Any tiling makes a twist

A square twist is a small square of paper that rotates while the sheet closes around it. Repeat it on a grid and the result is one of the best-known patterns in the subject: a field of little squares, each turned a few degrees, with pleats running between them — one vertex, repeated. It is beautiful, it folds, and everybody draws it the same way.

Drawing is the wrong verb. The pattern can be derived — and the derivation takes an arbitrary tiling of the plane as its input, which means the family of twist tessellations is enormous and nobody chose any of it.

The rule, stated once #

Take any tiling. At every vertex, put a polygon with one side facing each edge that meets there, with each side’s outward normal turned from its edge by the same angle. Along every edge of the tiling, the two polygons at its ends now face one another across a gap; join their facing sides corner to corner and that quadrilateral is a pleat. What is left — one region per tile — is uncreased paper.

That is the whole construction. Two numbers are free: how far the sides sit from their vertex, and the turn.

The description usually given is different and it is worth separating them. The usual account is rotate and shrink: take each tile, spin it about its centre, shrink it a little, and let the gaps between rotated tiles become the pleats. That produces the right picture for a square grid and it is a description of the answer rather than a rule, because it hides the fact that does the work.

The polygon’s shape is not a choice #

At a vertex where the tiling’s edges meet, consider two consecutive edges and the tile between them. The polygon has a side facing each of those two edges, and those two sides meet at a corner. Their outward normals differ by whatever angle the tiling has between the edges — call it T — because both normals were turned by the same amount from their own edges, and turning both by the same amount leaves the angle between them alone.

Two lines whose outward normals differ by T meet at an interior angle of 180° − T. So the polygon’s corner is forced.

This is why the polygon here is built as an intersection of half-planes and not as a regular polygon drawn on a circle. The two agree exactly when the tiling has the same angle at every corner of every tile round that vertex, which is true of the familiar tilings and stops being true the moment it is not. Building it the second way would work on the square grid, work on the triangular grid, work on the honeycomb, and then produce nonsense on the first tiling that is not vertex-transitive — and the nonsense would look like a drawing error rather than like a wrong rule.

The angles also have to add up, and they do without being asked. A vertex of degree k has k tile angles summing to a full turn, so the polygon’s k corners sum to 180°k − 360°, which is exactly the angle sum of a k-gon. Every vertex of every tiling admits a twist polygon. There is nothing to check.

Kawasaki, satisfied identically #

Now the pleasant part. At every corner of every polygon, four creases meet: two sides of the polygon, and one crease from each of the two pleats the corner belongs to. Going round, the four sectors are the polygon’s interior angle, a pleat sector, the tile face’s angle, and the other pleat sector.

Kawasaki’s condition asks the alternating sums to be equal — the first and third sectors together against the second and fourth. The first and third are the polygon’s angle and the tile’s angle, and those are supplementary by construction. The second and fourth are the two pleat sectors, and they are supplementary for the same reason. Both sums are a straight angle, at every corner, for every twist angle and every polygon size.

That word identically is doing real work. Nothing here bisects, iterates, or fits. The residual is not small; it is the alternating sum of two pairs of supplementary angles, which is zero the way 1 − 1 is zero. What is measured is the arithmetic that arrives at it: over four tilings, four twist angles and 7,396 polygon corners, the departure from the supplement relation is 3.1 × 10⁻¹⁵ radians; over 630 interior vertices of the finished patterns, read off the crease pattern after the clipping and the vertex merging, the worst alternating-sum residual is 8.9 × 10⁻¹⁵.

The pleat is where the condition lives #

Something has to have a condition in it, and it is not the polygon. It is the pleat.

The face angle in that list of four sectors is the angle of the tile face of the crease pattern, not the angle of the tiling. Kawasaki asks it to come out at T, the tiling’s own angle, and that happens exactly when the two sides facing each other across the pleat are the same length.

