The tiling the unit could not promise

Every twist this site draws arrives with the same sentence attached: the unit is verified, the tessellation it belongs to is not, and it cannot be, because deciding whether a whole crease pattern folds flat is NP-hard. That sentence has been true and it has also been doing more work than it should. It is a statement about what cannot be decided, and it has been read as a statement that nothing can be said.

Something can be said, in one direction only, and it costs one pass over the crease list.

The one thing that scales #

A crease decides which of its two panels lies higher. A valley brings the far panel over, a mountain takes it under, and turning the near panel over swaps what a reader sees — so once the letters are on the pattern, each crease is an inequality between two panels.

Collect them and the letters are a directed graph on the panels, with one arc per crease. If that graph has a cycle, no ordering of the panels can satisfy all the inequalities at once, and the pattern has no flat folded state. Not “no ordering was found”: none exists, because the crease rule is necessary and a crease where the paper turns the other way is not the crease that was drawn.

Everything else about layer order is a search. Enumerating the orderings of a sheet’s panels is factorial and gives out somewhere past a dozen panels; the patch in this essay has forty-nine. Finding a cycle is a depth-first walk over the crease list, and it is the same cost on a hundred panels as on nine.

So the answer available at tessellation scale is negative-only. A loop is a proof of failure. No loop is not a proof of anything.

What the twist tessellation actually had #

The square twist tiling drawn here is forty-nine panels, eighty-four creases and thirty-six interior vertices — a patch far past what any enumeration reaches, which is why its letters are found by propagating the vertex conditions rather than by listing labellings.

Propagation returns a solution. It has no reason to return a good one, and on this patch the solution it returned contained a loop of twenty-eight panels. Every vertex of that patch satisfied developability, Kawasaki, Maekawa and the big-little-big lemma. The patch had no flat folded state, and one pass over its own crease list said so.

That is the same failure as the one the printed twist unit had, arriving by a different route. The unit’s lettering was chosen for looking like a twist; the tiling’s was chosen by whichever branch the propagation took first. Neither criterion had anything to do with layers, and both selected a pattern that cannot be made.

Drawing again until the letters agree #

The repair is cheap because the test is cheap. Draw letterings independently — the same randomised propagation that samples a pattern’s folding set, which makes each draw a separate solution rather than the same walk twice — and take the first one whose letters do not contradict themselves.

On the square tiling, seven of forty draws are clean. On a small patch of the same tiling, nine panels rather than forty-nine, thirty-four of forty are. On a single square twist unit it is thirty-nine of forty, and on a single hexagon twist unit all forty.

The trend down that column is the essay. A single unit’s letters almost never contradict themselves — thirty-nine of forty on the square, forty of forty on the hexagon. A patch of nine still mostly does not, at thirty-four of forty. At forty-nine it is seven, which is one draw in six.

The three that never got as far as a loop #

The other three tilings in the family — triangular, hexagonal and rhombille — fail earlier, and the failure is worth more than the loop is.

Placing a folded state is a composition of reflections: the panel on the far side of a crease carries the near panel’s motion followed by reflection in that crease, and a walk over the panels places all of them from one starting panel. The walk uses a spanning tree, so it places every panel exactly once whatever the pattern says. Every other shared edge is then a second route to a panel already placed, and the two routes agree only if the reflections around each loop compose to the identity.

On the square tiling the largest disagreement between two routes is 5 × 10⁻¹⁵ of a sheet width, which is arithmetic noise. On the other three it is 1.84, 2.20 and 1.96 sheet widths.

A disagreement of two sheet widths is not a numerical problem. It means there is no consistent flat placement of those panels at all: the paper cannot be put down. And every one of the three satisfies every condition at every interior vertex — developability, Kawasaki, Maekawa and the big-little-big lemma, with zero failures reported across seventy-three, eighty-one and a hundred and twenty-five panels.

