Letters that agree get rarer

A twist tessellation patch is cut out of an infinite pattern, and until recently the interesting question about one was whether it had been cut out correctly. It had not been: assembling whole twist units on a square and running the outstanding pleats to the rim put creases across other creases, which is a pattern that cannot exist. Generating the tessellation over a larger region and clipping the drawing removes every one of them.

The patches then place perfectly — every panel lands where the composed reflections say, to five parts in a quadrillion — and every one of their interior vertices satisfies every condition the subject has. And their letters contradict themselves: the arrows one crease at a time force a chain of panels each of which must lie below the next, which is a proof that no flat folded state exists.

The question this leaves is not whether a patch folds. It is how many of its letterings do.

The count, on five patches #

Independent letterings are obtained by propagating the vertex conditions to a fixed point and, wherever propagation stalls, branching on a crease with the two letters tried in an order a stream decides. Every draw satisfies every condition at every vertex; the draws differ because the decisions differ. Two hundred of them per patch.

Twenty-six of two hundred on the square patch, five on the elongated, two on the hexagonal, none on the triangular, none on the rhombille. Thirteen per cent down to nothing across a family whose members differ only in which tiling they are built on.

A nought there is a nought of two hundred draws and not a proof. What can be said about the rhombille is that its own construction’s lettering closes a chain of sixteen panels, that two hundred independent attempts did no better, and that the tangle in every one of them covers most of the sheet.

What is being sampled, and what it licenses #

Before any of that is read as a probability it is worth saying what the sampler is and is not.

It is a sampler over solutions rather than a uniform draw over them. The obvious alternative — write random letters and test whether they pass — has an acceptance rate of about one in two to the power of the vertex count, which is one in thirty thousand on a Miura and one in a hundred million on a patch of thirty-six vertices. Nothing can be measured that way. So what is randomised is the search: the vertex conditions are propagated to a fixed point, and where propagation stalls the branch is taken in an order the stream decides. Every draw is a lettering; the draws differ because the decisions do.

A share obtained that way is a share of the letterings this procedure reaches, and dividing it by anything would be pretending otherwise.

Three agreements do not establish that the sampler is unbiased on a patch of two hundred and eighty-two creases. They establish that it is not obviously broken on the only cases where the question can be settled at all, and that is the whole warrant behind every share quoted below.

The dial that changes nothing #

The obvious first suspect is the twist angle. It is the family’s one continuous parameter, it is what makes a twist a twist, and it is fenced at both ends by geometry that the construction refuses to draw past.

It does nothing here at all. A hundred and twenty draws at each of seven turn angles from 0.15 radians to a full radian, and the count of consistent letterings is thirteen at every one of them.

That is worth more than it looks. The turn angle changes every angle in the pattern and therefore every evaluation of Kawasaki and of the big-little-big lemma — and the set of admissible letterings comes out the same size, with the same share of it self-consistent. Which means the thing being measured is not about angles at all. The dial that decides nothing made the same point about a different quantity, and this is the sharper version of it: a purely combinatorial property of the drawing, invariant under the one thing about the drawing a designer controls.

What it is not falling with, either #

Size is the second suspect and it is much better than the angle. The five patches run 49, 62, 77, 83 and 157 panels, and the share falls monotonically across them.

But size is not the quantity, and the printed shelf says so immediately. The Yoshimura pattern folds to sixty-five panels — more than the square patch’s forty-nine, more than the elongated’s sixty-two — and a hundred and ninety of its two hundred draws are consistent. Ninety-five per cent against thirteen, at a larger size.

So panel count is correlated with the answer inside one family and useless across families. Something else is doing the work.

It is not the twist angle and it is not the panel count. It is not the period either, which is the other dial the construction takes: a square patch drawn at half the period has eighty-one panels instead of forty-nine and its share falls, which looks like size until the hexagonal patch at seventy-seven panels comes in forty times lower again.

And it is not the tiling as such. The five patches are built on five different tilings and the ordering of their shares is not the ordering anybody would guess from looking at them — the hexagonal patch, whose polygons are the largest and fewest, scores below the elongated one, whose polygons are neither.

