The propagation that never had to work

Any tiling makes a twist hands the construction a tiling and gets back a crease pattern. At each vertex of the tiling it places a small polygon, turned by a common angle and set back from the vertex by a distance; between neighbouring polygons it draws a pleat. For the pleat to be paper rather than a drawing, the two sides facing each other across an edge have to be the same length, and that is one equation for every edge of the tiling.

Those equations are propagated. A seed vertex is given a distance, its neighbours’ distances follow, theirs follow in turn, and the walk covers the patch. Round a loop of edges the ratios multiply, and whether the product is one is a property of the tiling.

That machinery has been here since the tessellations were built, it is described where it runs, and until now nobody had asked what it was actually doing.

What the equation on an edge says #

The side of a twist polygon that faces a given neighbour runs between two corners, and each corner sits in a tile of the tiling. How long that side comes out is the set-back distance multiplied by a factor built from the interior angles of the two tiles at that corner — a sum of two tangents of half-angles, which the module computes and does not quote.

So the equation on an edge is a ratio of two such factors, one read at each end of the edge. And here is the sentence the whole essay turns on: the two tiles that meet along an edge are the same two tiles at both of its ends. Walking from one end of an edge to the other, the polygons on either side do not change.

If those polygons are regular, they have one interior angle each. So both ends read the same two numbers, the factors are equal, the ratio is exactly one, and the propagation carries the same distance everywhere it goes.

Measured on five tilings by regular polygons — the square grid, the triangular grid, the honeycomb, the elongated triangular tiling and the truncated square tiling — the worst departure from one, anywhere, is zero to the last bit a double holds. Not small. Zero.

That is not a fact about symmetry. It has nothing to do with the tiling being vertex-transitive, or with any group acting on it. It is a fact about regular polygons having one angle, and it holds on tilings whose vertices are not all alike as readily as on tilings whose vertices are.

It is worth pausing on how thoroughly that hides. The propagation is written, it runs on every tessellation the site builds, it reports a worst-loop figure, and that figure has been zero on every pattern published — which reads as the machinery working. It was the machinery having nothing to do. A quantity that is always zero and a quantity that is zero because it is being maintained look identical from the outside, and the only way to tell them apart is to break the thing that was supposedly maintaining it.

The one that did something #

The rhombille tiling is made of rhombi, and a rhombus has two different interior angles. So the two ends of an edge read different pairs, the factors differ, and the ratio is not one — it is three, or a third, depending on which way the edge is walked.

That is exactly what where two twists share a pleat found: hand the construction a tiling with two kinds of vertex and the pleat between a large twist and a small one fixes their sizes at three to one, and nothing else folds. The number came out of the propagation, and the propagation was doing work for the first and only time.

The second half of that is not automatic and had not been checked. Ratios that are not one may fail to close: go round a loop of edges multiplying, and there is no reason in general for the product to come back to one. On the rhombille it does, to 4 × 10⁻¹⁵, on every loop of every patch generated. So the rhombille is a tiling whose equations are non-trivial and consistent, which is a narrow and lucky place to be.

The tiling where they do not agree #

Take any of these tilings and move its vertices — a seeded displacement of a few per cent of an edge length, applied to every vertex. What comes back is still a tiling of the plane. The edges are still straight, the faces are still polygons, no two edges cross, and every vertex still has the degree it had. The twist construction accepts it exactly as it accepts a regular one, because nothing in the construction asks whether a tile is regular.

What it stops being is a tiling by regular polygons. Each tile now has several different interior angles, the two ends of an edge read different pairs, and the ratios are no longer one.

They do not. On the square grid at six per cent, the loops disagree by 38 per cent; at twelve per cent, by 90 per cent; on the honeycomb at twelve per cent, by 173 per cent. Two paths from the same seed to the same vertex demand twist sizes differing by nearly a factor of three.

A pattern that satisfies inconsistent equations does not exist. What the construction returns instead is a drawing.

The refusal, and an honest limit on it #

The refusal is total and it is worth being precise about what causes it. On a deformed tiling every interior vertex of the returned pattern fails Kawasaki, not merely the vertices near the edges whose equations disagreed. So the failure is not only the loop condition: the twist polygon’s own corners stop being flat-foldable too, because the construction sets every side back by the same turn from the edge it faces, and that turn produces a flat-foldable corner only when the tiling’s angles cooperate.

The loop condition is the part that matters here because it is the part no vertex can see. A corner that fails Kawasaki fails a check the site already runs, at a point, immediately. A loop that fails to close is a property of a cycle of edges, it involves no vertex at all, and there is nothing local to test. Two failures arrive together on a deformed tiling and only one of them has a detector.

Which theorem was checked, and how #

Three separate measurements, and the first is the one that makes the others mean anything.

The per-edge ratio is computed directly: for every edge of a patch, the side factor at each end, and their quotient. On the five tilings by regular polygons every quotient is one to machine precision, and if any of them had not been, the explanation offered above would be wrong rather than incomplete.

The loop closure is computed by the propagation itself, which walks a spanning tree from a seed and reports the largest relative disagreement it finds when it reaches a vertex it has already assigned. That is the only place a loop can announce itself, and the number it reports is a departure from one rather than a distance.

The refusal is computed by the site’s own checker on the returned pattern, vertex by vertex, with the residual reported in degrees rather than as a boolean — because the interesting fact is not that a deformed tiling fails but by how much, and nine degrees at every vertex is a different statement from a hundredth of a degree at one.

Regular is more than the argument needs #

The explanation above turns on regular polygons having one interior angle, and that is sufficient. It is not necessary, and the weaker condition is worth extracting because it says which other tilings are in the same comfortable position.

