The dial and the tiling that is not alike

A twist tessellation is a tiling with every polygon rotated in place and pleats run between them, and any tiling makes one provided the arithmetic at each vertex comes out. This collection builds them on five: the square, an elongated variant, a hexagonal one, a triangular one, and the rhombille.

For four of the five, the construction is exactly what a description of it would suggest. Every vertex of the tiling looks like every other, so every twist polygon can be given the same size, and the pattern is a single unit stamped out over the plane.

The fifth is not like that, and the difference reaches further than anything about its appearance.

The condition that has nothing to propagate #

A pleat between two twist polygons has to arrive at both ends correctly, and that puts a relation on the two polygons’ sizes. On a tiling whose vertices are all alike, the relation is satisfied by giving every polygon the same size, and a construction could simply assume it — the assumption would be invisible because it is true.

On the rhombille there are two kinds of vertex, three-fold and six-fold, and no single size satisfies the relation everywhere. Each polygon’s size has to be solved for, by propagating the relation from a starting vertex out across the tiling until every polygon has a size consistent with all its neighbours. The tiling the unit could not promise is where that was established, and its finding was that a construction which assumes uniformity produces, on this tiling, a drawing rather than a pattern.

So four tilings have a construction with a trivial step in it and one has a construction with a real one. That is a single line of difference, and everything that follows is downstream of it.

It is worth being clear that the difference is not one of complication. The rhombille’s tiling is not harder to draw, its polygons are not more numerous, and its patches are not larger — at the collection’s standard setting the rhombille patch has a hundred and fifty-seven panels against the square’s forty-nine, but that is a matter of pitch rather than of kind, and a square patch drawn at a finer pitch would have as many.

What differs is that a quantity which is constant on four tilings is a field on the fifth: a number that varies from place to place and whose values at neighbouring places are related. Constants and fields behave differently under almost every operation, and the rest of this essay is four instances of that one sentence.

The propagation is over-determined, and that is the theorem #

Calling the sizes a field understates what the walk has to do, and the arithmetic of the rhombille says by how much.

Take a patch of $F$ rhombi. Each rhombus has two 60° corners and two 120° corners; six of the first meet at a six-fold vertex and three of the second at a three-fold one, so there are $F/3$ six-fold vertices and $2F/3$ three-fold ones — $F$ vertices in all, and therefore $F$ twist polygons whose sizes have to be found. Each rhombus has four edges shared in pairs, so there are $2F$ edges, and every edge carries one relation between the two polygon sizes at its ends.

So the walk solves $2F$ equations in $F$ unknowns, one of which is an overall scale. The system is over-determined roughly two to one, and there is no reason on the face of it that it should have a solution at all.

That it does is the fact the whole construction rests on, and it is a fact the four uniform tilings cannot test. On a vertex-transitive tiling the constant solution satisfies every relation by symmetry, so the over-determination is real and vacuous — the equations are redundant because the tiling maps each onto the others. The rhombille has two kinds of vertex and no such map, so its $2F$ relations are genuinely $2F$ separate demands, and their agreement is a theorem rather than a symmetry.

Which is why a walk can start anywhere #

The starting-vertex result reads differently once that is in view.

If the system were merely determined, a walk from a three-fold vertex and a walk from a six-fold one would be two ways of picking a solution from a family, and their agreeing up to scale would be unremarkable. It is over-determined, so a solution is unique up to scale when it exists — and two walks agreeing up to scale is not a convenience of the implementation but the only thing that could have happened.

The earlier version that produced overlapping polygons from one starting vertex was therefore not solving the system; it was propagating a relation that was not the right relation, and the disagreement between starting points was the over-determination reporting the error. A system with slack would have absorbed the mistake and drawn something plausible from either end.

The only tiling whose search has a tail #

Sweeping a grid of turn angle and pleat width over the four uniform tilings gives ninety-six patches, and on every one of them the search for a consistent lettering is uneventful. A randomised search costs twenty-five to fifty-six steps; a deterministic one costs twenty-one to forty-seven. There is nothing between them worth reporting.

