A region with no lettering

Turning a dial on a construction and watching what happens is the cheapest experiment this collection has, and it is usually the one that produces the least. Most parameters change a drawing without changing anything a theorem cares about; a dial that decides nothing is the standing example, and the expected outcome of any sweep is that the answer is the same all the way along.

Sweeping the turn angle of a twist tessellation was not like that. Somewhere just above a fifth of a radian the triangular patch stops having any consistent lettering at all — proved by exhausting the search rather than by failing to find one — and just below that it has one found in forty-five steps. Same panels, same arcs, same table sizes at every vertex, and a different answer.

One sweep of one dial finds a threshold. Two dials find something with a shape.

Nine patches, in a corner #

Sweeping turn angle against pleat width over four tilings gives ninety-six patches. Eighty-seven of them have a consistent lettering, found in twenty-one to forty-seven steps. Nine have none, and every one of the nine is at a shallow turn and a narrow pleat.

They are not scattered. They occupy the corner of the plane where both parameters are small, on the elongated, hexagonal and triangular tilings, and the corner has the shape a region has rather than the shape a line has: at the narrowest pleat width the failure reaches out to a turn of 0.25 radians, and at the next width it reaches only to 0.15.

That is the correction the two-dimensional sweep makes to the one-dimensional one. The threshold found by turning a single dial was real and it was not a threshold — it was one section through a boundary, taken at whatever pleat width the sweep happened to be holding fixed, and the boundary moves when the other parameter moves.

The nine are a staircase, one cell per step #

The region’s shape can be reconstructed exactly from the description, and doing so says more than “a corner” does.

Three tilings fail — elongated, hexagonal and triangular — and the square never does. At the narrowest pleat width the failure reaches turns of 0.15 and 0.25; at the middle width it reaches only 0.15; at the widest it reaches nothing.

Multiply: $3 \times 2 + 3 \times 1 + 3 \times 0 = 9$, which is the count.

So the region is not a ragged corner. It is a staircase with steps of exactly one grid cell, and — the part worth noticing — the three failing tilings fail in exactly the same cells. Not one of them fails at a setting where another succeeds.

Which says the boundary is not about the tiling #

Three different tilings with three different vertex arrangements, three different panel counts and three different pleat geometries, and their boundaries land in the same two grid cells.

That is strong evidence for the mechanism the essay proposes. If the boundary were a property of each tiling’s own sectors, three tilings would have three boundaries and the grid would show a ragged edge. A common edge says the sectors of all three cross their threshold at the same combination of turn and pleat, which is what a shared construction with a shared sector formula would produce.

The square’s absence then reads as the control it is: the one tiling whose sectors never cross, failing nowhere at any setting.

And the grid catches only a corner of it #

The staircase also says which way the boundary runs, and it runs off the edge of the sweep.

As the pleat widens, the critical turn falls: 0.25 at the narrowest width, 0.15 at the middle, and something below 0.15 at the widest — below the smallest turn the grid samples.

So the region is a wedge whose boundary descends as the pleat widens, and the grid has caught its upper corner. At the widest pleat width the failures are there and the sweep does not reach them, which is a prediction rather than a speculation: turns of 0.05 and 0.10 at the widest width should refuse, and the construction will draw them.

That is the cheapest next measurement available — three more cells, on patches the sweep already knows how to make — and it would say whether the region is a wedge or a corner.

The square tiling has none of it #

Across every combination of turn angle and pleat width, the square tessellation patch always has a consistent lettering. Not once does it join the others.

The reason is the same one that produced the original threshold, and it is a fact about sectors rather than about searches. The big-little-big lemma forbids the two creases bounding a strictly smallest sector at a vertex from carrying the same letter, so which pair of creases the lemma constrains depends on which sector is smallest — and as a sector angle crosses sixty degrees the smallest sector changes identity, swapping the constrained pair at every vertex of the patch at once.

A square twist’s vertices have sectors that never pass through that crossing, whatever the turn and whatever the pleat. There is no swap to undergo, so there is no configuration in which the lemma’s demands at neighbouring vertices become jointly impossible, and the square patch letters at every setting.