So the ordering question is not merely undecided on those patches. It does not arise, because there is no folded state whose layers could be ordered, and the machinery refuses them rather than ordering panels that are not anywhere.

Two proofs, both one pass long #

The two failures are different and it is worth keeping them apart, because they are refused at different stages and by different arithmetic.

The placement fails when the reflections round a loop of panels do not compose to the identity. That is checked before any letter is read — it depends on the angles — and it is the condition the placement routine was written to have arrive from a direction it was not put in. A pattern that fails it has no folded state whatever its letters say.

The order fails when the letters demand a loop of panels. That is checked after the letters are on and depends on nothing else. A pattern that fails it has a consistent placement and no way to stack it.

Both cost one pass. Both say no and never yes. And both were available on this site’s own patterns for years, computed and never read: the placement routine has reported the placement disagreement since it was written, as a number nothing looked at.

Why a bigger patch is worse #

The mechanism is visible in the small case and it is the loop the square twist unit has when its ring reads as one letter: each corner region between two consecutive pleats has to lie above one pleat and below the other, and going round the ring the requirement closes on itself.

A tessellation is that structure repeated and joined. Every twist polygon has a ring of pleats round it and every pleat is shared with a neighbour, so the arcs a single unit contributes are woven into a much larger graph — and a graph with more arcs per node has more cycles in it. One twist unit carries twelve arcs over nine panels; the tiling carries eighty-four over forty-nine, and every extra arc is another chance to close a loop.

That is why the share of clean draws falls rather than staying put. It is not that large patterns are harder to letter correctly; it is that a randomly chosen correct-at-every-vertex lettering of a large patch is much more likely to contain a contradiction somewhere, and there is a lot of somewhere.

How fast the clean draws run out #

The column of clean draws falls from thirty-nine in forty to seven, and the essay reads that as a trend without giving it a rate. There is a rate, it is computable from the patterns’ own crease counts, and it turns the closing prediction into a number.

The panels and the creases make a graph, and the number of independent loops in a connected graph is its creases less its panels plus one. One square twist: twelve less nine plus one, which is four. The hexagon twist: six. The nine-panel patch of the tiling: four. The forty-nine-panel tiling: eighty-four less forty-nine plus one, which is thirty-six.

Now suppose each independent loop is its own coin — that a draw is clean when none of the loops closes badly, and that the loops are roughly independent of one another. Then the clean share is $p^{c}$ for some per-loop probability $p$ and the loop count $c$, and $p$ can be read off any one row and used to predict another.

Read it off the nine-panel patch: thirty-four in forty over four loops gives $p = 0.85^{1/4} = 0.960$. Carry that to the full tiling’s thirty-six loops: $0.960^{36} = 0.230$, which is nine draws in forty. Seven were observed, and with forty draws the spread on a count like that is nearly three either way.

So the model holds across a fivefold change in the patch, which is worth more than the agreement of any single number. Reading $p$ off the large patch instead gives 0.953, against 0.960 from the small one — the same coin, near enough, with the two twist units sitting a little above it because a lone unit’s four loops all run round one polygon and are not independent at all.

Which puts a ceiling on the repair #

The practical consequence is the one the closing section asks for and does not have. Redrawing until the letters agree works, and it works for exactly as long as the clean share stays above the number of draws anybody will make.

That share is $0.96$ raised to the loop count, and the loop count grows with the area of the patch — a tessellation adds a loop for roughly every panel it adds, since each new panel arrives with more than one new crease. So the clean share decays exponentially in the number of panels. Thirty-six loops give one draw in four or five. A patch of a hundred panels carries something like eighty loops and gives one in thirty. Two hundred panels gives one in a thousand, and at that point drawing again is not a repair; it is a search with no better prospect than any other.

That is a definite and unwelcome answer. The cheap test scales — one pass over the crease list, whatever the size — and the cheap repair does not, because it relies on stumbling into a consistent lettering rather than constructing one. Somewhere between fifty and two hundred panels the redrawing stops paying, and past there the honest position is that a large twist tessellation has a lettering nobody has found rather than one that has been checked.