The length of the side facing edge i is L = d · (tan(T[i−1]/2) + tan(T[i]/2)), where d is how far the sides sit from the vertex. On a tiling whose vertices are all alike, giving every vertex the same d makes both sides the same length automatically, and the condition has never needed stating — which is why the usual account can omit it and still produce working patterns.

On a tiling with two kinds of vertex it is an equation. It relates the d at one end of an edge to the d at the other, and propagating it across the tiling either closes or does not — a property of the tiling and nothing to do with how the pattern was drawn. The rhombille tiling, six rhombi round a lattice point and three round a triangle’s centre, comes out at exactly 3 to 1 and closes around every loop to 3.8 × 10⁻¹⁵. That case is the pattern this construction turns out to contain, and it deserves its own account.

Three tilings, three tessellations #

The regular tilings give the three twist tessellations everybody already knows, and the construction makes their relationship plain rather than coincidental.

A square grid has four edges at a vertex and gives square twists. A triangular grid has six and gives hexagonal twists. A honeycomb has three and gives triangular twists. So the twist polygon’s shape is the vertex figure of the tiling turned inside out, and the reason there are three familiar twist tessellations is that there are three regular tilings — which is a fact about the plane rather than about paper.

The honeycomb is the interesting one to look at, because its twists are triangles and the tiles between them are hexagons — the reverse of the triangular grid, which has hexagonal twists and triangular tiles. The two patterns are related by exchanging the roles of tile and vertex, which is what the construction does when it is handed a tiling and its dual.

The census of which polygons admit a twist at all asked a related question from the other end and got the same three answers, by a different route: a twist unit exists for every regular polygon, and only three of them have an interior angle dividing a full turn, so only three can meet.

What the assignment does, and why it had to be found #

The mountains and valleys are not part of the construction. The geometry decides the angles; something else has to decide the letters, and at this scale the site’s usual method is unavailable.

Elsewhere on this site an assignment is found by enumeration: every labelling of the free creases is tried and the ones passing every condition are kept. That is exact and it stops at twenty-two creases. A patch of twist tessellation has several hundred.

So the letters are propagated instead. Kawasaki is already settled by the angles, which leaves Maekawa and the big-little-big lemma, and both are conditions at a single vertex — enumerate the labellings of one vertex’s own four creases that pass, and the pattern is the intersection of those lists over every vertex that shares a crease. Propagate to a fixed point; where propagation stops, branch on the vertex with the fewest labellings left.

The rule that suggests itself instead is worth recording because it fails. Make every twist polygon’s ring a single letter, and let the two creases of each pleat take one of each: at every corner three creases are the ring’s letter and one is the other, which is exactly Maekawa’s three-to-one. It passes Kawasaki everywhere. It passes Maekawa everywhere. It fails the big-little-big lemma at 32 of 36 vertices, because the smallest sector at a corner is a pleat sector and the two creases bounding that have been given the same letter. The repair that fixes one corner asks the crease at its far end for the opposite letter, and the two demands are inconsistent. There is no local rule; there is a constraint problem.

Where the model stops #

Nothing here decides that the sheet folds. Every condition checked is local — a statement about the creases at one vertex — and deciding flat-foldability for a pattern as a whole is NP-hard. A twist tessellation that passes at every vertex is a candidate, and a unit that folds is not a tessellation. Two neighbouring twists still have to agree about layer order in the pleat between them, and that is precisely the question nobody can answer in general.

The patch is clipped to a square sheet. A twist polygon is drawn whole or not at all, because a partial one would leave corners inside the paper with three creases at them, and three creases at an interior vertex is the one thing this subject’s patterns never have. Pleat creases aimed at a polygon that did not make the cut are truncated at the edge of the paper, which is where a fold line stops anyway.

The tiling is periodic and it did not have to be. Everything above uses the tiling’s angles at a vertex and the lengths of its edges; nothing uses repetition. An aperiodic tiling would go through the construction unchanged, and whether the side-matching condition closes on one is a question this file can ask and this essay does not answer. Which rotations a repeating pattern of the plane may have, and the enumeration of the groups that follow, are crystal-symmetry.com’s ground and nothing here derives any of it.

