Closing the loops is not folding

The propagation that never had to work looked inside the twist construction and found it idle. Every edge of a tiling carries one equation relating the sizes of the twists at its two ends; on every tiling anybody draws twists on, both ends of every edge read the same two tile angles, the equation is exactly one, and the propagation that sets the twist sizes does nothing. The rhombille was the exception, its tiles not regular, its ratios three and a third — and still multiplying to one round every loop.

It ended with a search it did not run. Over tilings by irregular polygons, which have consistent loop products? The rhombille shows the set is not empty and a random deformation shows it is not everything, and the boundary between them is a condition on a tiling nobody had written down.

A family of irregular tilings that is easy to generate and hard to dismiss is the images of the familiar ones under a linear map. The map keeps every edge straight and every tile a polygon, and turns regular tiles into irregular ones. Asking the question of those images gives a clean answer about the loops — and a second answer, about folding, that the question had assumed would follow.

The equation on an edge, again #

At a vertex of a tiling, the twist polygon has a side facing each edge, and that side’s length is the side distance times a factor built from the two tile angles flanking the edge at that vertex: $\tan(\alpha/2) + \tan(\beta/2)$. A pleat across the edge needs the two facing sides equal, so the edge demands that the twist sizes at its two ends be in the inverse ratio of their two factors.

Round a loop of the tiling the ratios multiply, and a propagation that sets twist sizes vertex by vertex is consistent only if every product is one. That is the whole of what the construction’s propagation can fail at, and it is a different kind of condition from the ones a found pattern satisfies: patterns nobody designed arrive already folded and satisfy the vertex conditions because they folded, while a constructed pattern has to be checked for them after the construction has made its choices.

The condition round one tile #

A loop of the tiling can be broken into loops round single tiles, so it is enough to ask what the product is round one tile, and that product has a compact form. Walk round a tile’s corners in order. At each corner the edge arriving and the edge leaving both belong to the tile, and the tile’s own angle $\theta$ sits between them; the arriving edge is flanked on its other side by a neighbour’s angle $\lambda$, the leaving edge by a neighbour’s angle $\rho$. The ratio the walk picks up at that corner is the leaving factor over the arriving one, and round the whole tile

$$\prod_{\text{corners}} \frac{\tan(\theta/2) + \tan(\rho/2)}{\tan(\lambda/2) + \tan(\theta/2)} = 1$$

is the condition. It is a condition on angles alone, which is why a random deformation of a tiling, moving vertices and so changing every angle, breaks it almost everywhere, and why a map that changes the angles only in a structured way can keep it.

Written that way it also shows the trivial case at once. If every edge is flanked by the same pair of angles at both of its ends, each corner’s leaving factor is the next corner’s arriving factor, the product telescopes, and every tile’s product is one without any angle needing a particular value. The question is which tilings guarantee that pairing, and which keep it when their tiles are distorted.

What a linear map keeps #

The square grid, the triangular grid and the honeycomb close every loop under every map tried, and they close it trivially: every edge’s equation is exactly one at both ends, just as it was before the map. A sheared honeycomb has hexagons with no two angles alike at a vertex, and its equations are still all one.

The reason is a symmetry, and it is one a linear map cannot destroy. Each of those three tilings has a half-turn about the midpoint of every edge — a rotation by a half turn that carries the tiling onto itself and swaps the edge’s two ends. It swaps the two tiles on either side of the edge as well, so the pair of tile angles flanking the edge at one end is exactly the pair flanking it at the other, taken in the other order. The two factors are the same sum, and the equation is one.

A linear map sends a half-turn about a point to a half-turn about the image of that point, because a half-turn is the map $x \mapsto 2c - x$ and a linear map commutes with negation. So every linear image of a tiling with half-turns about its edges has them too, and every such image closes its loops for the same trivial reason, however irregular its tiles have become.