The quantity that does #

A lettering contradicts itself by closing a chain of panels — a sequence each joined to the next by a crease, returning to where it started, with every arrow round it pointing the same way. So the natural thing to count is how many independent chains a pattern’s panels admit at all, and that count is arithmetic rather than a search: edges, less nodes, plus pieces, on the graph whose nodes are panels and whose edges are creases.

On twelve of the thirteen patterns measured here that number comes out equal to the pattern’s count of interior vertices. That is Euler’s relation rather than a discovery — the chains round single vertices generate every other chain — but it makes the quantity free to compute and easy to say. The exception is the hexagonal patch, which has sixty interior vertices and fifty-four chains, because six of its vertices sit where the clip has taken the paper away on one side and their panels no longer close a ring.

Fold-and-cut outlines: one to three chains, and every one of two hundred and eighty draws consistent. The printed shelf: one to twenty-five chains, between eighty-seven and a hundred per cent consistent. The patches: thirty-six to a hundred and twenty-six chains, thirteen per cent down to nothing.

The separation is complete at the ends. Every pattern here with six chains or fewer is more consistent than every pattern with thirty-six or more, and nothing in the middle crosses.

Why more chains is worse, exactly #

Each of the generating chains goes round one interior vertex, and Maekawa closes every one of them: orienting a chain round a single point requires the letters to alternate strictly, and an alternation has equal counts of mountain and valley where a flat-foldable vertex needs a difference of two.

What is not closed is a combination. Two of those chains sharing a panel boundary combine into one longer chain enclosing both vertices, and nothing local stands in its way. Three combine into a longer one still. The number of combinations available grows as two to the power of the chain count — so a pattern with three chains offers a handful of opportunities for the letters to close one, and a patch with a hundred and twenty-six offers a number with thirty-eight digits.

A lettering has to avoid all of them. That is why the share collapses rather than declining: each additional chain is another independent way to fail, and a share that has to survive many independent hazards falls off a cliff.

The length distribution confirms the mechanism from the other side. If contradictions were closing round large combinations, the chains reported would be long; they are overwhelmingly six and eight, which are two and three vertices’ worth. The failures are the smallest combinations, and a tessellation is made of them: every pleat between two twist polygons is a six-panel chain.

A second consequence follows and it is worth separating, because it explains something about the printed shelf that had no explanation.

The two printed patterns with the lowest shares are the tapered corrugation at eighty-seven per cent and the Miura at ninety and a half — and the Miura has fifteen chains against the Yoshimura’s twenty-two and the waterbomb’s twenty-five, both of which score higher. So the chain count is a first-order predictor and not a law. Something about how the chains sit relative to one another matters too: a Miura’s vertices are all alike and its chains all interlock in one grid, while a Yoshimura’s are arranged in rows that share less.

The families make the same point more loudly. Nine sheets creased by folding them at random carry between five and twenty-nine chains, and at twenty-three to twenty-nine chains their shares are twenty-eight to sixty-eight per cent — well below the printed corrugations at a comparable count. A regular pattern does better than an irregular one with the same amount of room to fail in, and that is a finding this collection has met before in other clothes.

Chain count is half of it, and the other half is the same size #

The claim that the chain count is a first-order predictor rather than a law can be sharpened into a decomposition, and the decomposition says the two halves contribute equally.

Model each chain as closing with probability $\varepsilon$, independently, so the consistent share is $(1-\varepsilon)^{V}$. Fit $\varepsilon$ to each patch separately from its own measured share and its own chain count:

The square patch, thirteen per cent at thirty-six chains, gives $\varepsilon = 5.5$ per cent. The elongated, two and a half per cent at forty-five chains, gives 7.9. The hexagonal, one per cent at fifty-four, gives 8.2.

The rate is not the same across the family. A chain on a hexagonal patch is half again as likely to close as a chain on a square one, and the exponent that decides the share is the product of the two.

Which is why the ordering is so steep #

Going from the square patch to the hexagonal, the chain count rises from thirty-six to fifty-four — a factor of 1.5 — and the per-chain rate rises from 0.055 to 0.082, which is also a factor of 1.5.