On the rhombille, twenty of twenty-four patches have a heavy-tailed cost: the same search on the same pattern taking eighty-six steps or fifteen thousand depending only on a seed, with a quarter of runs not finishing at all. That tail belongs to the search’s own coin rather than to the pattern — but the coin is present on all five tilings, and only one of them notices.

The mechanism is the propagation again. Writing a letter on one crease forces letters on others, those force more, and how far the forcing runs decides how expensive a wrong guess is. On a uniform tiling the forcing is short and local, so a wrong letter is contradicted nearby and costs a few steps. On the rhombille, where each polygon’s size depends on its neighbours’, the forcing runs further, and a wrong letter can commit the search to a large region before anything objects.

The only tiling that is expensive to draw #

The same shape appears one level down, in the construction itself, and it appears as time.

A crease pattern here never has its letters stated. They are found, by propagating the vertex conditions and branching where they stop deciding, because an assignment somebody wrote down is an assertion of the thing being established and the collection’s habit is to compute rather than assert.

On the four uniform tilings that search is instantaneous at every setting of every dial. On the rhombille at a shallow turn and a narrow pleat it takes six and a half minutes for a pattern of two hundred and seventy creases — and the pattern it eventually returns is perfectly ordinary, satisfying every condition at every vertex.

That is the same phenomenon as the tail, in a search that has no coin in it at all: this one tries one letter first, always. Its cost is entirely in its choice of which crease to decide next, and at the shallow end of the rhombille that choice leaves a long stack of independent decisions above whatever eventually objects.

There is a third appearance of it that is less dramatic and more useful, because it happens at the collection’s ordinary settings rather than at an extreme.

On the four uniform tilings, drawing at random letterings and asking how many agree with themselves gives shares that fall smoothly with patch size: twenty-six of two hundred on the smallest, five on the next, two on the third. On the rhombille the share is nought, at two hundred draws and at two thousand — and that nought turned out to be a sampling accident on one patch and a genuine emptiness on another, which is precisely the ambiguity a field-valued quantity introduces into a sampler.

The pattern across all three appearances is consistent: whatever the rhombille does, it does more strongly and less predictably, and the reason is always that its constraints reach further.

What the dials actually change #

With the tilings separated, the dials themselves become readable, and most of them do very little.

Pleat width changes how much of the sheet is polygon and how much is pleat. It moves the shrinkage, it moves the crease density, and on every uniform tiling it does not change any verdict at all — except at the narrowest setting, where it participates in the region with no letterings.

Turn angle changes the picture most and the mathematics least, until it goes shallow. Between 0.35 and 1.3 radians every patch on every tiling behaves the same way, and the searches cost within a factor of two of one another throughout. Below 0.3 the sectors at a vertex begin to reorder, and a region opens where no consistent lettering exists.

Pitch changes how many polygons fit on the sheet, and it does so in integer jumps rather than smoothly, which is how one printed patch came to sit on a knife edge.

So of three dials, two do almost nothing over most of their range and become interesting only at one end, and the third changes a count rather than a property. That is a fair summary of the whole family, and it is only visible once the fifth tiling is out of the average.

The one dial that behaves the same everywhere #

Against all that, it is worth naming the parameter that genuinely does not distinguish the tilings, because a family in which everything differed would be five families rather than one.

The turn direction — whether the polygons rotate one way or the other — changes nothing anywhere. Every count, every verdict, every search cost is identical on a patch and its reflection, on all five tilings, at every setting. That is not surprising once stated, since reflecting a crease pattern reflects everything about it, and it is worth recording precisely because it is the kind of symmetry that gets assumed rather than checked.

The same is true of which vertex the size propagation starts from on the rhombille, up to an overall scale. Starting the walk at a three-fold vertex and starting it at a six-fold one produce the same pattern with all its distances multiplied by a constant, which the construction then normalises away by asking how much of the available room the pleats should take. That was not always true — an earlier version of the walk produced a pattern with paper in it from one starting vertex and a pattern whose polygons overlapped from another — and the normalisation is what makes the starting choice irrelevant rather than a hidden parameter.

Why averaging the five was the mistake #

A number quoted across all five tilings is a number in which one member is unlike the others in a way that correlates with what is being measured. Every such number is a mixture, and the mixture’s proportions are an accident of how many tilings the construction happens to support.

