the arithmetic says its sheet was consumed by hinges long before it got there.

So the flatness does not rescue the convergence claim; it sharpens the refusal. A window of ±32% is exactly wide enough to say that two lineages whose materials differ by a third could share a fold count without either suffering, and exactly narrow enough to say that two differing by a factor of ten cannot.

What a finer fold buys on the way open #

Everything above prices the closed state, and every structure in this field spends most of its life somewhere else. A leaf in a bud is opening; a wing is being stowed or deployed; an array is unfolding once and then staying.

The fold count enters the opening differently and it is worth separating, because it is the only place where a finer pattern is unambiguously better.

A corrugation’s exposed span is the total panel length times the sine of the fold angle, so it does not depend on the fold count at all: sixteen folds and four folds sweep out the same span at the same angle. What the fold count changes is the depth the structure occupies while doing it. A four-fold corrugation at a half-open angle is a deep, loose thing; a sixteen-fold one at the same angle is shallow.

So on the way open the fold count buys clearance rather than span, and clearance is what a structure confined in a bud or under a wing case actually needs. That is a second objective with the same variable in it, it points the same way as the packing objective — more folds is better — and it has no hinge cost in it, because the hinges are already paid for.

Which means the trade-off this rung computes is the whole of the argument against a finer pattern. Nothing else pushes back.

What this does to the convergence argument #

The rung below this one showed that agreeing on a corrugation is not a choice, because at a stated ratio there is one corrugation. This rung shows that the ratio itself is not shared either.

Read together the two make a specific and unusual claim. Four lineages arriving at the same pattern have converged on something forced. Four lineages arriving at the same fold count would be converging on something that is not forced — that varies with the material by an order of magnitude — and that would be a genuinely striking observation.

Nobody reports fold counts side by side. The published descriptions of these four structures give packing ratios, because a packing ratio is what a folded thing is for. A ratio is a layer count, a layer count is a fold count, and a fold count would be exactly the comparison that could distinguish the lineages — so the number that is reported and the number that would be informative are the same number, and it is reported in the units that make the comparison impossible to run.

Which is the awkward consequence and it deserves stating without softening. The observation “four lineages reach ratios of thirty” is arithmetic. The observation “four lineages, with hinge radii spanning a factor of ten, all reach ratios of thirty” would be extraordinary — it would mean three of them are far from their own optima — and it is the same observation.

Which theorem was checked and how #

The closed form is checked against a search rather than derived and drawn. For each material the delivered ratio is evaluated at every whole fold count up to a bound, the largest is taken, and that argmax is required to sit within half a fold of S⁄2(π − 2)ρ. A search that agreed with the algebra by construction would prove nothing; this one is over the same function the curve is drawn from and could disagree.

The reciprocal law is checked as an invariant. ρ × k* has to be the same number on every material, and the figure refuses if it varies at all. That is the statement “twice the radius is half the fold count” in a form that a wrong exponent breaks.

The half-share is checked at every optimum. It is the one quantity the four have in common, and asserting it is what stops the figure from being four unrelated parabolas that happen to have peaks.

And the loss per fold is the same (π − 2)ρ this collection has used since it first gave a crease a radius. No new material model is introduced here, and none of the numbers depends on anything measured from an organism.

Where the model stops #

The hinge radius is taken as fixed and it is not. A compliant hinge’s radius depends on how far the material is bent, and a corrugation folded flat bends its hinges further than one left part-open. So a real optimum is the solution of a harder problem in which the cost per fold rises with the fold angle, and the effect of that is to move the optimum down rather than up.

The hinges are taken as identical across the sheet. In a tapered structure they are not, and the sum k(π − 2)ρ becomes a sum over hinges with different radii. The optimum then depends on the distribution rather than on the mean, and a structure with a few fat hinges pays for them the way a corrugation with one long panel pays for that.

Nothing here is a measurement of a hinge. The four radii are stated as a range spanning what these structures plausibly cover and the essay says so; what is computed is the shape of the trade-off, which is the same shape at any radius.

And the optimum is an optimum of packing alone. A structure that also has to be stiff, or to survive being opened a thousand times, is solving a different problem with the same variables, and the fold count that wins it is not this one.

What the picture cannot show #

The parabolas are curves through a design space and nothing in them says any lineage is on the curve. An organism might sit anywhere along its own parabola, including far from the peak, and the figure has no way to mark where.

Nor can it show where any of them sits on its own curve, which is the quantity a selective argument would actually need. A lineage at the peak and a lineage at half the peak are drawn at the same place on this page, because the page is a design space and not a census of structures in it.

The one thing the figure does establish is negative and firm: there is no fold count that is best for more than one of the four, so a shared fold count would need explaining rather than being the null.

The idealisation, named #

The sheet has an extent S and no thickness, so the only thing a fold costs is the surface its own hinge consumes. Give the sheet a thickness and there is a second cost — the stack fills the space it is folding into — and that is the correction the surface-in-a-volume ladder is built on, which produces a parabola of its own with a peak at S⁄2t.

The two corrections are structurally identical and they compose in the obvious way: the fold costs (π − 2)ρ of surface and t of depth, and the optimum is set by whichever bound binds first. For a leaf, whose hinges are wide relative to its thickness, the hinge bound binds. For a stack of thin panels with engineered joints, the thickness bound does.

Naming that is the honest limit of this rung. The closed form here is the answer to a problem with one cost in it, and a real structure has at least two.

Where the ladder goes next #

This ladder has two rungs left in it and both are about the same gap.

The immediate one is the measurement this rung asks for and cannot make. A census of hinge widths against panel widths, across the structures this field names, would test the half-share prediction directly — and unlike a packing ratio it is a quantity a photograph carries. That is not a computation and it is not in this collection’s reach; it is a morphometric survey, and it is the one measurement that would turn the convergence story from a description into a claim that could fail.

The other direction is upward, into what a fold count is for besides packing. Every structure in this field opens as well as closing — a leaf with nothing to pull it most of all — and the ladder has priced the closing throughout. What a finer corrugation buys on the way open — how the exposed span grows, how fast, and whether the growth is usable — is a different objective with the same variable in it, and it is the one an organism actually experiences.

The habit worth carrying is the one this rung’s arithmetic hands over for free. When a benefit and a cost are both proportional to the same count, look for the half. A trade-off of that shape optimises where half the budget has gone, whatever the budget is and whatever the constants are — so a structure’s position in its own trade-off can be read off without knowing any of them, which is a rare thing to be able to do.