Origami & Folding
what one sheet can do
Generator

The the square grid's twist tessellation

A generator in the tessellations library, called 33 times across 12 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

tiling-twist is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

The the square grid's twist tessellationThe crease pattern the offset construction produces, with its assignment found by propagating the two vertex conditions rather than drawn on. Every interior vertex has four creases and passes developability, Kawasaki, Maekawa and the big-little-big lemma.what the construction produced9 twists, 36 interior verticesturned 24.1° from the tiling's edgespleats 0.118 to 0.118 wide2.20× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge
square twist tessellation — sheet 165×165 mm — 41 mountain, 43 valley, 2180.74 mm of crease

kind: "square", theta: 0.42, view: "construction"

The tiling, and the polygon it puts at every vertexThe tiling is drawn as dashed lines and the twist polygons over it. Each polygon has one side per edge of the tiling meeting there, and every side is turned by the same angle from the edge it faces. Nothing about the polygon is chosen except how far its sides sit from the vertex and how far they are turned.the square grid — four squares at every vertex9 twist polygons, turned 24.1° from the edges they facethe dashed lines are the tiling and are not creases

kind: "square", theta: 0.42, view: "corner"

Four sectors, two of them supplementary by constructionOne corner of one twist polygon, with the four angles measured off the finished pattern. The angle inside the polygon and the angle of the tile face opposite it add to a straight angle because the polygon's sides were turned from the tiling's own edges; the two pleat sectors add to a straight angle for the same reason. Kawasaki is then an identity rather than something the construction had to solve for.90.00°40.40°90.00°139.60°at every cornerthe polygon's own angle — 90.00°a pleat sector — 40.40°the tile's angle — 90.00°the other pleat sector — 139.60°alternating sums 180.00° and 180.00°which is Kawasaki, satisfied identicallymountainvalley

kind: "square", theta: 0.75, view: "pattern"

The the square grid's twist tessellationThe crease pattern the offset construction produces, with its assignment found by propagating the two vertex conditions rather than drawn on. Every interior vertex has four creases and passes developability, Kawasaki, Maekawa and the big-little-big lemma.what the construction produced9 twists, 36 interior verticesturned 43.0° from the tiling's edgespleats 0.095 to 0.095 wide1.54× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge
square twist tessellation — sheet 165×165 mm — 41 mountain, 43 valley, 2359.31 mm of crease

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

Any tiling makes a twist

A twist tessellation is usually drawn, admired and copied. It can be derived instead: hand the construction any tiling of the plane and it returns a crease pattern that folds flat, with the twist polygons' shapes forced by the tiling's own angles and nothing left to choose but how large and how turned.

Cutting a patch out of a plane

A tessellation is infinite and a sheet is not, so every picture of one is a decision about where the paper stops. Assembling whole twist units on a square and running the outstanding pleats to the rim puts 12, 18, 12 and 5 creases across other creases on four of five tilings; generating the pattern over a larger region and clipping it puts none. The panels then place exactly — and what is waiting behind the repair is a different refusal that could not be asked about before.

Fenced at both ends

The twist angle of a tessellation looks like a free dial, and it is fenced twice. Turn too far and the pleats have no paper left. Turn too little and something stranger happens: every angle condition in the subject goes on holding and the pattern loses its mountain-valley assignment entirely.

Four ways to draw a pattern

Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.

Letters that agree get rarer

Two hundred letterings drawn independently from a square twist tessellation patch, and twenty-six of them have letters that do not contradict themselves. On the next patch up it is five, then two, then none, then none. What the share falls with is not the size of the patch and not the angle of its twist: it is the number of independent closed chains its panels form, which is Euler's relation on the drawing and is fixed before a single letter is chosen.

The dial and the tiling that is not alike

Four of the five tilings a twist tessellation can be built on behave identically under every dial the construction has. The fifth has two kinds of vertex, and everything about it is different: it is the only one whose search has a tail, the only one whose shallow patches take minutes to draw, and the only one where a distance has to be solved rather than assumed.

The dial that decides nothing

Turn a twist tessellation's angle from one fence to the other and every measurable thing about it changes: the smallest sector goes from 88 degrees to under one, the pleats swallow a quarter of the sheet and then almost none of it, the folded footprint changes by a third. The number of ways it can be creased does not change at all — sixteen, at every angle tested — because the lemma reads which sector is smallest and never how small.

The propagation that never had to work

The twist construction carries one equation per edge of its tiling and propagates the twist sizes outward from a seed. On every tiling anybody has drawn a twist on, every one of those equations is satisfied trivially — both ends of an edge read the same two numbers, because a regular polygon has one interior angle. The construction has been running and doing nothing, and the one tiling where it did something is the one whose tiles are not regular.

The ring is the loop

The square twist's central polygon is four creases enclosing one panel, and a lettering that gives all four the same letter has no folded state. That was established by enumerating the orderings of nine panels. It can now be read off the crease list in one pass, because the eight panels the letters send round in a circle are exactly the ring — the twist's own defining feature, contradicting itself.

The tiling the unit could not promise

Every twist on this site carries the same caveat: the unit is verified and the plane is not, because deciding a whole pattern is intractable. There is one thing about a whole pattern that costs a single pass over its crease list, and it says no. The square twist tiling was drawn with a lettering that contains a loop of twenty-eight panels, so the patch on this site had no flat folded state at all — and only seven of forty independent redraws avoid one.

Where a sector crosses sixty

Turn the twist polygons of a tessellation patch a hundredth of a radian further and the pattern goes from having no mountain-valley labelling at all to having one immediately. Nothing about its graph changes across the transition — the same eighty-three panels, the same hundred and forty-two creases, the same four labellings at every one of its sixty vertices. What changes is which sector at a vertex is the smallest one.

Where two twists share a pleat

Every twist tessellation the tradition draws has one size of twist, because every tiling it is drawn on has one kind of vertex. Hand the construction a tiling with two, and the pleat between a large twist and a small one turns out to fix their sizes exactly — three to one, and nothing else folds.

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