The ring is the loop

A square that turns is the simplest twist there is: a square panel in the middle, four pleats running out from its sides, and the whole thing rotating as the paper gathers. Nine panels, twelve creases, four interior vertices. It is the unit every twist tessellation in this collection is built from, and it is small enough that every one of its four thousand and ninety-six letterings can be written down.

Two hundred and fifty-six of them satisfy every condition at every vertex. Eight have a folded state. The lettering that folds nowhere is the account of the gap, and its sharpest finding was about a particular group: the thirty-two letterings whose central ring reads as one letter — all four creases of the middle square the same — have no folded state at all, and that is the group a designer draws.

Establishing it took an exhaustive search over the orderings of nine panels, seven and a half thousand nodes at a time, thirty-two times.

Four of those thirty-two can now be settled in one pass. And what settles them says where in the pattern the trouble is.

The circle is the ring #

Each crease of a flat-folded pattern fixes which of the two panels it joins lies above the other — a valley brings the far panel over, a mountain takes it under, and turning the near panel over on the way swaps which a reader sees. A circle in those statements is a proof that no order of the panels satisfies all of them.

The circle is eight panels long. Trace it and it is the central square, then a pleat panel, then the next quarter of the surround, then the next pleat panel, all the way round and back — which is to say it is the ring: the closed chain of panels that the twist polygon’s own four creases and the four pleats between them make.

That is the twist’s defining feature contradicting itself. A twist is a polygon that rotates, and the panels that let it rotate are the ones that go round it. Give them letters that all point the same way and the paper is being asked to spiral without end.

Reading the circle by hand #

The chain is short enough to check with a pencil, and doing so once is worth more than the picture.

Number the central square’s four creases going round. Each joins the middle panel to one of the four quarter-panels of the surround; each quarter-panel is joined to the next by a pleat. So the ring alternates: middle, quarter, middle, quarter — except that the middle panel is only one panel, so the chain that goes all the way round is not through the centre but through the pleats, which is why it has eight panels rather than eight visits to five.

Now the orientations. The central panel is face up; every panel one crease away from it is face down; every panel two creases away is face up again. Round the eight-panel chain the orientation alternates, so an arrow points forward at each step only if the letter flips at each step too — and the ring’s four creases all carrying the same letter is exactly the flip pattern that produces agreement all the way round when combined with the pleats’ letters. Which pleat letters those are is what the other twenty-eight uniform-ring letterings differ in, and it is why only four of the thirty-two close.

Why a designer draws it #

The uniform ring is not an obscure choice. It is the natural one and there is a reason for it that survives the fact that it does not work.

The rule that suggests itself for a twist is that the polygon in the middle folds one way and the pleats alternate — every crease of the central ring a mountain, say, with the pleat creases taking valleys by turns. It is the rule anyone would state, it is symmetric under the rotation the twist is built on, and it satisfies Kawasaki and Maekawa at every vertex.

Both extremes are empty. A ring that reads as one letter has no folded state; a ring that alternates at every step has none either; the eight that work all change letter exactly twice going round. So the symmetric answers are the wrong ones on both sides, and the symmetry the letters cannot keep is the standing account of why a pattern with a rotational symmetry so often has no lettering that shares it.

The other polygons #

The square is the smallest twist and the argument is not about squares. Every polygon that can twist has a ring, and the ring is the chain the letters can close.

Which polygons twist settled which shapes admit the construction at all: the polygon’s sides must be able to take pleats that meet the neighbours’ at the right angles, which is a condition on the tiling rather than on the polygon alone. Every one that passes it has a closed ring of creases round a central panel, so every one of them is exposed to this failure — and the larger the polygon, the longer the ring and the more letters have to conspire to close it.

That cuts both ways. A hexagon twist’s ring is six creases and a twelve-panel chain, and two hundred draws of it are consistent in all two hundred: the longer chain is harder to close, not easier. So a larger unit is safer and a patch of many small units is not, which is the opposite of what a designer worried about complexity would guess.