That is the population question, in its most concrete available form. Four populations that could not be told apart was the same problem with the members too similar; this is the same problem with one member too different. Both are failures of a set assembled because it was to hand.

The repair is not to drop the rhombille. It is to report it separately, with the structural reason for the separation stated — two kinds of vertex, so a propagation rather than an assumption — so that a reader can see which side of the line any claim falls on.

What a folder would say #

Very little of the above is visible in the hand, and one part of it is.

The four uniform tessellations fold the way a tessellation is supposed to fold: crease the polygons, crease the pleats, collapse, and the sheet gathers. The rhombille folds the same way and is fussier in a specific respect — its polygons are of two sizes, so the pleats between them are of two widths, and a folder who has internalised the rhythm of a uniform tessellation will get the second width wrong the first time.

That is the hand version of a constant became a field, and it is worth noticing that a folder discovers it immediately while a construction that assumes uniformity does not discover it at all. The paper refuses; a drawing does not. The pattern is the object here in the most literal way — the sheet is the instrument that catches the assumption.

Which theorem was checked, and how #

Every patch in every sweep is checked by three instruments that share no code. Kawasaki, Maekawa and the big-little-big lemma are read off the drawn sectors at every interior vertex. The panels’ closure is computed by composing reflections round every loop of the panel graph, which knows nothing about letters and returns a distance rather than a verdict. And a lettering is searched for and then written back onto the pattern to be checked by the first two.

The rhombille’s separation is not asserted from its appearance. It is checked at the point where it matters: on a rhombille patch the propagated side distances genuinely differ from vertex to vertex, and a construction that assumes them equal produces a pattern whose panels fail to close — which is measured rather than described, and is the same check that established the distinction in the first place.

What the picture cannot show #

The propagation. Every figure here shows a finished pattern, and the thing that separates the rhombille from the others is a computation that happens before the drawing exists — a walk over the tiling assigning sizes, whose result is a set of numbers that the picture then embodies without displaying.

Nor does five tilings make a law. The claim that a tiling with two kinds of vertex will behave this way is supported by one tiling with two kinds of vertex, and the honest form of it names the mechanism rather than the count: where a construction has to solve rather than assume, its patterns’ letters are more entangled, and entanglement is what makes a search’s early decisions expensive. Testing that properly needs a tiling with three kinds of vertex, and the construction does not currently support one.

What it costs to keep the fifth tiling #

There is a case for dropping the rhombille from the family, and it is worth answering rather than ignoring, because everything above reads as a list of ways it is inconvenient.

The case is real. It is the tiling that costs minutes to draw at the shallow end, the tiling whose search has to be treated separately, the tiling that cannot be averaged with the others, and the tiling that made a heavy-tailed run-time distribution look like a property of crease patterns for a long time. A family of four uniform tilings would be simpler in every direction.

It would also be a family with nothing in it. The three quantities the construction computes — the polygon sizes, the pleat widths, the propagation that reconciles them — are all trivial on a uniform tiling, and a construction whose interesting step never fires is a construction nobody would have written. The rhombille is the reason there is a propagation at all, and it is the only member that can test whether the propagation is right.

So the fifth tiling is not an inconvenient member of a clean family. It is the member that gives the family content, and the other four are the controls.

The reading that generalises #

Strip the tessellations out and what is left is a statement about constructions rather than about paper, and it is short.

A construction has a step that is either an assumption or a computation. Where it is an assumption, the construction’s outputs are rigid: they vary smoothly with the parameters, their properties are uniform across the object, and everything downstream is cheap. Where it is a computation, the outputs carry a solved field, the field’s values at nearby places are coupled, and everything downstream — searching, sampling, drawing — inherits the coupling.

Four tilings sit on one side of that and one sits on the other, and every difference measured in this essay is the same difference read through a different instrument. That is why the list is a list rather than a collection of coincidences, and it is why the honest description of the rhombille is not the awkward one but the one where the construction does something.

Where the ladder goes next #

The uniform tilings, having turned out to be uninteresting under every dial, are interesting for exactly that reason: a corrugation never backtracks puts a number on how uninteresting, and finds it to be one step per panel exactly, at every size the construction reaches.