The census returns one

The same corrugation turns up in four unrelated places, and the rung above that one ends by naming a measurement it does not make. For a stated packing ratio and a stated deployment requirement, list 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 census is out of reach in general, because enumerating patterns that deploy means deciding rigid foldability. It is entirely in reach for the corrugations — parallel creases, one degree of freedom, a ratio fixed — and this is it.

It comes back with one member.

What a corrugation at a stated ratio actually is #

The family has to be written down before it can be counted, and writing it down is most of the work.

A corrugation is a set of parallel creases across a sheet. Between consecutive creases is a panel, and the panels have lengths — call them ℓ₁ through ℓₙ, measured across the sheet in the direction the creases run perpendicular to. Nothing requires them to be equal. A sheet may be creased at any set of positions whatever, and the result is still a corrugation with parallel creases and a single degree of freedom.

Fold it flat. Each panel turns through half a turn relative to its neighbour, and the panels come to rest lying on one another. Where do they lie?

They lie on top of one another aligned at the creases, which is the part that decides everything. Panel one runs from the left edge to crease one. Panel two starts at crease one and runs back, in the folded state, to wherever its own far end lands. If panel two is longer than panel one, its far end sticks out past the left edge. If it is shorter, it stops short.

So the folded footprint is not the average panel and it is not the shortest. It is the longest — the extent of the pile is set by whichever panel reaches furthest, and every other panel lies inside that extent.

That single observation does the census.

The ratio is a quotient, and the quotient has a unique maximum #

The packing ratio is the sheet’s area divided by the folded footprint’s area. Both are measured in the same units and the direction along the creases is unchanged by folding, so the ratio is a length quotient: the total panel length divided by the footprint width.

The total is ℓ₁ + ℓ₂ + ⋯ + ℓₙ. The footprint is max ℓᵢ. So

ratio = (Σ ℓᵢ) ⁄ (max ℓᵢ)

and that is the whole of it. The quotient is at most n, because a sum of n terms each no larger than the maximum cannot exceed n times the maximum. It equals n exactly when every term equals the maximum — which is to say when every panel is the same length.

So for a corrugation of n panels the top ratio is n, and there is exactly one arrangement reaching it. Every deviation from uniform spacing lowers the ratio, and lowers it by exactly the shortfall of the short panels against the long one. A corrugation with one panel three times the others does not pack three times better anywhere; it packs worse everywhere, because the whole pile now has to be as wide as that one panel.

The census the ladder asked for therefore has a one-line answer for the corrugations, and it is not the answer a convergence argument wants. There was nothing to choose.

What the losing arrangements do with the paper #

A ratio is a summary and the layer profile is what it summarises, so it is worth looking at what the shortfall consists of rather than only at how big it is.

Take the alternating rule — panels of one, two, one, two, and so on. The footprint is two units wide. Over the inner unit every panel contributes a layer, so the pile there is n deep. Over the outer unit only the long panels reach, so the pile there is n⁄2 deep. The mean over the footprint is three quarters of n, which is exactly the ratio the arithmetic gives.

That is the general shape. The layer profile is the survival function of the panel lengths: over a point at distance x from the fold line, the number of layers is the number of panels longer than x. A uniform corrugation has a rectangular profile, n deep everywhere and zero past the edge. Any other spacing has a profile that falls away, and the area under it is fixed at the total paper — so a profile that is thin at the outside is a profile that had to be no thicker at the inside than the count allows. It is the same accounting every corrugation on this site is measured by, read one column at a time.

The mean of that profile is the ratio. That is not a coincidence and it is not an approximation: it is the conservation identity read over a footprint that is no longer uniform.

Which sharpens the census rather than softening it. Non-uniform corrugations are not worse designs reaching the same ratio by a different route. They reach a different ratio, and the difference is visible in a picture of where the paper went.

Widening the census past parallel creases #

One family with one member is a thin census, and the honest response is to widen it until it has some members in it. The obvious widening is to drop the requirement that the creases be parallel.

There are three other things a developable sheet can do to occupy less room, and they have been counted in the rung below this one: corrugate, roll, and give itself creases by buckling. Adding a second crease direction gives the Miura family; adding a common point gives a fan; taking the creases away entirely gives a roll.

Each of those is its own family and each has its own identity between ratio and pattern. A fan of k sectors about one point packs to two over k, because the sectors sweep a half-turn as the fan closes. A roll of k turns packs to one over k, because the length is wrapped k times round a circumference. A Miura shrinks in both directions at once, so its ratio is the square of a corrugation’s at the same fold angle.

Four families, then, and the census’s answer changes from one to four. That is still not many.