Read the ratio on an edge again. It compares a factor built at one end against a factor built at the other, and each factor is built from the interior angles that the edge’s two tiles contribute at that corner. So the ratio is one when each of the two tiles offers the same angle at both ends of the edge.

Now ask what a polygon must be like for that to hold on every one of its edges. Its angle at the first end of an edge must equal its angle at the second, for every edge — which chains round the boundary and makes every interior angle equal to every other. The tile has to be equiangular, and nothing about its side lengths enters at all.

Equiangular is genuinely weaker than regular. A rectangle that is not a square is equiangular; so is a hexagon with all its angles at a hundred and twenty degrees and its sides of six different lengths. Every such tile offers one angle at every corner, exactly as a regular one does, and every edge between two of them carries a ratio of one.

Which makes a prediction worth running #

That converts the finding from an observation about five tilings into a claim with a test attached, and the test is cheap.

Hand the construction a tiling by identical non-square rectangles, edge to edge — a grid stretched in one direction. It is a tiling by irregular polygons, so if regularity is what makes the propagation trivial, the ratios should leave one. If equiangularity is what does it, every ratio should be one to the last bit, exactly as on the square grid.

The argument above says the second, and it says so before the measurement, which is the only useful order for a prediction to arrive in. It also says which stretched tilings would not behave: stretch a triangular grid and the triangles stop being equiangular immediately, so that one should break where the rectangles do not.

The distinction matters for more than tidiness. The seeded deformation used to break the tilings above moves every vertex, which destroys equiangularity and regularity together — so it cannot separate the two explanations, and the essay’s own experiment is silent on which of them was doing the work. A stretched rectangular grid separates them in one run, because it destroys exactly one.

If the prediction fails, the account here is wrong and the tighter condition really is regularity. If it holds, the family of tilings on which the pleat equations do nothing is considerably larger than the Archimedean repertoire the literature works in — and the rhombille is not merely the odd one out among five, but the odd one out among a family nobody has enumerated.

Where the model stops #

The deformation is a blunt instrument. It breaks regularity, and regularity was buying several things at once, so a deformed tiling is not a clean experiment isolating the loop condition. A cleaner one would be a tiling by irregular polygons whose loop products are deliberately consistent, and whether such tilings exist beyond the rhombille is a question this essay does not answer.

Nothing here says anything about tilings by regular polygons that are not edge-to-edge — where a long edge meets two short ones — and the argument above assumes edge-to-edge throughout, because “the two tiles that meet along an edge” is not well defined otherwise.

And the whole account is about the distances, which is one of the twist construction’s two free numbers. The other is the turn, and it is common to every vertex by construction; whether a tessellation could be built with different turns at different vertices is a larger question with a different set of equations.

What the picture cannot show #

The ratio figures write a number on every edge, which is the only way to show that a quantity is one everywhere without drawing nothing at all. A picture of a tiling whose equations are trivially satisfied looks exactly like a picture of a tiling whose equations are hard-won, and that is the whole difficulty this essay is about: the propagation leaves no trace when it succeeds easily.

Nor can a figure show a loop failing. The disagreement is between two paths, and a path is not a thing on the page; what is drawn instead is the pattern that results, which fails for two reasons at once.

The generalisation #

There is a habit of mind this is a case of, and it is worth naming because any body of checking machinery is full of opportunities for it.

A procedure that maintains an invariant is doing one of two things: enforcing it against pressure, or reporting it in a situation where it could not have failed. From inside, the two are indistinguishable — the invariant holds, the check passes, the log is clean. The only way to find out which is to construct a case where the invariant should fail and see whether the machinery notices.

That is what a refusal is for, and it is why every module on this site is required to produce one. The refusals that matter are not the ones that catch a typo; they are the ones that establish that a check has ever said no. Until the measurement above, the twist construction’s propagation had never said anything but yes, and had never been given an input that could make it say otherwise.

The generalisation to the subject rather than the code: a constraint that is satisfied by the symmetry of every example anybody uses is not a constraint anybody has tested. Origami’s canonical objects are extremely symmetric — squares, grids, regular polygons, halved angles — and symmetry makes conditions hold for reasons other than the ones they were written for. The big-little-big lemma is silent at a tie for the same kind of reason, and the local conditions are exact at a symmetric vertex for a third.

Who found it, and when #

Twist tessellations are Ron Resch’s, from the 1960s, and Shuzo Fujimoto’s independently in the 1970s; the systematic derivation from a tiling rather than from a drawing is much more recent and belongs to the computational tessellation literature. That literature works almost entirely on the Archimedean tilings, and this essay is an explanation of why that has been so comfortable: on those tilings the hardest-looking part of the construction is free.

The rhombille result is this collection’s own. What is added here is why it was the only one — and the answer is not that the rhombille is exotic but that it is the only tiling in the standard repertoire whose tiles have more than one angle.

One practical note follows from all of this, and it is about how a tessellation should be specified. The construction takes a tiling, a turn and a fill fraction, and returns a pattern; the distances are computed. On the tilings anybody uses, the distances are all equal, so a designer who simply sets every twist to the same size gets the right answer and never learns that the equations exist. That shortcut is correct on four of the five tilings measured here and wrong on the fifth, where it produces a pattern whose pleats do not match and whose vertices fail — which is what the construction’s unmatched setting exists to demonstrate.

Where the ladder goes next #

The obvious continuation is a search rather than an essay: over tilings by irregular polygons, which ones have consistent loop products? The rhombille shows the set is not empty and the deformations show it is not everything, and the boundary between them is a condition on a tiling that nobody has written down.

The other direction is the second free number. The distances are fixed up to a scale by the propagation; the turn is fenced at both ends; and what a folder can choose after both of those have had their say turns out to be nothing at all about the letters.