Where the loops are #

The loops are not scattered. Every one found here runs through the corner regions between consecutive pleats, which are exactly the panels a twist tessellation has most of.

On a single square twist the loop is eight panels long — four pleats and the four corners between them — and it is the loop that made every uniform-ring lettering unfoldable. On the nine-panel patch of the tiling it is eight again. On the forty-nine-panel patch the propagation’s own answer produced twenty-eight, which is the same structure running through several polygons before it closes.

Loop length is not severity. A loop of eight and a loop of twenty-eight are equally fatal; what the length says is how far the contradiction had to travel to close. A short loop means a single polygon’s own ring is inconsistent; a long one means several polygons agreed locally and disagreed once the chain came round.

That distinction is what makes the repair work at all. The square tiling’s loops are long, so a lettering that differs anywhere along a twenty-eight-panel chain breaks it — which is why seven of forty independent draws come back clean rather than none of them.

What the unit could and could not promise #

A unit that folds is not a tessellation made this argument with counts of surviving repeating rules. This is the same argument with a proof in it.

A verified unit promises exactly one thing: that its own vertices satisfy the conditions. Tiling it copies those vertices and adds new ones where the copies meet, and the new vertices are checked too — but nothing in any of that looks at the order of the layers, and the order is where a tessellation fails.

The failure is not local and cannot be made local. The loop of twenty-eight on the square tiling passes through panels from several different twist units. There is no unit to examine and no neighbourhood to check that would reveal it; the loop only exists once the pieces are joined into a ring.

Three limits worth naming #

The test says no and never yes. A patch whose letters are consistent has passed one necessary condition. Forty-nine panels is far past the point where the orderings can be enumerated, so “no loop” is where this essay’s certainty stops, and it stops there deliberately rather than being rounded up.

The draws are not uniform. They come from propagation with a randomised branch order, which samples solutions and does not sample them evenly. “Seven of forty” is a statement about that sampler on that patch, not an estimate of the fraction of clean letterings — and it is quoted as a count for that reason.

And the patch is not the plane. Every number here is about a finite patch with a boundary, and the boundary is where the letters have room to move: a crease reaching the raw edge of the sheet only constrains one interior vertex. A larger patch of the same tiling has proportionally less boundary, so the share of clean draws should keep falling. That is a prediction and this essay does not test it.

Why nobody noticed #

The failure had years to be found in and was not, and the reason is instructive rather than embarrassing.

Nothing draws the folded state of a tessellation. The figures on this site draw the pattern — a grid of lines with colours on them — because that is what a reader folds from and what a designer publishes. A pattern with a loop in its forced order looks exactly like a pattern without one: the same lines in the same places, the same count of mountains and valleys, the same sector angles at every vertex.

And every check reads the pattern. The flat-folding assertion evaluates four conditions at each interior vertex and the vertex sweep collects them; the drawing checks measure labels, boxes and fills. Not one of them composes a reflection, and the two quantities that would have caught this — the disagreement between two routes to a panel, and a loop in the order the letters force — are one function call and one graph walk that nobody had written a reason to make.

That is the shape of every defect the new instrument found: not a wrong calculation, but a question with no instrument pointed at it.

What a folder should take from it #

A tessellation printed with mountains and valleys on it has been checked at its vertices and, until now, nowhere else. That is true of every published crease pattern of a twist tessellation, this site’s included, and the check that was missing is one pass long.

If a large patch will not collapse, look for a ring. The failure has a shape: a closed chain of regions each of which must sit above the next. It is the same shape at every scale, and on a tessellation it can run through a dozen units.

And a unit that folds is a promise about the unit. Any tiling makes a twist is a statement about geometry, and the geometry is fine — the sector angles hold as an identity rather than by design, at every corner of every tiling in the family. What the tiling does not inherit is the order.