What the one-pass proof adds, and what it does not #

Four of the thirty-two, not thirty-two of thirty-two. It is worth being exact about that, because the temptation is to say the cheap test has replaced the search and it has not.

Of the thirty-two uniform-ring letterings, four close a circle in their letters and twenty-eight do not. The twenty-eight have letters that agree with themselves perfectly and no folded state, and finding that out still costs the full search over orderings — refused by the rule that a panel may not lie between the two panels a crease joins, or by the rule that two folds in the same place may not interleave, both of which are statements about where the paper landed rather than about what the letters said.

So the improvement is in cost and in explanation rather than in coverage. Four letterings that used to be refused by an exhaustive computation are now refused by a sentence naming eight panels, and the sentence points at the ring. Twenty-eight are refused as before.

It is worth noticing what the middle group is. A ring changing letter exactly twice going round divides the four creases into two consecutive pairs — two mountains then two valleys — which is not a symmetric arrangement and cannot be described by any rule of the form the polygon folds this way. It is a choice of where to put the break, and there are four places to put it, times the pleat letters that go with each. Nothing about the twist suggests it, and it is the only thing that works.

Why this matters more on a patch than on a unit #

On a nine-panel unit, none of this is needed. The search finishes in milliseconds and gives a complete answer.

On a tessellation patch it is the only answer available. A patch is many twist units sharing pleats, and every one of them carries a ring — so the failure the square twist can have once, a patch can have at every polygon, and the search cannot be run on it at all.

Twenty-six of two hundred draws are consistent on a square patch of nine polygons; a hundred and ninety-eight of two hundred on the single unit. Nine copies of a pattern that fails one time in a hundred fail together most of the time, because the rings interact: a pleat is shared between two polygons, and a letter that suits one ring is a letter the other has to live with.

A result that moved from a search to a sentence #

There is a general point here about what it means for a finding to be superseded, and this is a clean instance of it.

The original result was obtained by running a complete decision procedure on a small enough case: enumerate the thirty-two uniform-ring letterings, place the nine panels for each, and search every ordering. That is unimpeachable and it is also opaque — it says no thirty-two times and never says why. Nothing in it generalises: the same computation on a pattern of twenty-four panels does not finish.

The new result covers four of the thirty-two and says something about all of them. It names the chain, it says the chain is the ring, and the naming transfers: any twist, at any polygon, on any tiling, has a ring, and the ring is where to look. It also predicts a thing the search could not — that a longer ring is harder to close, which is why the hexagon twist never fails this test and the square one occasionally does.

Neither result replaces the other, and the shape of the pair is worth carrying: an exhaustive answer on a small case, and a partial answer with a mechanism in it that survives the case being scaled up. The second is what makes the first mean something on a patch.

The trap this construction fell into #

There is a piece of history here that is directly downstream of the ring being the failure, and it is worth recording because it went unnoticed for a long time.

The tessellation builder does not draw the letters on. It propagates the vertex conditions to a fixed point and branches where propagation stalls, which returns a solution — and a propagation returns whichever solution its branch order reaches first, not a good one. What it reached first on the square patch was a lettering that satisfied every condition at every vertex and forced a circle of twenty-eight panels.

So the patch that was drawn, printed and measured for several rounds of work had no flat folded state and could be proved to have none in one pass over its own crease list. Nothing noticed, because nothing was reading the crease list that way.

The builder now checks, and where it finds a circle it redraws the letters with the branch order randomised until it finds a lettering without one — seven of forty independent draws are clean on the square tiling, so it is cheap. A patch with no clean draw keeps the first solution and says so rather than pretending, which is the case the rhombille is in.

Why the rings pull together rather than apart #

The coupling has a direction and it is worth asking why it runs the unhelpful way, since two constraints sharing a variable could in principle make each other easier.

