Fenced at both ends

The construction two rungs down leaves two numbers free: how large the twists are, and how far they are turned. The turn is the one that makes a twist look like a twist, and throughout that essay it was dialled from eleven degrees to forty-five without comment, on the grounds that Kawasaki holds identically and nothing else was asking.

Something else is asking. At each end of the dial the construction stops working, and the two ends fail for reasons that have almost nothing in common.

The ceiling is arithmetic about paper #

The upper fence is easy and it is worth doing first, because it is the one anybody would predict.

Between two neighbouring twists there is a pleat, and its width is the distance between the two facing sides. If the twists’ sides sit at distances d and d′ from their vertices and the tiling edge between them is L long, that width is L · cos θ − d − d′, which falls as the turn grows, because turning a side away from the edge it faces moves it closer to the neighbour. It reaches zero at θ = arccos((d + d′) / L), and past that the two twists want the same paper. With the pleats using about three-fifths of the room available — the setting every figure in this ladder uses — that is 55.52°.

There is nothing deep here. It is the geometric statement that a construction runs out of room, and the only reason to measure it rather than assert it is that the flat-folding conditions do not notice. Kawasaki is a statement about a full turn of paper at one point, and a point does not know that the paper around it has been used twice. At 60°, past the fence, every alternating sum at every corner is still exactly a straight angle. The pattern is nonsense and the theorem is content.

The floor is not arithmetic about anything #

Now turn the other way. Below about twelve degrees on the triangular grid, the construction produces a crease pattern whose angles satisfy every condition in the subject — developability, Kawasaki, both of them exactly — and which has no mountain-valley assignment whatever.

That is a strange sentence and it is worth being careful about what it says. Kawasaki is a condition on angles and it is satisfied. Maekawa and the big-little-big lemma are conditions on letters given the angles, and at any single vertex both are satisfiable — there are labellings of four creases that work. What fails is the attempt to satisfy them at every vertex at once. The conditions at neighbouring corners share creases, and below the floor the shared demands are contradictory.

The search that establishes this is exhaustive rather than heuristic. Kawasaki is already settled by the geometry, so what remains is a constraint problem over the free creases: enumerate the labellings of each vertex’s own four creases that pass Maekawa and big-little-big, propagate to a fixed point, and branch where propagation stops. A run that returns nothing has tried everything. That distinction matters more here than usual, because a negative answer from a search looks exactly like a result — and this site has already been caught by a checker whose refusal was about the checker rather than about the paper.

What changes at the crossing #

The floor is not arbitrary. It sits exactly where the smallest sector at a corner changes which one it is.

At a corner the four sectors are the twist polygon’s own interior angle, a pleat sector, the tile face’s angle, and the other pleat sector. On the triangular grid the polygon’s angle is 120° and the tile’s is 60°, both fixed; the two pleat sectors move with the turn, one growing and one shrinking as they must, since they are supplementary.

At a large turn the shrinking pleat sector is the smallest of the four. It lies between a polygon side and a pleat crease, so the big-little-big lemma demands that those two have opposite letters — one demand per corner, and satisfiable.

At a small turn the pleat sector has grown past 60° and the tile’s own angle is the smallest. That sector lies between the two pleat creases, so the lemma now demands that those two differ. Each pleat crease belongs to two corners, one at each end, and the two ends make demands that cannot both be met. The system has no solution, and the pattern has no assignment.

So the fence at the bottom is a combinatorial fact, produced by a geometric one. The angles decide which sector is smallest; the smallest sector decides which pair of creases must differ; that pair decides whether a global labelling exists. Three steps, and only the first of them is about paper.

What the search actually does #

It is worth saying plainly what the labelling problem is, because “no assignment exists” is the kind of claim that deserves to have its method on the table.

