Where two twists share a pleat

The construction at the rung below takes a tiling and hands back a twist tessellation. Run it on the square grid, the triangular grid or the honeycomb and it returns the three patterns everybody already folds, with the twist polygon’s shape forced by the tiling’s own angles and only its size and its turn left free.

Those three tilings have something in common that is easy not to notice: every vertex of each of them looks like every other. Four squares here, four squares there; six triangles everywhere; three hexagons everywhere. So every twist polygon in the finished pattern is the same shape and the same size, the pleats are all alike, and the size of a twist really is free.

Hand the construction a tiling where that is not true and a condition appears.

Two kinds of vertex #

The rhombille tiling is what a heap of cubes drawn in isometric projection looks like: rhombi in three orientations, six of them round some vertices and three round others. It is the dual of the trihexagonal tiling and it is as ordinary as a tiling gets. What matters here is only that it has two sorts of vertex — degree six and degree three — and that no symmetry of the tiling carries one sort to the other.

The construction does not mind. At a degree-six vertex it puts a hexagon, because six edges need six sides; at a degree-three vertex a triangle. The angles are forced as before: the tile angle at a degree-six vertex is 60°, so the hexagon’s corner is 120°; the tile angle at a degree-three vertex is 120°, so the triangle’s corner is 60°. Both are supplements, both arrive without being asked for, and Kawasaki is satisfied at a corner as an identity exactly as it was before.

The sizes are another matter.

The pleat is between two unlike things #

Along a tiling edge the two polygons face one another and are joined corner to corner. That quadrilateral is the pleat, and the reason the whole construction folds is that the pleat’s two long sides — the facing side of one polygon and the facing side of the other — are the same length.

On a vertex-transitive tiling that is automatic. Here it is an equation. The side facing an edge has length L = d · (tan(T/2) + tan(T′/2)), where d is how far the sides sit from the vertex and T, T′ are the tiling’s angles either side of that edge. For the hexagon every angle is 60°, so its side is 2d·tan 30°. For the triangle every angle is 120°, so its side is 2d·tan 60°. Requiring them equal gives d(hexagon) ÷ d(triangle) = tan 60° ÷ tan 30° = 3, exactly. The hexagon’s sides must sit three times further out than the triangle’s, and the ratio is not a preference or an aesthetic choice; it is the only value at which the pattern folds.

That refusal is the most useful thing in this essay. A construction that silently produced a wrong pattern for a tiling it had not been tested on would be exactly the failure this site keeps guarding against, so the wrong construction is built deliberately and measured: give every twist the same size and all ninety vertices fail, with a worst alternating-sum residual of 103.18°. Nothing subtle happens. The pattern is not nearly foldable; it is not a crease pattern at all.

The ratio, reached twice by different routes #

The number three is arrived at here by a walk across the tiling. Starting from one vertex with an arbitrary distance, every edge propagates the matching condition to its neighbour: multiply by the ratio of the two side factors, move on. It never forms a tangent of 60° or of 30°; it forms a ratio of sums of tangents at each edge and multiplies them along.

The closed form is the other route, and the two agreeing to 10⁻¹⁵ is the check. It matters because the walk can also disagree with itself: go round a loop of the tiling and the ratios multiply, and whether that product comes back to one is a property of the tiling. On the rhombille it closes to 3.8 × 10⁻¹⁵.

A tiling on which it does not close has no twist tessellation of this kind. That is a stronger statement than anything in the rung below, and it is the reason the propagation is a piece of machinery rather than an arithmetic step: the interesting output is not the distances but whether they exist.

The three has a formula behind it #

The ratio is derived above from two particular tangents, and the same two lines give a rule covering every tiling of a whole class — which is worth having, because it turns the propagation from a walk into a lookup.

