Twists — the series
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A square that turns
A twist is a small polygon that rotates as the sheet closes around it. The geometry is forced rather than designed — Kawasaki fixes one sector, the big-little-big lemma forbids a strictly smallest one, and what is left is the pattern Ron Resch was drawing in the 1960s.
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Which polygons twist
Twist tessellations come in three kinds — triangle, square, hexagon — and it is natural to read that as a fact about twists. It is not. A twist can be built around any regular polygon and every one of them folds; what stops at three is the tiling, and the tiling is a fact about the plane.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Four easy patches and one that is not
Run the same search a hundred and twenty times on each of five tessellation patches, changing nothing but the order the letters are tried in. Four of them answer in between twenty-five and fifty-three steps every single time. The fifth answers in eighty-four steps at best, a hundred and sixty-six in the middle, and does not answer at all in forty-eight runs of the hundred and twenty.
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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.
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The most decided vertex here
Sixteen ways to letter four creases; Maekawa allows eight; the big-little-big lemma allows four. A twist polygon's corner is one of the few vertices in this collection where the second cut applies, so it keeps four labellings where a grid, a leaf, a Miura and a crumple all keep eight — and the family the collection long called difficult turns out to be the one whose conditions decide the most.
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A tessellation on a cylinder
A twist tessellation has been drawn here as a patch and as a torus, and never as anything in between. Gluing one pair of a cell's edges gives the family its first sheet with exactly two edges — the shape every folded tube actually has, and the only object in the collection that can say whether the rim's cost is linear in how much rim there is.
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The seam that is not a symmetry
Gluing a cell's edges looks like a symmetry of the drawing and is not. It is an instruction about which points of the paper are the same point, the drawing has to agree with it along the whole of a glued edge, and a rectangle that is not a period of the pattern does not glue at all — which turns out to be the only real restriction on which cylinders exist.