Each free crease is a variable with two values. Each interior vertex has four creases and a list of the labellings of them that pass — sixteen candidates, of which Maekawa’s three-to-one split leaves eight, and the big-little-big lemma leaves rather fewer. Two vertices that share a crease share a variable, so the pattern is the intersection of those lists over the whole sheet.

Propagation does most of the work: wherever every surviving labelling of a vertex agrees about one of its creases, that crease is fixed, and fixing it prunes its neighbours. On these patterns the propagation alone usually finishes. Where it stalls, the search branches on the vertex with the fewest labellings left — which sounds like a detail and is not, since branching on the first unlabelled crease instead makes the same patterns unsolvable in any reasonable time. The creases are numbered in the order the pattern was drawn, so a search that takes them in order walks across the sheet planting decisions nothing contradicts until it is a long way from the corner that will refuse.

Where the two fences move #

The floor and the ceiling do not sit still. Both depend on how large the twists are, and — pleasingly — they move in opposite directions.

Larger twists leave less room, so the pleats close at a smaller turn and the ceiling comes down. Larger twists also change the pleat sectors, and the crossing they are heading for arrives at a smaller turn too, so the floor comes down further and faster: on the triangular grid it is 24.79° when the twists are small and 6.59° when they nearly fill the space available.

The band between them is therefore never empty, but it is not the same band at every twist size, and a pattern chosen near either edge is a pattern whose properties are about to change discontinuously. That is the practical form of this result: a designer moving a twist angle a couple of degrees for aesthetic reasons can walk a working pattern out of existence, and nothing about the picture warns of it.

The square has no floor, and that is the tell #

One of the four tilings behaves differently, and the reason confirms the account above rather than complicating it.

On the square grid the twist polygon’s interior angle is 90° and the tile face’s angle is 90°. They are equal, because the square is self-supplementary in the sense that matters here. So the smallest of the four sectors is always one of the two pleat sectors — one of them is below 90° whenever the turn is anything but zero — and the situation that produces the floor never arises.

Measured down to a fifth of a degree, the square grid’s pattern is assignable everywhere. That is a prediction the account makes and a measurement confirms, and it is the difference between an explanation and a coincidence.

One twist has no floor either #

The most useful control is the smallest one. Take the triangular grid at half the floor angle — an angle where a patch of eleven twists has no assignment at all — and build the same construction on a patch large enough for exactly one twist.

It solves immediately.

So the floor is a property of the tessellation and not of the twist. That is precisely the distinction the waterbomb’s rule family made first, arriving here in a family nobody wrote it for, and by a different mechanism: there, a small patch simply did not contain all four kinds of vertex the pattern makes; here, every patch contains the same one kind of vertex, and what a patch of one lacks is the sharing.

The refusal has a name #

The account above says the demands at the two ends of a pleat crease cannot both be met. That is correct, and it is doing more work than it looks, so the structure is worth naming — because it is a structure this collection already has a ladder about.

Every demand the big-little-big lemma makes takes one form: these two creases must differ. It never says which letter either of them takes; it constrains the pair. A collection of such demands is a graph — a node for each crease, an edge for each demand — and a labelling satisfying all of them exists exactly when that graph can be two-coloured.

A graph is two-colourable precisely when it contains no odd cycle. So below the floor there is an odd cycle among the demands and above it there is not. That is the whole obstruction, stated in the vocabulary the subject already keeps for the two-colouring of a folded sheet’s own faces — a different graph, the same criterion.

Two consequences fall out at once, and both match what was measured rather than being fitted to it.

A single twist cannot fail. Its demands form a tree: each pleat crease runs off to the edge of the sheet and has a corner at one end only, so there is no cycle of any length to be odd. That is why the one-twist control solves at every turn, and it is a stronger statement than the version above it — not merely that a lone twist has no neighbours, but that a system of difference constraints with no cycle in it cannot be refused whatever its angles do.

Two twists are enough. A cycle needs a crease with a corner at each end and a path back between them, which is exactly what one shared pleat provides. Nothing larger is required to produce the floor, and the patches of seven, eleven and twenty-three twists add copies of the same cycle rather than new kinds of cycle.

