One number where the corners wanted four

Closing the loops is not folding ended with a construction that had run out of excuses. Twelve linear images of the familiar tilings were put to it; six of them satisfied the consistency condition the construction propagates, exactly and for a reason — a half-turn about every edge survives any linear map — and the construction drew a pattern on every one of the twelve that failed Kawasaki at every vertex tried. The diagnosis there was that the polygons were the wrong shape, and the evidence for it was a theorem from outside: those images are spider webs, so each of them carries a flat twist tessellation, and the construction was not finding it.

That leaves a question with a definite answer. What shape would be right, and what stopped the construction from drawing it?

The answer turns out not to be a better construction. It is that the construction had been solving a smaller problem than the conditions pose, and the smaller problem was never written down anywhere — not as an assumption, not as a simplification, not as a choice. It arrived in the phrase everybody uses to describe these patterns.

The rule that was never written down #

A twist tessellation is described as rotate-and-shrink: take each tile of a tiling, spin it about its centre, let the gaps between the spun tiles become pleats. Any tiling makes a twist replaced that picture with a rule — at every vertex, a polygon whose sides face the incident edges, each side’s outward normal turned from its edge by a common angle — and showed that the polygon’s angles are then forced. A corner between the sides facing two consecutive edges has interior angle $\pi - T$, where $T$ is the tiling’s own sector between those two edges, because two lines whose normals differ by $T$ meet at $\pi - T$. That is a fact about half-planes and it holds however the sides are placed.

The essay then said that nothing is left to choose but how large the polygon is and how far it is turned. The first half of that sentence is where the assumption lives.

A polygon of degree $K$ with prescribed side directions is not determined by one number. It is determined by $K$ — one distance from the vertex to each side. Fixing a single distance and using it for every side picks out one polygon from a $K$-parameter family, and on a tiling whose sectors are all equal that polygon is the natural one: equal distances give equal side lengths, the object looks like the regular polygon everybody draws, and there is nothing to notice. On a tiling whose sectors are not equal, equal distances give sides of unequal length in a pattern the tiling’s angles dictate — and there is still nothing to notice, because the construction returns a drawing either way.

The propagation that never had to work found the first consequence of the restriction without recognising it as one. Under the single-distance rule, the matching condition across a pleat — the two facing sides equal in length — becomes a ratio between the two vertices’ single numbers, one equation per edge, propagated across the tiling and required to close round every loop. That propagation exists only because each vertex has one number to propagate. Give a vertex a number per side and the matching condition stops being a relation between vertices at all.

What a corner actually asks #

At a corner of a twist polygon four creases meet: two sides of the polygon, and one crease from each of the two pleats that corner belongs to. The four sectors are the polygon’s interior, one pleat, the tile face, the other pleat. Kawasaki asks the alternating sums to be equal, which for degree four means opposite sectors are supplementary. The polygon’s interior is $\pi - T$ whatever the distances are, so the condition reduces to one statement about the face: the angle between the two pleat creases leaving that corner must equal $T$, the tiling’s own sector there.

Run that round a vertex and it says something compact. The pleat creases leaving the corners of one twist polygon must make, with each other, exactly the angles the tiling’s edges make with each other. A set of directions with the same successive angles as another set is that set rotated, so the condition is that every pleat crease is its own tiling edge turned by one common angle. Call that angle the pleat angle. It is shared by the whole pattern, because the crease along an edge is one segment and both of its vertices read the same rotation from it.

That is the condition in full, and it is worth saying what is not in it. It mentions no lengths. It does not require the polygons to be similar, or congruent, or centred anywhere. It does not care whether the tiling is uniform, or regular, or periodic. And it constrains the side distances only through the pleat creases, which depend on the distances at both ends of an edge and on where along each side the creases start.

The equation, and what is in it #

Put the pleat along an edge of length $L$ between a side at distance $h_v$ from one end and a side at distance $h_u$ from the other, and let $\Sigma$ record where along those two sides the crease begins and ends — a quantity built from the neighbouring distances at each vertex and the sectors between them. The crease then runs along

$$e^{ia}\left[\,L - (h_v + h_u + i\Sigma)\,e^{i\theta}\right]$$

where $a$ is the edge’s own direction and $\theta$ the turn. Asking that this make the angle $\psi$ with the edge means asking that the bracket have argument $\psi$, and an argument condition on a complex number is one real equation once the modulus is cleared:

$$L\sin\psi + (h_v + h_u)\sin(\theta - \psi) + \Sigma\cos(\theta - \psi) = 0$$

Linear in the distances. That is the whole of why the problem was the wrong size. A condition that looks like it needs a search — find the polygon that makes four creases meet correctly — is a linear equation once the polygon is described by the numbers it actually has. A pleat has two creases and they are constrained separately, so an edge contributes two of these; their being parallel is the matching condition the single-distance rule had to propagate, and here it falls out of the same system instead of being imposed ahead of it.