It is worth being careful about what “crosses sixty degrees” means, because the number is not a coincidence and it is not universal either. At a degree-four vertex the four sectors sum to a full turn, so their average is ninety; at degree six the average is sixty, and a sector passing through sixty on a degree-six vertex is a sector passing through the average. The tilings that fail here are the ones whose twist polygons produce degree-six vertices with sectors that straddle their own average as the construction’s parameters move. The square twist’s vertices are degree four and its sectors straddle nothing.

So the boundary is not a magic angle in the subject. It is the place where a particular family’s particular sectors reorder, and a different family would have its boundary somewhere else — or nowhere, like the square.

Why the failure is global and the condition is local #

A reader who knows the lemma might expect the failure to be visible at a single vertex, and it is not. Every one of the nine patches has vertices that are individually fine: each has admissible labellings, and a table of them can be written down.

What fails is the joint problem. The lemma’s constrained pair at one vertex shares a crease with the constrained pair at its neighbour, and once the identity of those pairs has swapped everywhere at once, the constraints chain around the patch and close on themselves. There is no assignment satisfying all of them simultaneously, and there is no single vertex to point at as the culprit.

This is local rules and global behaviour in its cleanest available form. The conditions are local, the drawing is uniform, every vertex is satisfiable, and the sheet is not. The gap between those two statements is what makes flat-foldability hard in general, and here it is visible as a region on a two-parameter plane.

There is a reading of this that makes the region much less surprising, and it is the right one. A tessellation patch is a rigidly regular object: every vertex is a copy of a few templates, laid out on a lattice, so any statement true at one vertex is true at dozens. That is what makes tessellations tractable, and it is also what makes their failures total. A patch does not fail a little — when its vertex template moves across the crossing, every instance of that template moves at once, and the constraints they impose on one another go from jointly satisfiable to jointly impossible in one step.

An irregular pattern would degrade instead. Move a parameter and a few vertices change their smallest sector while the rest do not, the constraints partially reorganise, and there is usually still an assignment somewhere. Regularity buys structure and pays for it in brittleness, which is a trade the subject makes everywhere and rarely gets to see measured.

What proving it costs, and why that varies wildly #

Proving that a pattern has no lettering means exhausting a search, and the cost of that turns out to be an unreliable guide to anything.

On the nine uniform patches the standard branching rule refuses in fifteen steps apiece — cheaper than finding a lettering anywhere in the collection, because the propagation reaches a vertex with nothing left almost immediately. On the same nine, a rule that branches on the patterns’ independent circuits needs between 879 and 524,287.

So the region is cheap to map with one rule and nearly impossible with another, and neither is the good rule. That matters for reading the map: the eight-by-four grid above is affordable only because the standard rule happens to be the right one on these patterns, and a finer grid over a family where it is not would be a different proposition entirely.

The rhombille, which is not on the map #

The four tilings above are the ones whose vertices are all alike. The rhombille has two kinds of vertex, so the matching condition that gives each vertex its own side distance has something to propagate there and nothing to propagate on the others — a genuine structural difference rather than a fifth sample of one thing.

It has the same region. At a turn of 0.15 with either of the two narrower pleat widths, and at 0.25 with the narrowest, the rhombille patch has no consistent lettering either. Those three are the patches that stood undecided, because the standard rule needs a quarter to half a million steps on them and the budget was two hundred thousand.

They are settled now, and they extend the region across the fifth tiling — which is worth having, because a region that stopped exactly at the boundary of the tilings somebody chose to sweep would be a suspicious region.

There is one more asymmetry hidden in those costs and it is worth naming. Fifteen steps is not merely cheap — it is cheaper than the cheapest positive answer anywhere in this collection, and the ordinary intuition about search says that should be impossible, because a no has to look everywhere and a yes only has to look once.

The intuition is about the size of the space and the measurement is about where the contradiction sits. Here the contradiction is at the root: a handful of forced letters, a vertex with no surviving labelling, and the space that would have had to be searched never opens. A no is expensive when the refusal is deep and cheap when it is shallow, and nothing about the answer decides which.