What that says about the count #

The closing section asks how many assignments a pattern inside the band has, and the two-colouring reading answers most of that before any search is run.

A consistent system of differ constraints has exactly $2^c$ solutions, where $c$ counts the connected components of the demand graph: choose a letter for one crease in each component and everything else in it follows. So the count is a power of two, and which power is decided by the pattern’s connectivity rather than by its angles.

Two things qualify that and neither of them removes it. The lemma constrains only the two creases flanking a strictly smallest sector, so a crease the lemma never mentions is a component of size one and doubles the count on its own. And Maekawa is not a difference constraint at all — it is a count, so it cuts the two-colouring’s solutions down from outside rather than being expressible among them.

What survives is a prediction with a definite shape. The number of assignments should be a power of two reduced by whatever Maekawa removes; it should not vary continuously with the twist angle; and it should step at exactly the angles where the smallest sector changes which one it is. That is the same crossing the floor sits at. So the count is expected to hold constant across most of the band and to jump at its edges — a far more specific claim than nobody has looked, and one the search already written could settle in an afternoon.

Where the model stops #

The floor was found and not derived. The account above says why the constraint changes at the crossing and it does not prove that the changed constraint is unsatisfiable — that is an observation about the search’s output on patches of seven, eleven and twenty-three twists. A proof would be a statement about an infinite pattern and this is a statement about three patches.

Nothing here decides that the sheet folds. Every condition used is local. Deciding flat-foldability in general is NP-hard, so a pattern inside the band is a candidate and not a certificate, and one outside the band is genuinely refused rather than merely unproven — which is the useful asymmetry: a no here is a real no.

The band is drawn at one pleat setting. Everything above holds the pleats at a fixed fraction of the room available. The two-parameter family is a surface rather than a curve, and the figures cut it along one line.

The ceiling is about this construction, not about paper. A pattern past 55.52° could be rescued by making the twists smaller, which is a different point of the family rather than a different pattern. What cannot be rescued is a pattern below the floor: shrinking the twists moves the floor down, so a pattern that has no assignment can be brought back into existence by changing its twists — which makes the floor a statement about a point in the family and not about the tiling.

Why nobody hit this #

Folders have been making twist tessellations for fifty years and this fence is not in the literature, which usually means one of two things: it is wrong, or nobody was ever near it.

It is the second. Twist tessellations are folded from paper, and a twist turned by five degrees is a twist whose pleats are two millimetres wide on a sheet the size of a book. It cannot be folded, it cannot be seen, and it looks like nothing at all — so the region below the floor is a region no folder has any reason to visit. The patterns people make sit at twenty to forty degrees, comfortably inside the band, and the dial has never been turned far enough down for the question to come up.

That is a satisfying reason and it is also a warning about this whole approach. A construction that is run by a program will visit parameter values that a person making things would never choose, and most of what it finds there is uninteresting. Occasionally the uninteresting region contains a fact about the interesting one — here, that the labelling of a twist tessellation is a constraint problem with a solution rather than a rule anybody could write down, which is true at every angle and only visible at the small ones. The same shape of finding has turned up before on this site, in the assignment nobody would have drawn for the preliminary base and in the rules a small waterbomb patch cannot see.

Where the ladder goes next #

The band has been mapped for four tilings and the natural next question is which tilings have a band at all. The matching condition that fixes two sizes of twist against each other has to close around every loop before there is a pattern to have fences on, and a tiling where it does not close is refused before the twist angle is even considered.

The other direction is the assignment itself. It has been treated here as something a search either finds or does not, and the count of how many it finds is a quantity nobody has looked at: a pattern with one labelling and a pattern with thousands are both “assignable”, and the difference between them is the difference between a pattern that folds one way and a pattern whose folder has a decision to make at every twist.