What a linear map breaks #

The rhombille closes as drawn and fails under every map. Its edges join a vertex where three rhombi meet to one where six do, so no symmetry swaps an edge’s two ends, and its equations are not one — they are three and a third. They multiply to one round every loop because of the exact angles of its regular rhombi, and a shear of 0.3 is enough to change them: the loop that multiplied to one now multiplies to 2.76.

The elongated triangular tiling sits between. As drawn its equations are all one, but not because of half-turns — a horizontal edge between a row of squares and a row of triangles has none — but because of a reflection across each edge’s perpendicular bisector, which also swaps the ends and keeps the flanking angles. A stretch along the strips keeps that reflection and the loops still close trivially; a shear breaks it, and the loops fail by more than three.

So the condition the earlier essay asked for has a first answer. An edge’s equation is exactly one whenever some symmetry of the tiling swaps its ends and keeps its two tiles’ angles, and whether a family of irregular tilings keeps its loops is decided by whether the family keeps those symmetries. Half-turns survive every linear map; reflections survive only stretches along their own axis; the rhombille’s exact balance survives nothing.

And then the construction #

Closing the loops was supposed to be what a tiling needed to carry a twist tessellation. The sheared lattices close theirs exactly, so the construction should run on them. It runs, and it does not fold.

On all five tilings as drawn, the construction folds at every turn tried. On the twelve images — including the six whose loops close exactly — it folds at none. The worst vertex on a sheared square grid misses the angle condition by at least 0.08 radians at the best turn tried, on a stretched one by 0.31, on a sheared honeycomb by 0.20; searched more finely over turns from 0.05 to 1.2 radians and over how much of the room the pleats fill, the residual approaches zero only where the turn approaches the ends of its range and the pattern degenerates.

The simplest case is the most telling. A square grid stretched by half along one axis is a grid of rectangles, every angle a right angle, every edge’s equation exactly one, every twist the same size. The construction draws square twists in it and joins them with pleats, and the pleats across the long edges and the short edges meet each twist’s corner at different angles. The corner then has four creases whose alternate angles do not sum to a half turn, and Kawasaki’s condition fails at every corner of every twist.

What the loops were #

The loops were never the whole condition. The propagation fixes the twist sizes so that every pleat’s two facing sides are equal; it says nothing about the angle at which a pleat meets a twist’s corner, and that angle depends on the edge’s length and direction as well as on the angles of the tiles. On a tiling whose edges are all alike, all the corners see the same pleats and the angle condition holds whenever it holds at one corner. On a tiling whose edges differ, the same construction produces corners that see different pleats on either side.

Where two twists share a pleat found the first case of a pleat fixing something the tiling’s angles did not: two twist sizes in the ratio three to one on a tiling with two kinds of vertex. The sheared lattices are the second case, and harder. There the pleat fixes the twists’ sizes consistently, and what it cannot fix is their shape: every twist here is a polygon whose sides are turned by the same angle from the edges they face, and on an irregular tiling that shape is not the one the corners need.

Fenced at both ends found the turn bounded above by paper running out between twists and below by the pattern losing its lettering. On the sheared lattices there is no fence to find: no turn in the open range folds, so the range the construction can use is empty.

Where the true boundary is written down #

The question of which tilings carry a flat twist tessellation has been answered elsewhere, by a different construction. Robert Lang and Alex Bateman showed in 2018 that a tiling carries a flat twist tessellation when it is a spider web — when its edges can carry tensions that balance at every vertex, which is the same as the tiling having a reciprocal figure whose edges are perpendicular to its own. Their construction builds each twist from the reciprocal figure’s cell rather than by turning a fixed polygon, so the twists on an irregular tiling are irregular in exactly the way its corners require.

The spider-web property is kept by linear maps, because balanced tensions stay balanced when every edge vector is transformed by the same map. So the sheared honeycomb, the sheared triangular grid and the grid of rectangles are all spider webs, and by that theorem each has a flat twist tessellation. They do not have one from this construction, because this construction’s polygons are the wrong shape for them.

