A corrugation agrees with itself

A crease pattern’s letters can contradict themselves, and how much room they have to do it in is a number the drawing fixes: the independent closed chains its panels form, which is edges less nodes plus pieces on the panel graph and comes out equal to the count of interior vertices.

That number is worth a sentence on its own, because it is unusually cheap for something that carries this much. It requires no folding, no letters and no search — only the drawing, walked once. A pattern’s drawing is settled long before anybody decides which creases are mountains, so the room for the letters to go wrong is fixed at the moment the lines are placed, and everything after that is spending it well or badly.

Across the tessellation patches that number tracks the answer well — thirty-six chains at thirteen per cent consistent, a hundred and twenty-six at nothing. Across families it does not. The Yoshimura pattern has twenty-two chains and is consistent in ninety-five per cent of its draws; the square tessellation patch has thirty-six and manages thirteen. Something else is at work and it is worth finding out what.

Three ladders on one axis #

The way to separate a family effect from a size effect is to grow each family along its own size parameter and put all of them on the same axis.

The Miura runs from a two-by-two of four panels at one chain, consistent in all two hundred draws, out to an eight-by-six of forty-eight panels at thirty-five chains, consistent in a hundred and twenty-eight. The Yoshimura runs from five chains at ninety-nine per cent to thirty-three chains at ninety-three. The patches run from thirty-six chains at thirteen per cent to a hundred and twenty-six at nothing.

All three fall. They do not fall together, and at the one place where all three are measurable — around thirty-four chains — they are ninety-three per cent, sixty-four per cent and thirteen per cent.

What a corrugation is doing right #

The three families differ in how their chains sit relative to one another, and that turns out to be the thing.

A Yoshimura is rows of triangles: a chain round one vertex shares a panel with the chains next to it along its own row, and shares much less with the row above. The chains are nearly laid out in strips, and a combination of chains that spans two rows has to pass through the few panels the rows have in common.

A Miura is a grid, and a grid’s chains interlock in both directions: every chain shares panels with the one above, below, left and right. Combinations are available in every direction, which is more room than a strip layout gives.

A twist patch is worse than either, and for a reason that is visible in the drawing. Its chains meet at pleats, and a pleat is a pair of creases shared between two twist polygons — so two rings are joined by a chain of only six panels, which is the shortest combination there is. A patch is built out of exactly the combination most likely to close.

The shortest combination, counted #

That last claim is checkable rather than merely plausible, and the length distribution of the contradictions checks it.

A contradiction closes round some combination of chains, and the number of panels in it says how many vertices the combination enclosed: six panels is two adjacent vertices, eight is two or three, and so on upward. If the twist patches were failing round large combinations their contradictions would be long. They are not.

Six panels, three hundred and nineteen times; eight, three hundred and twenty-nine. The failures are overwhelmingly the smallest combinations there are, which is exactly what a pattern built out of pleats between adjacent rings would be expected to produce, and not what a pattern whose chains meet sparingly would.

The Miura’s own lettering never fails #

None of this is about the pattern the Miura is normally drawn with. That one is consistent at every size, and so is the Yoshimura’s, and so is the waterbomb tessellation’s.

That is expected and it is not evidence of much. A construction that produces a lettering produces one that works, because it was written by somebody who folded the result. What the shares above measure is the alternatives: letterings that satisfy every condition at every vertex and that a folder might reasonably draw, one in three of which is contradictory on a large Miura.

The gap between those two facts is where the interest is. The Miura is a pattern with exactly one sensible lettering and a great many admissible ones, and the family it belongs to is a family of patterns all of which have that shape.

What the sensible one is doing #

The Miura’s lettering has a description: the creases running one way alternate row by row, and the creases running the other way all take the same letter within a column. That is a periodic rule, and periodicity is what keeps it consistent.

Every chain in a Miura goes round a vertex, and every vertex is like every other — the pattern has one vertex repeated, which is the whole of what a Miura is. So a rule that satisfies one vertex satisfies all of them, and the combinations of chains inherit the same regularity: the arrows round any combination come out in the same relative arrangement wherever in the sheet it is taken, and if that arrangement does not close once it never closes.