A pleat is one crease with one letter, and it belongs to the rings of the two polygons either side of it. Each ring wants its four letters not to agree all the way round — that is what avoiding a circle means — so each ring is asking its four creases to break somewhere. A pleat is one of the four for two different rings at once, and a letter that provides the break one ring needs is, at the neighbouring ring, one more letter agreeing with its predecessor.

So a shared crease is a resource two rings compete for rather than one they share. Breaking one ring at a pleat spends the break, and the ring on the other side has to find its break elsewhere — among creases that are themselves shared. On a patch every pleat is shared, so every break a lettering makes is a break withheld from a neighbour, and the constraints tighten each other instead of relaxing.

That also says which patches should be worst, and the prediction is checkable. A tiling whose polygons have many neighbours puts more rings on each pleat’s competition, so the coupling should be stronger the higher the tiling’s degree — the rhombille, with its degree-six vertices, worse than the square grid at degree four. The rhombille is the patch on which no draw comes back clean, which is consistent and is not proof.

And it explains the essay’s own surprise about polygon size. A larger polygon has a longer ring, which needs one break among more creases and is therefore easier to satisfy — and it also has more neighbours, which is harder. On the twists measured the first effect wins, which is why a hexagon twist never fails and a square one occasionally does. Which of the two dominates on a tiling nobody has drawn is a question about the tiling’s degree against its polygon’s size, and it is exactly the trade a designer choosing a tessellation is making without knowing it.

What a folder would notice #

None of this is visible in the drawing, and that is the part a reader with paper should be warned about.

A crease pattern for a square twist looks the same whichever of the two hundred and fifty-six admissible letterings it carries: the same lines in the same places, differing only in which are dashed and which are solid. Thirty-two of those printings cannot be folded by anyone, and there is nothing on the sheet to say so — the vertices all check out, the paper is the right shape, and a folder discovers the problem by running out of ways to put the layers.

That is the ordinary experience of trying to fold a twist from a pattern found somewhere, and it has usually been attributed to the folder. It is worth knowing that a third of the letterings that pass every published condition are impossible, that the natural one is among them, and that on four of them the impossibility is a chain of eight panels anybody can trace with a finger.

The unit and the tiling, once more #

There is a pattern to the mistakes this family has produced, and this is the fourth of them.

A unit that folds is not a tessellation was the first: a waterbomb base folds, and a sheet of them need not. The tiling the unit could not promise was the second, where the side distances that make one polygon work do not propagate. Cutting a patch out of a plane was the third, where the way the patch was taken from the plane put creases across one another.

This is the fourth and it is the same shape: a property of the unit — that a good lettering exists — does not survive being repeated. The unit has eight good letterings out of two hundred and fifty-six. Nine units sharing pleats have twenty-six out of two hundred draws, and the same construction on a denser tiling has none.

The count that separates them is the number of independent closed chains, which is four on the unit and thirty-six on the patch — and that number grows linearly, four per twist polygon, so nine units carry nine times as many chains and not two to the ninth.

What grows exponentially is the share that fails, because a lettering has to keep every chain from closing at once. If the chains were independent coins, the share consistent at nine units would be the unit’s share raised to the ninth: 198 in 200 becomes about 183 in 200, which is a mild decline.

The measured figure is twenty-six in two hundred, which is seven times worse than that. So the chains are not independent: they share pleats, a letter that suits one ring is a letter its neighbour has to live with, and the coupling is positive — nine rings sharing edges are considerably harder to satisfy together than nine rings that share nothing.

That is the sharper version of a unit is not a tessellation, and it has a number in it. Repetition does not merely multiply the opportunities to fail; it correlates them, and the correlation costs a factor of seven at nine units on this pattern. Whether that factor grows with the patch or settles is a measurement nobody has taken, and it is the one that would say whether a large tessellation is difficult or impossible.

What is different this time is that the failure is not in the drawing. The drawing is right; it has been right since the clipping repair. The letters are what is wrong, and the letters were never drawn by anybody — they were the first thing a propagation happened to return.