Nothing published is affected, and that is not luck #

Every twist tessellation patch this collection prints is drawn at a turn of 0.35 radians or above, which is outside the region on every tiling. That was not a precaution against a phenomenon nobody knew about; it is because a shallow turn produces a patch that looks wrong — the twists barely turn, the pleats are nearly parallel, and the picture stops being a picture of a twist.

So an aesthetic preference kept the collection out of a region that would have produced unfoldable published patterns. That is worth noticing without drawing a moral from it: the preference had no connection to the mathematics, it happened to align, and there is no argument that it would have aligned on a different construction.

What the region does affect is any future sweep. A parameter grid over this family now has nine cells that will refuse, and refuse correctly, and any process that treats a refusal as an error rather than an answer will report nine failures that are not failures.

What a folder would see #

A patch from inside the region, printed, looks like an ordinary crease pattern. The lines are where the construction puts them, the twists turn, the pleats run between them, and nothing about the drawing says it cannot be folded.

A folder attempting it would get some way in. Creasing the polygons and the pleats is unproblematic, and the sheet would gather in the usual way — and then, somewhere, two creases would need to be the same letter and would need to be different letters, and the paper would refuse. Not tear, and not obviously fail: a sheet that will not lie flat holds itself a little open, and the usual response is to press harder.

That is worth stating because it is the practical content of the whole result. The region is not a curiosity about a search. It is a set of crease patterns that look right, print right, and cannot be folded, and the only way to know in advance is to ask the question the search asks.

Which theorem was checked, and how #

The verdict on every patch is an exhaustion, not a failure to find. The search reports three outcomes — a witness, a completed search with nothing in it, and a budget that ran out — and only the second is used here.

Every verdict is also checked under more than one order. An exhaustion visits the whole tree, so two orders that both finish must agree; that agreement is asserted across the grid rather than assumed, since an ordering rule that changed a verdict would mean the search was not exploring what it claims to. And every witness on the other side of the boundary is written back onto its pattern and put past the four conditions and a folded sheet rebuilt from scratch.

What the picture cannot show #

The boundary. The grid has eight turn angles and three pleat widths, so the failure’s edge is known to within one grid step in each direction and no better. Whether the region’s boundary is a smooth curve, a staircase, or something with structure in it is not answered by anything here, and a finer grid is the obvious next measurement.

Nor does the map say anything about tilings not on it. Four uniform tilings and the rhombille are five tilings, chosen because the construction supports them, and which polygons twist at all is a shorter list than anybody expects. The region may well be a general feature of twist tessellations; five families is not enough to say so.

And a region of no letterings is not a region of no folded states. The arcs are a necessary condition, so a pattern failing them fails — but the converse never held, and patterns whose letters agree and whose panels still cannot be stacked are why the two questions are kept apart.

The measurement that should be made next #

The obvious one is a finer grid, and it is worth saying what it would buy and what it would cost.

It would buy the boundary’s shape. At present the region is known to within one grid step in each direction, which is enough to say it is a region rather than a line and not enough to say whether its edge is smooth. If the account above is right, the edge should be exactly the locus where a particular sector of the twist’s vertex template equals sixty degrees — a curve computable in closed form from the construction’s own geometry, with no searching at all. Comparing a computed curve against a searched grid would be the strongest available check on the mechanism, because the two share no code and no reasoning.

It would cost very little on the four uniform tilings, where a verdict is fifteen to forty-seven steps and a patch takes a fraction of a second to draw. It would cost a great deal on the rhombille, where drawing a single shallow patch takes minutes before any search begins — because the construction finds its own letters by the same kind of search, and at the shallow end that search is itself expensive.

So the sensible next measurement is a fine grid on the four cheap tilings, a closed-form curve derived independently, and the rhombille checked at a handful of points rather than swept. That is a shape this collection reaches for often: the expensive member of a family gets sampled while the cheap ones get mapped, and the map is what the sampling is checked against.

Where the ladder goes next #

Two directions. The population these nine were found in was assembled from a grid rather than by hand, which is the only reason they were found at all — none of the five patterns anybody had chosen is anywhere near the region. And the family they belong to has a second dial with a much stranger reading, on the one tiling of the five whose vertices are not all the same.