A drawn lettering has no such protection. It satisfies each vertex separately, by whatever letters propagation happened to reach, and the arrangement round a combination is then a different accident in each part of the sheet. Consistency has to hold everywhere at once and nothing is arranging for it.

Two vertices, and the letters they can share #

There is a way to make the mechanism concrete without any of the plotting, and it is worth doing once.

Take two adjacent interior vertices of a Miura. They share a crease, and the chain of panels round each of them shares two panels with the chain round the other. The combination — the chain enclosing both — is six panels, and whether its arrows agree all the way round is decided by six letters, of which one is shared.

Now count. Each vertex admits eight letterings that satisfy the conditions, so the pair admits at most sixty-four, less those the shared crease disagrees about. Some fraction of the survivors close the combination. That fraction is a small number and it is the same small number at every adjacent pair in the Miura, because the vertices are identical.

On a Yoshimura the adjacent pairs are not all alike: two vertices in the same row share differently from two in adjacent rows, and one of those two arrangements has fewer ways to close. On a twist patch the shared thing is a pleat — two creases, not one — which constrains more and, it turns out, closes more easily.

That is a calculation nobody here has done in closed form, and the shares above are what stands in for it. Stating it is worth the paragraph anyway, because it says what a closed-form answer would have to be about: not the pattern’s size and not its symmetry, but the arrangement of its adjacent pairs.

One number per family #

The closed form is out of reach and a cruder model is not, and the cruder model turns out to carry the whole figure.

Suppose each of a pattern’s $V$ chains closes with some probability $\varepsilon$, independently of the others. The share of drawn letterings that agree with themselves is then $(1-\varepsilon)^{V}$, and $\varepsilon$ can be read off one point of a ladder and tested against the others.

Fit it to the far end of each ladder. The Miura at thirty-five chains is consistent in 128 draws of 200, which gives $\varepsilon = 0.0127$. The Yoshimura at thirty-three chains is 93 per cent, giving $\varepsilon = 0.0022$. The twist patches at thirty-six chains are 13 per cent, giving $\varepsilon = 0.055$.

Now check them where they were not fitted. The Yoshimura at five chains is predicted at 98.9 per cent and measured at 99. The Miura at twenty chains is predicted at 77.5 and measured at 81. The patches at a hundred and twenty-six chains are predicted at 0.08 per cent — under one draw in a thousand, so zero in two hundred, which is what the ladder reports.

So each family is one number. Not a curve, not a shape, not an interaction between chains: a per-chain probability of closing, raised to the chain count.

What that number is, and what it says #

The three fitted rates are 1 in 455 for the Yoshimura, 1 in 79 for the Miura and 1 in 18 for a twist patch — a ratio of about one to six to twenty-five.

That is the quantity the whole essay has been circling. The family effect is not a different law for each family; it is the same law with a different chance per chain, and the chance per chain is set by exactly what the row-against-grid argument says it should be — how much a chain shares with its neighbours, and therefore how short the combinations available to it are.

The crumples fall where the argument predicts too. Nine of them between five and twenty-nine chains, with the worst at 28 per cent at twenty-nine, give $\varepsilon = 0.043$: three and a half times the Miura’s rate and just under a patch’s. An irregular pattern is not a different kind of object on this measure; it is a regular one with a worse coefficient.

The model earns one caution and it is the same one the essay ends with. Independence is certainly false — chains share panels, so two adjacent ones cannot close in unrelated ways — and the fit succeeding anyway means the dependence is weak at these sizes rather than absent. What would expose it is a ladder run far enough for the predicted share to disagree with the measured one, and the twist patches, whose predicted 0.08 per cent is indistinguishable from the measured zero, are the family where that test has the least power.

Where a crumple sits #

The comparison that settles the regularity argument is not another designed pattern. It is a sheet creased by folding it at random — a pattern with no periodicity at all, whose vertices are all different, and which is nonetheless developable and flat-foldable by construction because it was made by actually folding.

Nine crumples at between five and twenty-nine chains come in at between twenty-eight and a hundred per cent, and at the top of that range — twenty-three to twenty-nine chains — they are between twenty-eight and sixty-eight per cent. A Miura at twenty chains is eighty-one and a Yoshimura at twenty-two is ninety-five.