That is the precise sense in which the loops are necessary and not sufficient. The rotated-polygon construction assumes more than the tiling needs, and on tilings with alike edges the extra assumption is harmless. On a linear image of those tilings the loops still close, the tiling still carries a tessellation, and the construction — not the tiling — is what fails.

What it says about the patterns people fold #

The tilings people draw twist tessellations on are, almost without exception, the three with half-turns about every edge and a handful of uniform tilings with every edge alike. The same vertex, found four times found the same flat-foldable vertices recurring across the subject’s patterns, and the twist tessellations are a clear instance: a designer choosing among tilings has been choosing, without saying so, among tilings on which the rotated-polygon construction cannot go wrong.

That has a cost already measured from another side. Folding it flat is one similarity found that a collapsed twist tessellation on five different tilings is the same scale and turn applied to the plane, and a shrink is two numbers found the twists drawing in equally in both directions. Both are properties of tilings with alike edges and alike vertices. A twist tessellation on a sheared honeycomb, built by the construction that does fold it, would shrink by different amounts in different directions and would be the first twist pattern here whose collapse was not a single similarity — a pattern a designer might want precisely because it is anisotropic.

The irregular twist tessellations are not unfoldable; they are unbuilt, because the construction that everyone uses, including the one here, cannot reach them.

What the census assumes #

The tilings are generated patches, three units across, with complete vertices only; a loop that crosses the patch’s edge is not tested, and the claims about “every loop” are claims about every loop in the patch.

The maps are three, a shear, an axis stretch and one general matrix. The half-turn argument covers every linear map; the reflection argument covers stretches along the reflection’s axis; neither is a statement the three maps prove by themselves, and the table is their check.

Folding means passing every vertex condition on the constructed pattern — developability, Kawasaki, Maekawa and the smallest-sector lemma — at one of four turns, and the residual reported is the worst Kawasaki difference at the best of them. A pattern could pass all four and still fail to fold for a layer-order reason, which is not tested because none passes.

What the tables cannot show #

They do not construct the Lang–Bateman tessellations. That the sheared lattices carry flat twist tessellations is the theorem’s claim applied to them, not a pattern drawn here; building the reciprocal-figure construction and checking its patterns on these images is the measurement that would confirm it.

They do not search tilings that are not linear images. The first figure’s boundary — symmetry about edges — is a boundary within that family, and the rhombille shows a tiling can close its loops without any such symmetry. Which other tilings do, and what they share, is still open.

They test each tiling at one period. The construction scales every tiling to the same period on the sheet, and a sheared tiling’s edges come out at several lengths relative to it; a different period changes how many twists fit but not the angles at a corner, which is where every failure here is found.

And they say nothing about the turn’s fences on the tilings that fold, which fenced at both ends and the dial that decides nothing examine on the regular ones.

Still open: the tiling with three kinds of vertex #

The construction’s failure on sheared lattices narrows the next question rather than closing it. On tilings whose edges are all alike, the loops are the whole condition, because every corner sees the same pleats. The tilings with more than two kinds of vertex — the ones where two twists share a pleat pointed at, with twist sizes forced into ratios — can still have all their edges alike, and on those the rotated-polygon construction’s only obstruction is the loop products.

Whether such a tiling closes its loops is a computation on its angles alone, and it is the right next measurement: it would say whether the ratio three to one that two vertex types forced extends to a chain of ratios over three types, or whether three types cannot be reconciled at all.

Sideways from here, the half-turn argument applies well beyond twists. Any construction that propagates a quantity along edges by a ratio of local factors is trivial on a tiling with a half-turn about every edge, because the factors at the two ends are the same numbers read in the other order. That is worth checking before believing any propagation on such a tiling has done work — the earlier essay found one that had not, and linear images of the same tilings are a family on which it still has not.

The habit worth carrying is about conditions that close. A consistency condition tells a construction where it cannot run; it never tells it where it can. Loops that close are the construction’s permission to try, and whether what it draws folds is a second question the loops have no way to answer.