The deployment requirement does not narrow the list #

Rung two proposed a second condition to make the census mean something: the pattern has to open, monotonically, from a single input. A leaf in a bud has nothing to pull it and no way to reverse, so a pattern that had to narrow before it widened is unavailable.

That condition is real, it is checkable, and it turns out not to narrow this list at all.

A corrugation’s exposed span is the total panel length times the sine of the fold angle, and the sine rises monotonically from a closed fold to a flat sheet. Every corrugation therefore opens monotonically, whatever its spacings — the spacings scale the span and cannot change its direction. The Miura does the same in two directions at once. The fan sweeps its sectors apart. The roll unwraps.

So the deployment requirement is satisfied by every member of every family, and using it as a filter selects nothing.

What does narrow it, and it is a count rather than a shape #

There is a condition with teeth, and it is not the one about opening. It is the one about how many things have to pull.

A pattern with one degree of freedom is determined everywhere by a single number, so one thing pulling it opens all of it. A pattern with several needs a driver for each, or a mechanism to sequence them. An organism with muscles only at the base of a wing, or with growth as its only actuator, has one input and needs a pattern that will accept one.

Most patterns will not. A generic crease pattern does not fold rigidly at all, and among the ones that do, having exactly one freedom is a structural property rather than a common accident. That is a genuine restriction, it is the restriction a convergence argument would have to invoke to be doing work, and it is not the restriction anybody invokes — because a folded organ is described by its ratio, and a ratio has nothing in it about freedoms.

The second column of the census, which is also fixed #

Rung two asked for the alternatives with their crease densities, and the density is the half that was supposed to be free. It is not.

A corrugation of n panels across a sheet of width W has n − 1 creases, and every one of them runs the full width. So the total length of crease is (n − 1)·W, whatever the spacings are. Bunch the creases at one end, space them geometrically, alternate wide and narrow cells: the count does not change, the width of each does not change, and the total does not change either.

The census’s second column is therefore constant down the family, in exactly the way the first column is not. And since the top ratio is n, the crease density at that ratio is (n − 1)⁄n creases per unit of paper, which is a function of the ratio alone.

That closes the question rung two left open, and it closes it in the direction that makes the convergence argument weaker rather than stronger. Two lineages reaching the same ratio with parallel creases have the same layer count, by conservation; the same crease count, by the identity above; and the same total creasing, by the same identity. There is no column left in which they could have differed.

The exception is instructive. A pattern that reaches its ratio in two directions — the Miura family — has a crease density that is genuinely free of its ratio, because the two directions can be traded against each other. That is one place where a comparison across the free axis would say something, and it is a comparison between families rather than within one.

A ratio that is not a whole number #

The census above is stated at a ratio of n and the ratios an organism needs are not whole numbers. A bud that is a twenty-seventh of the open leaf is a bud — and the bud is what does the choosing — and no corrugation of a whole number of equal panels reaches exactly 27 unless it has 27 panels.

Take the ratio as given and ask which corrugations reach it. With n panels the ratio is (Σ ℓᵢ)⁄(max ℓᵢ), which runs continuously from 1 — every panel but one vanishingly short — up to n. So a ratio of 27 is reachable by 27 panels in exactly one way, by 28 panels in a great many, by 29 in more still, and by 26 in none at all.

The census at a non-integer ratio is therefore infinite rather than empty, and every member of it is a corrugation with more panels than the ratio and some of them short. Which is to say: the extra members are all patterns carrying panels that do not reach the edge of the pile and are therefore not paying for themselves.

That is a real design distinction and it is the one nobody reports. Two corrugations at ratio 27, one with 27 equal panels and one with 34 unequal ones, have the same ratio, the same mean layer count and different crease densities — the second has seven more creases doing nothing for the ratio. The free axis rung two was looking for exists, and it exists only off the top of the family, among the arrangements that have already given up some of their ratio.

So the sharp version of this rung’s finding is a conditional. At the top ratio the census returns one member and every column is determined. Below the top ratio it returns infinitely many, they differ in crease density, and they are all paying for panels that do not reach.

Which theorem was checked and how #

The identity is checked over the profile rather than asserted from the algebra. Every spacing rule’s layer count is sampled across its own footprint at two hundred and forty points, and the mean of those samples has to reproduce that rule’s ratio. It does, to a few hundredths of a layer, the residue being the sampling.

The uniqueness is checked by exhibition rather than by the inequality. The arithmetic says the uniform rule is the unique maximiser, and the figure computes five rules and requires the uniform one to be strictly above all the others. A sign error in the footprint would put a non-uniform rule on top and the figure would refuse to draw.