Take a tiling in which the tile angles round any one vertex are all equal. A vertex of degree $n$ then has $n$ angles of $360°/n$ apiece, so both angles flanking any edge at that vertex are $360°/n$, and the facing side’s length is

$$L = 2\,d\,\tan\!\left(\frac{180°}{n}\right).$$

Two polygons facing each other across an edge must give the same $L$, so the distances at a vertex of degree $n$ and one of degree $m$ satisfy $d_n \tan(180°/n) = d_m \tan(180°/m)$. Which means the whole assignment can be written down at once:

$$d_v \;\propto\; \cot\!\left(\frac{180°}{\deg v}\right).$$

Put the rhombille’s degrees in and the cotangents are $\cot 30° = 1.732$ at the hexagon and $\cot 60° = 0.577$ at the triangle. Their ratio is three, exactly, which is the number the walk arrives at.

Which settles the loop condition for the whole class #

That is the more useful half. The walk’s interesting output is not the distances but whether they exist — whether multiplying the edge ratios round a loop of the tiling comes back to one.

A formula assigning a distance to each vertex from that vertex alone satisfies every edge equation simultaneously, and a solution that exists cannot fail to close. So for every tiling whose angles are equal round each vertex, the loop condition holds automatically and no propagation is needed: read off each vertex’s degree, take the cotangent, and the pattern folds.

That covers the rhombille and it covers the three vertex-transitive tilings the rung below uses, where all the degrees are equal and the cotangent is a constant — which is exactly why the size looked free there. The size is free on those tilings, because a constant times any constant is still a solution.

Where the condition can genuinely fail is a tiling whose angles are unequal round a vertex. Then the two angles flanking one edge differ from the two flanking another at the same vertex, the side length is no longer a function of the degree alone, and there is no per-vertex formula to write down. The propagation has something to do, and whether it closes is a real question with no general answer.

So the family splits cleanly. Equal angles at each vertex means a closed form and guaranteed closure; unequal angles means a walk that may or may not come back to one. The rhombille sits on the easy side of that line despite having two kinds of vertex — which is why it is the one tiling outside the standard three where the construction works and the sizes are still forced.

The pattern this turns out to be #

Look at what the construction has produced: a triangular arrangement of hexagonal twists, with a small triangular twist in each of the gaps between them, and pleats running from every hexagon to the three triangles around it.

That is the structure of the tessellation Ron Resch worked out in the 1960s and patented in 1968 — a triangulated sheet that collapses with hexagonal twists at the grid points and triangles between them, and one of the very few origami patterns that has ever been the subject of a patent rather than a diagram. Resch was an artist and a computer graphics researcher, and he arrived at it by making things: paper, then aluminium, then the folded structures he spent the rest of his career on.

The site’s rule about other people’s designs applies and is worth restating. What is printed here is this repository’s own output — the construction was handed a tiling and returned a pattern, and no diagram of anybody’s was consulted. The point of saying whose it is anyway is not caution but accuracy: a construction that lands on a known pattern is evidence about the construction, and the interesting fact is that a rule about angles and a person making things out of aluminium converged on the same object from opposite ends. That is a thing this site has already found happening four separate times at a single vertex.

The corner does not care which polygon it is on #

It is worth looking at a corner of the small twist and a corner of the large one side by side, because the thing that makes the construction work is that they are the same kind of object.

At a hexagon’s corner the sectors are 120° for the polygon and 60° for the tile face; at a triangle’s they are 60° and 120°. The pair has been exchanged and the sum has not moved, which is the whole reason a pattern with two sizes of twist can satisfy the same condition everywhere. Kawasaki does not know how big anything is. It knows the angles, the angles came from the tiling, and the tiling supplied both kinds at once.

What the sizes do is decide whether the two facing sides of a pleat meet, and that is a length condition rather than an angle one — which is exactly why it is invisible to every check a vertex can make on its own. A pattern with unmatched sizes has perfectly reasonable-looking polygons, perfectly reasonable pleats, and vertices whose angles are wrong by a hundred degrees, and the wrongness is only visible after the pleats have been drawn.

What the two sizes cost #

The pattern’s arithmetic is not the same as a single-twist tessellation’s, and the difference is worth a number.

At a turn of 24° and pleats using three-fifths of the room available, the rhombille pattern’s paper closes to 1.58 times smaller than the sheet, against 1.65 for the triangular grid’s and 2.20 for the square grid’s at the same settings. The rhombille is the least economical of the three, and the reason is visible in the picture: it has more pleats per unit area, because it has more vertices per unit area, because every triangle of the underlying grid has acquired a twist of its own.