So an irregular pattern with the same room to fail does markedly worse than a regular one. That is the argument’s other end, and it is the one that could have come out the other way: a crumple’s vertices are all different, and there was no obvious reason for that to hurt rather than help.

The one the Miura loses to #

There is an ordering here that is worth stating plainly because it is not the one a folder would predict: on this measure the Yoshimura beats the Miura at every comparable size.

That is surprising because the Miura is by far the more constrained pattern in every other respect. It is the only quadrilateral mesh in its family that folds rigidly; it has one degree of freedom where a general developable mesh has none; its vertices are identical. Constraint of that kind usually shows up as fewer admissible letterings, and it does — but fewer is not the same as better arranged, and what this measure responds to is the arrangement.

The Yoshimura is the looser pattern and its chains are in rows. The Miura is the tighter pattern and its chains are in a grid. Rows win.

What this says about scaling a corrugation #

The practical form of all this concerns anybody making a corrugation larger, which is what corrugations are for.

The chain count grows with the vertex count, which grows with the area. The paper a pattern asks for is the other side of the same arithmetic, and it is the side that gets attention. So the share of admissible letterings that work falls as a corrugation is scaled, and falls fast: the Miura goes from every draw at four panels to two thirds at forty-eight. Extrapolating that curve is not something the measurement supports, but its direction is not in doubt.

What saves a real corrugation is that nobody draws its letters by hand. The pattern is generated, the generator applies a periodic rule, and the rule works at every size because it works at one vertex. A material made of creases is exactly a pattern being scaled to thousands of cells, and it is only usable because its lettering is a formula rather than a choice.

The twist tessellations are the case where that fails, and now it can be said exactly why. Their letters are not given by a rule at all — no local labelling works, so the letters are found by propagating the vertex conditions and taking what comes back — and what comes back is one of the letterings the shares above are about. On the square patch that is a one-in-eight chance of a lettering that folds, and the construction has to redraw until it gets one.

What a periodic rule costs #

There is a price for the protection a periodic lettering gives, and it shows up wherever periodicity has to stop.

A rule that works at one vertex works at every vertex of the same kind. A corrugation grown by repeating a cell has one kind of vertex in the middle and different ones at the rim, and the rule says nothing about the rim — which is why the rim is where a patch’s construction had to make decisions and where its first repair was needed.

That is a general shape and the twists show it most clearly: the periodic part of a tessellation is safe by construction, the boundary is not covered by any rule, and a finite patch is mostly boundary.

What it does not explain #

Two things sit outside this account and both are worth naming rather than smoothed over.

The tapered corrugation is the lowest of the printed patterns at eighty-seven per cent, and it has eighteen chains — fewer than the Yoshimura’s twenty-two, which scores ninety-five. It is a corrugation with rows, so the row argument ought to apply, and it does not do as well as one. The difference between it and a plain corrugation is that its creases have different lengths in different rows, and that taper is the whole point of it, so the natural guess is that the taper is what costs it. That is a guess and it is the subject of its own essay.

The waterbomb tessellation is at ninety-three and a half per cent with twenty-five chains, which puts it between the Miura and the Yoshimura on a pattern that resembles neither. Its vertices come in two kinds and its chains are arranged in a grid of units rather than in rows, and there is no story here that predicts where it should land.

Two unexplained points out of eight is the right number to be honest about. The separation between the families is large and holds everywhere the ladders overlap; the ordering within a family’s neighbourhood is not something this measure decides.

What the axis is worth #

One caution about the plot, since a figure with three curves on one axis invites more than it supports.

The chain count is a genuine common measure: it is computed the same way for all three families, from the drawing, before any letter is chosen, and it is the number of independent chains a lettering has to keep from closing. Placing the families on it is legitimate.

What is not legitimate is reading a curve as a law. Each is six or fewer points from one construction, and the constructions differ in more than one respect at a time — a Miura grown wider is not just a Miura with more chains, it is a Miura with a different aspect ratio and a different rim share. The figure supports the separation between families, which is large and holds at every overlapping count, and it does not support a functional form for any one of them.