There is one thing the algebra hides and it matters. Clearing the modulus also clears its sign, so the equation is satisfied by a crease running along the required direction and by one running back along it. The second is not a pattern — the corner’s four creases then lie in the wrong cyclic order and Kawasaki fails by something near a straight angle. The linear system is therefore necessary and not sufficient, and every solution it returns has to be drawn and checked rather than trusted.

Four numbers where there was one #

The system can be solved for the tiling rather than for a patch of it. Two vertices related by a translation of the tiling should carry the same distances, so the unknowns are one per class of vertex per incident edge, and a patch supplies representatives rather than degrees of freedom. On the square grid that is four unknowns; the single-distance rule had one. Two of the equations are independent and the rest repeat them.

Two of the free directions are never a new pattern, and they have to be taken out before the arithmetic means anything. Sliding every twist polygon by the same vector satisfies every corner condition exactly — checked here rather than assumed, to within a part in $10^{16}$ — because the tiling is scaffolding and not ink, so what comes out is the previous crease pattern moved. Once those two are removed, three of the five tilings have nothing left to choose: the turn and the pleat angle fix the pattern outright. The rhombille and the elongated triangular tiling have exactly one number left, and spending it trades side lengths between the two kinds of vertex without moving the ratio of their sizes, so the three-to-one that where two twists share a pleat found is not a casualty of the extra freedom.

That is a better answer than a large family would have been. A construction with a wide family is one where something else decides the pattern; a construction with two numbers and no residue is a construction.

The arithmetic also has a check in it that could have failed and did not. Run the general solve on the five tilings as drawn and it returns the same distance at every side of every vertex — to within a part in $10^{14}$ on all five, which is the accuracy of the solve rather than a tolerance anybody chose. The general construction therefore contains the older one instead of replacing it: where the sectors at a vertex are all equal, the extra numbers are spent on nothing and the polygon that comes back is the one everybody draws. The rhombille is the sharper case, because its two kinds of vertex get 0.2609 and 0.0870 — a ratio of three to one, to four figures, arrived at by a solve that was never told the ratio existed. The dial and the tiling that is not alike treated that tiling as the awkward one precisely because it was the only one where a distance had to be solved for rather than assumed; under the general rule every tiling solves for its distances, and the rhombille stops being exceptional.

Twelve images that now fold #

Solved that way at a turn of 0.42 radians and a pleat angle of $-0.5$, and drawn as an ordinary crease pattern, all twenty of the tilings and their images pass every vertex condition — developability, Kawasaki, Maekawa and the smallest-sector lemma — and every one of them admits a mountain-valley assignment. The single-distance construction passes five. The twelve images that would not fold under the older rule are twelve of the fifteen that now fold, and the three others were folding already.

The count deserves a sentence of its own, because it is the kind of number that looks like a summary and is not. Twenty tilings-and-images, four vertex conditions each, every interior vertex of every patch: the five that folded before fold still, and the fifteen images divide into three that were already folding and twelve that were not. Nothing was relaxed to get there. The same four theorems are applied to the same kind of drawing at the same turn, on patches generated the same way, and the only thing that changed is how many numbers the solver was allowed to place the polygons with.

It is worth being exact about what the twelve failures were. They were never a statement about the tilings. Each one was the construction reporting that no member of a one-parameter family met a condition that a four-parameter family meets comfortably, and reporting it in the only way it could — by drawing something and failing it afterwards. The consistency condition on the loops, which the earlier account spent itself on, belongs to the propagation and therefore to the restriction; with a distance per side there is no propagation and no loop to close.

The angle that replaced the old dial #

The family used to be indexed by how much of the available room the pleats were given — a scale on the distances, since the single-distance construction fixes them only up to a common factor. That factor is gone. The distances are now fixed by the edge lengths and by $\psi$, and $\psi$ is the dial.

It behaves like the old one in the way that matters: it is fenced. Run it from near zero and the pattern folds over a range and then stops, and what stops it is the pleat closing to nothing — the two facing sides meeting, which is paper running out rather than a condition failing. On sixteen of the twenty the window runs from about $-0.05$ to $-1.15$ radians. On four it is half as wide, and those four are the stretched and general images of the honeycomb and the rhombille.

The near end is a different kind of edge. As $\psi$ approaches zero the pleat creases approach their own tiling edges, the polygons swell to fill the plane, and the construction runs into the same wall from the other side. Fenced at both ends found the turn bounded above by paper and below by the pattern losing its assignment entirely, and only one of those two fences was geometry. The pleat angle’s two fences have not been separated that way here, and they may not be the same pair.