What it costs the paper #

The construction takes paper. Each pleat is folded twice, so a pleat of width w along an edge of length L consumes about 2wL of sheet, and the flat pattern’s remaining area is what the folded object is made of.

On a square grid at a turn of 24°, with the pleats using about three-fifths of the room available between neighbouring twists, the paper left is 2.20 times smaller than the sheet. At 11.5° it is 2.69 times smaller and at 45.8° it is 1.46 times, so the turn trades directly against the shrink — and it trades the way round that a reader would not guess. A bigger turn shrinks less.

The mechanism is not that a large turn spends more paper on pleats; it spends less. Turning a polygon’s side away from the edge it faces brings it closer to its neighbour, so the room available between two twists falls as the turn grows, and a pleat filling a fixed share of a smaller gap is a narrower pleat consuming less sheet. The paper the pleats take goes down, the paper left in the twist polygons goes up, and the folded object is a larger fraction of what it came from.

That is worth a second look because the turn is what makes a twist look like a twist, and the more it looks like one the less it packs.

Which makes the dial a two-sided choice #

The corrected direction turns the turn from a free parameter into a decision with a cost on each side, and it is worth setting out because neither side is the one the picture suggests.

A small turn packs best. At eleven and a half degrees the sheet folds to a little over a third of itself, which is the largest compaction the family offers — and the polygons are barely rotated, so the pattern looks like a grid with a hint of a twist rather than like a twist tessellation.

A large turn packs worst. At forty-six degrees the folded object is two thirds of the sheet, which is barely a fold at all by the standards of the corrugations this collection measures — and it is the setting at which the rotation is unmistakable and the pattern is at its most striking.

So the aesthetic and the packing run in opposite directions across the whole band, and there is no interior optimum to find. A designer choosing a turn is choosing where on that line to sit, and the choice is not between a good number and a bad one: it is between a pattern that does what a corrugation is for and a pattern that shows what a twist is.

That also explains something about the published repertoire without any appeal to taste. Twist tessellations are folded to be looked at rather than to pack anything, so the angles that get drawn sit toward the large end — and the family’s compaction, which is the quantity this collection measures corrugations by, is correspondingly poor. A twist at a comfortable viewing angle shrinks by about two; a Yoshimura shrinks by sixty.

The two are not competing at the same job, and the numbers say so cleanly once the direction of the trade is right. A corrugation converts area into depth and is judged by how much. A twist converts a flat sheet into a field of rotations and is judged by how visible they are, and the more of the second it delivers the less of the first it has left.

Who drew these, and when #

Twist tessellations arrived in the folding tradition as objects rather than as a construction. Shuzo Fujimoto worked them out in Japan in the 1970s and published them as patterns to fold; the same shapes were being made independently in Europe and America within a decade, and Eric Gjerde’s collection of them in 2008 is where most folders now meet the family. What was passed around was diagrams — this polygon, this rotation, these pleats.

That is not a criticism of anybody. A crease pattern is a complete instruction and a tessellation is a repeating one, so a diagram of one unit cell is a full description of the object, and there was no practical reason to have a rule for producing them. The rule is worth having anyway, for the reason this site keeps running into: a pattern that is drawn is a fact about one pattern, and a pattern that is derived is a fact about a family. The derivation says which tilings work — all of them — and it says what is forced and what is free, which no amount of looking at a finished square twist would ever reveal.

Where the ladder goes next #

Two directions leave here, and both are measurements rather than pictures.

The first is the tiling with two kinds of vertex, where the side-matching condition stops being free and becomes an equation between two twists of different sizes. That case turns out to be a pattern with a name and a designer, arrived at from a direction that had nothing to do with either, and it is the next rung.

The second is the twist angle itself. It has been treated here as a free dial, and it is not free: past a certain turn the pleats have no paper left, and below a certain turn something stranger happens — the pattern loses its mountain-valley assignment entirely, while every angle condition in the subject goes on holding. The two fences are different kinds of thing, and only one of them has anything to do with the geometry.