That is the trade the second family of twists buys. What it buys for is not in this measurement — a sheet with twists at two scales behaves differently under load and folds into a very different three-dimensional object — and neither of those is a geometric claim, so neither is made here.

The other half of the price is creases, and it is the half a folder feels. Twenty-three twists on a patch of this size means twenty-three rings and ninety crossings, against seven rings and forty-two crossings for the triangular grid’s pattern at the same period. Whatever the second family of twists is worth, it is bought at roughly three times the folding.

Where the model stops #

This is a flat-folding claim and the Resch pattern is famous for not being flat. What Resch’s structure is admired for is its partly-folded state: a rigid, deeply textured surface with the triangles standing out of the plane, which is what makes it a structural object rather than a picture. Everything measured here is about the crease pattern and the flat state it admits, and the pattern is the verified artefact while a folded state is a second view. Whether this pattern has a rigid folding — a continuous motion with flat panels and hinges — is a different question with a different test, and nothing above answers it.

Every condition checked is local. Ninety vertices passing does not decide that the sheet folds; deciding that in general is NP-hard, and two neighbouring twists still have to agree about layer order in the pleat between them.

The assignment solver does not check Kawasaki, and this pattern is where that shows. Handed the deliberately unmatched construction, the solver still returns a labelling — Maekawa and the big-little-big lemma are conditions on letters given the angles, and they are satisfiable at vertices whose angles are hopeless. A pattern with a valid-looking assignment and no valid geometry is exactly the object a reader would trust and should not, and it is the reason the two checks are reported separately here rather than as one verdict.

Nothing here is a claim about plane groups. The rhombille has a symmetry group, the pattern built on it has another, and which repeating patterns of the plane are possible at all is crystal-symmetry.com’s subject. What is used above is one number per vertex and one angle per tile corner; no group is formed, named or enumerated anywhere in the construction.

The tiling is one of many. Two kinds of vertex is the smallest interesting case, not the general one. A tiling with three kinds gives three equations, a tiling with unequal edge lengths gives more, and whether the propagation closes is decided case by case. Nothing here says which tilings admit twist tessellations, and that is a genuinely open question this file could be pointed at.

What a diagram could not have said #

There is a reason this ratio was never a famous fact. Anybody folding the Resch pattern works from a crease pattern, and a crease pattern has the ratio in it — the hexagons are drawn three times the size of the triangles because that is how the diagram was drawn, and a folder copying it correctly never encounters the question. The sizes are not two numbers a person chooses; they are two lengths on a page.

The construction has to choose, which is what makes the condition visible. It is handed a tiling with no sizes at all and it must produce some, and the moment there are two kinds of vertex the “some” is not free. This is the ordinary reason a derivation is worth having when a drawing already exists, and this site has hit it repeatedly: the axioms of folding were a list of moves anybody could do long before anybody asked how many there were, and the answer to how many was not available from doing them. The same gap opened over which polygons admit a twist at all, and over the assignment of the preliminary base, which nobody had ever needed to derive because everybody had been shown it.

The same argument runs the other way as a warning. A construction that has only ever been run on tilings whose vertices are alike has never been asked the question this tiling asks, so it can carry a wrong rule indefinitely and produce correct pictures the whole time. Rotate-and-shrink is that wrong rule, it is what almost every written account of these patterns says, and on three tilings out of three it is right.

Where the ladder goes next #

The obvious continuation is the tiling that has more than two kinds of vertex, or none of its vertices alike at all, and the question of whether the matching condition closes on it. That is a question about tilings rather than about paper, and answering it would say which tilings this construction can be handed — a considerably more interesting statement than any single pattern.

Nearer to hand is the parameter that has been treated as free throughout both rungs. The turn has been dialled from 11° to 45° without comment, and it is fenced at both ends: past one angle there is no paper left between the twists, and below another the pattern loses its mountain-valley assignment entirely while every angle condition goes on holding. One of those fences is geometry and the other is not, and the difference between them is worth an essay.