The four families are built rather than quoted. Each one’s packed fraction is computed from its own parameters — a corrugation’s from its fold angle, a Miura’s from a pattern with its interior vertices counted, a fan’s from its sector count, a roll’s from its turn count.

And the census is a census of the model rather than of the world. Nothing here is a measurement of any organ. What is enumerated is the set of geometries this collection can build and check, and the essay’s claim is about what a convergence argument licenses given that set.

Where the model stops #

The corrugation family is the one with parallel creases and nothing else. Allow the creases to be nearly parallel, or to end partway across the sheet, and the family is larger and the identity is only approximately true. What survives is the direction: a pattern whose panels do not all reach the same distance pays for its longest one.

The footprint is the pile’s extent and not its convex hull. For a corrugation those are the same thing. For a pattern with creases in two directions they are not, and the ratio computed from the hull is optimistic.

Nothing here forbids a lineage from being at a ratio no family reaches exactly. The census answers “which patterns reach ratio r” for the ratios the families produce; between them a real organ sits at whatever fold angle it has got to, which is a separate matter with its own arithmetic.

And a census of four families is a census of the four this collection knows how to build. A fifth would change the count. What it would not change is that the count is small, which is the argument, and that the corrugation family has one member at each ratio, which is arithmetic rather than a survey.

What the picture cannot show #

No figure here shows a lineage, a selective pressure or a history. The census draws the option set and says how large it is; whether anything chose among the options, and on what, is not in it and could not be.

Nor can a figure show a family the collection has not thought of. A census is only ever a census of what was enumerated, and the honest statement of this one is that the corrugations are complete — the parametrisation by panel lengths is exhaustive — while the list of families is a list of what a developable sheet has been observed to do.

The layer profiles have a further limit worth naming. They are drawn as step functions with the panels lying exactly on one another, which is the zero-thickness model. A real pile has layers that push each other apart, and a profile that steps down by one layer at a point is a profile that changes thickness at a place where the material has to bend around the step.

The idealisation, named #

Every sheet here is a plane of zero thickness that folds on lines. Three things follow from that and all three are used.

The panels lie on one another with nothing between them, so the pile’s depth is a layer count rather than a length. The creases have no width, so a panel’s length is the distance between two lines rather than between two curved regions. And the fold is exact, so the panels align at the crease rather than approximately.

Removing any of the three changes the numbers and none of them changes the census’s shape. Give the creases a radius and every panel loses a fixed length to its own hinge, so the ratio falls and the uniform arrangement is still the best one. Give the sheet a thickness and the pile has a depth as well as a count, so the footprint grows and the ratio falls again. The maximiser does not move, because none of the corrections depends on which panel is which.

That robustness is worth more than the exact numbers. The census’s answer is one member, and it is one member for a reason — the objective is a quotient of a sum by a maximum — that does not care what the sheet is made of.

The generalisation #

Where an objective is a sum divided by a maximum, the optimum is uniform and it is unique. That sentence is the whole of this rung, and it has nothing to do with paper. A quantity spread over parts, divided by the largest part, is maximised by making the parts equal — which is why a corrugation packs best when its panels match, why a load spread over supports is best carried when the supports share it, and why a schedule finishes soonest when its stages are balanced.

The consequence for a convergence argument is the uncomfortable half. An objective with a unique optimum produces convergence by construction: every finder that gets close to the optimum is at nearly the same place, and their agreement says nothing about how they got there. Convergence is evidence in proportion to the size of the option set that was not taken, and an objective of this shape has an option set of one.

The version worth carrying out of the subject is a question to ask of any convergence claim before believing it. Is the objective’s optimum unique? If it is, then the finders had nowhere else to go, the agreement is arithmetic, and the interesting question is a different one — how each of them got close, and what it cost them.

Where the ladder goes next #

The census answers the question rung two asked and leaves a sharper one behind it, which is the next rung.

Four lineages agree on the pattern. They do not have the same materials, and the correction that matters most to a folded organ is the one that comes from the material: a hinge has a radius, the radius consumes surface in proportion to the number of folds, and the radius is not the same in a leaf, a wing, a gut lining and a metal array.

So agreement on the geometry is not agreement on the number of folds to put in it. Each material has its own best fold count, that count has a closed form, and the four are not close. Four materials, four optima makes that computation, and it turns the convergence claim over: the four lineages did converge on a pattern, and the question of whether they converged on a design is a question about a number none of them shares.

The habit worth carrying from this rung is smaller and more portable. When a set of solutions is offered as evidence that a particular one was chosen, write down the objective and ask what its argmax looks like. If the answer is a point, the evidence is arithmetic wearing the clothes of a selection argument, and no number of additional examples will change that.