So the exponent $\varepsilon V$ goes from 2.04 to 4.61, a factor of 2.26, and the two contributions are equal to within a per cent of each other. The share falls by a factor of thirteen across those two patches, and half of the fall is more chains while the other half is worse chains.

That resolves the essay’s own puzzle about the Miura and the Yoshimura without needing a new quantity. A Miura at fifteen chains scoring below a Yoshimura at twenty-two is not chain count failing; it is chain count being the smaller of two factors on that particular comparison. The rate is doing the work there and the count is doing the work here, and they are the same two numbers throughout.

So the honest statement is that consistency is decided by $\varepsilon V$ and not by $V$, and the tiling fixes both of them at once — which strengthens the essay’s design conclusion rather than weakening it. The choice that is taken first and cannot be revisited settles not one quantity but the product of two, and the two happen to move together.

What remains unexplained is the rate itself. Something about how a hexagonal patch’s chains overlap makes each of them half again as likely to close as a square patch’s, and nothing here measures overlap. That is the next quantity, and unlike the chain count it will not come out of Euler’s relation.

Which tiling, and why the order comes out as it does #

Within the family the ordering by chain count is square 36, elongated 45, hexagonal 54, triangular 60, rhombille 126, and the consistency ordering follows it exactly — thirteen per cent, two and a half, one, nought, nought.

That is a fact about the tilings rather than about twists. A tiling with more vertices per unit area, drawn over the same sheet at the same period, gives more twist polygons and more pleats between them. The rhombille has three kinds of vertex arrangement and packs the most in; the propagation that makes it fold at all is the same density showing up as a constraint on the side distances.

The tangle, and why a small failure is not what happens #

One more measurement bears on what these shares mean, and it changes what a low share is a low share of.

When a lettering does contradict itself, the contradiction is not a small local knot. The chain a search reports is six to twenty-four panels, but the set of panels lying on some chain is between ten and eighty-four per cent of the patch, with a mean around half. A patch that fails does not fail slightly.

So the picture is not of a family of patterns edging toward the boundary of foldability as they grow. It is of two states: a lettering works, or half the sheet has no consistent reading. The share is how often the first happens, and the second is the same size whenever it happens.

What a designer would have to decide #

There is a design reading of all this and it is not the one a tessellation designer would expect.

The parameters a patch’s designer chooses are the tiling, the period, the turn and the side distance. Three of those are about shape: they decide what the patch looks like, how much paper it takes up, how much it shrinks and whether it can be drawn at all. Only the first is about the quantity that decides whether the letters can agree, and it decides it entirely — through the vertex count, which the tiling and the period fix between them before the turn has been chosen.

So the decision that settles whether a patch is likely to have a consistent lettering is taken first, is taken for reasons that have nothing to do with letters, and cannot be revisited afterwards without redrawing everything. That is an unusual shape for a design constraint and it is worth naming: the dial that matters is not continuous, is not adjustable, and is not the one anybody thinks they are turning.

What a patch would need #

The practical reading is uncomfortable and worth stating.

A patch is not repaired by drawing it more carefully, because the drawing is already correct. It is not repaired by twisting less, because the angle does nothing. It is not repaired by cutting a crease, because a cut gives the sheet a freedom and the patch stops having a folded state to order at all — of four hundred and seventy-four single cuts across four patches, sixteen leave panels that still place and none of the sixteen helps. And it is not repaired by a small change to its letters, because the letters a local move can reach are the ones with an end on the boundary, and on the square patch that is twenty-four creases of eighty-four.

What is left is a different lettering altogether — and on the square patch twenty-six in two hundred are available, so the honest description of that patch is not it does not fold but it folds for one lettering in eight, and the construction’s own is not one of them, and there is no path from here to there.

For the rhombille, two hundred attempts found none. Whether one exists is exactly the question the general problem makes hard, and a patch with two hundred and eighty-two creases is not a small instance of it.

What has changed is that the question is now stated in the right currency. It is not about the drawing, the angle or the size. It is about how many chains the panels form, and that number was fixed the moment the tiling was chosen.