What the new dial does not do is the thing the old one did. The dial that decides nothing ran the turn from fence to fence and found every measurable property of the pattern moving while the number of ways it could be lettered stayed at sixteen throughout, because the smallest-sector lemma reads which sector is smallest and never how small. The pleat angle moves the same quantities and is not interchangeable with the turn: the turn rotates every side about its own vertex and leaves the distances free, while the pleat angle leaves the sides’ directions alone and fixes the distances outright. Two dials that both shrink the polygons are not the same dial, and a family indexed by both is two-dimensional rather than one.

What the tables rest on #

The tilings are generated patches three units across, with the conditions imposed at vertices whose whole neighbourhood is present. A claim about “every vertex” is a claim about every vertex of the patch, and the solve is by class, so what it actually states is a fact about the tiling that the patch is a witness to.

Folding means passing every vertex condition on the drawn pattern and finding a mountain-valley assignment for it. That is the same standard used throughout and it is not a folded state: a pattern can pass at every vertex and still have no consistent ordering of its layers, which is a global question and a much dearer one.

The turn is one number and the pleat angle is another. The tables fix the turn at 0.42 radians throughout, which is comfortably inside every fence found here, and vary the pleat angle only where the window is being measured. Whether the window’s width depends on the turn is not tested.

The solve is least-norm. Where a family remains, the member reported is the one nearest the origin in the distances, which is a choice made by the solver and not by the subject. On the three tilings with no freedom left that choice is vacuous; on the rhombille and the elongated tiling it is a choice, and a different member is a different pattern.

What the pictures do not settle #

They do not say that the general construction reaches every tiling. It reaches these twenty. Nothing here bounds the solutions away from zero on a tiling not tried, and a side distance that comes out negative puts a side on the wrong side of its own vertex, which is a drawing and not a pattern.

They do not say the new patterns are worth folding. A crease pattern that passes at every vertex can be ugly, can have creases too close together to crease, and can have twists so small that the pleats dominate the sheet. Nothing here reads the drawing as a drawing. A unit that folds is not a tessellation is the standing warning on that point from the other direction, and it applies here unchanged.

They do not identify the patterns with anybody’s. The theorem that says these images carry flat twist tessellations builds them a different way, from the reciprocal figure of a tension field. Whether the patterns here are those patterns is a comparison nobody has made, and it is the sort of comparison that can be made exactly rather than argued about.

They say nothing about layer order. Every pattern here is checked at its vertices and lettered, and the loop that a lettering can force is exactly the failure that the vertex conditions cannot see. Whether these twelve new patterns fold as sheets rather than as vertex sets is untested.

And the equation’s sign is the one place the algebra is weaker than the check. Clearing a modulus loses a sign, so the linear system admits reversed creases; the drawing-and-checking step is what rejects them, and a construction that trusted the algebra would report patterns that fail at every corner.

Still open: what the distances are, rather than what they satisfy #

The conditions have been solved and the solutions have not been read. Each side of each twist polygon now has a length that the equations chose, and those lengths are numbers about the tiling rather than about the drawing — they are what the construction knows and cannot say.

The obvious thing to do with them is to divide each side’s length by the length of the edge it faces. That quotient is dimensionless, it is the same from both ends of an edge because the matching condition says so, and the polygon closing says the quotients weighted onto the edges balance at the vertex. A set of positive numbers on the edges of a graph that balances at every vertex is not a new object; it is exactly what a tiling has when it is the plan of a spider web, and the theorem quoted from outside earlier is a theorem about precisely that. Whether the construction has been writing one all along is a measurement, not an argument, and it would turn a borrowed theorem into a derivation.

There is a second reading of the same quotients, and it is what makes the measurement worth making rather than merely tidy. Which polygons twist found that a twist can be built around any regular polygon and that what stops at three is the tiling rather than the paper — a restriction about the plane. If the quotients turn out to be a balance condition, then the restriction is about something more specific still: not which polygons tile, but which tilings carry a set of positive weights that balance, and those are not the same list. One vertex, repeated is the essay that treats a tiling as a way of making a material; a balance condition would say which materials are available.

Sideways from here, the same shift of unknowns is worth trying wherever a construction propagates one quantity per vertex. The propagation in the propagation that never had to work was idle on every tiling anybody draws and became a real obstruction on the images — and it now turns out to have been an artefact of the parameterisation in both cases. A quantity that has to be propagated is a quantity somebody decided there would be only one of.

The habit worth carrying is about the shape of a failure. When a construction refuses, ask how many numbers it was allowed to try before believing the refusal is about the object. A one-parameter family failing a condition that a four-parameter family satisfies is not evidence about the condition, and the arithmetic that would have said so — count the unknowns, count the independent equations — is cheaper than any of the searches that were run instead.