Which polygons twist

Every book about twist tessellations shows three of them. A triangle twist, a square twist, a hexagon twist, and the impression left — never quite stated, which is why it survives — is that those are the twists there are.

They are the twists that tile. The twist itself has no such restriction, and the difference is worth several minutes because the reason is not where anybody looks for it.

The twist vertex, whatever the polygon #

A square that turns derived the square twist from the two vertex conditions and found the geometry forced. Repeat the derivation for a K-gon and the forcing is identical, which is the first surprise.

Around a K-gon of interior angle A, each twist vertex has four creases: two along the sides of the central polygon and two pleats running out to the paper’s edge. The sectors come out as A, A, 180° − A, 180° − A, in that cyclic order.

Kawasaki asks whether the alternating sums are equal. They are: A + (180° − A) on one side, and A + (180° − A) on the other, both exactly 180°, whatever A is. The condition is satisfied identically, not by choosing the angles well.

That is worth pausing on, because it is a trap of the same kind the Yoshimura’s row height turned out to be. Kawasaki holding for every polygon makes the family look completely free, and something else has to be what constrains it.

What the local conditions leave #

Maekawa is the next condition and it also holds: a degree-four vertex needs three of one letter and one of the other, and the search finds such an assignment for every polygon here.

Big-little-big is the interesting one. It bites where a sector is strictly smaller than both its neighbours, and the sector order A, A, 180° − A, 180° − A never has that shape: the two equal sectors sit next to each other, so neither is strictly smaller than both neighbours whichever of the two values is smaller. All three local conditions pass identically, for every K.

So the twist is locally unconstrained. A five-sided twist folds. A seven-sided twist folds. Nothing about the polygon’s number of sides enters the vertex conditions at all.

Where the restriction actually is #

The three familiar twists are the ones whose polygons tile the plane, and that is the whole of it.

A tessellation of twists needs the twists to meet. Around a point where several central polygons come together, their interior angles have to sum to 360°, which means the interior angle must divide 360° exactly. For a regular K-gon the interior angle is 180°(K − 2)/K, and 360° divided by that is 2K/(K − 2) — a whole number only for K = 3, 4 and 6, giving six triangles, four squares or three hexagons around a point.

Five sides gives an interior angle of 108°, and 360/108 is 10/3. Seven gives about 128.6°, and 360 divided by it is not a whole number either. The pentagons and heptagons cannot meet.

Reading 2K/(K − 2) as K grows is the quickest way to see why the list is so short. At K = 3 it is 6, at K = 4 it is 4, at K = 5 it is 10/3, at K = 6 it is 3, and from there it falls steadily toward 2 without ever reaching a whole number again. Three polygons around a point is the last integer available, so the hexagon is the last polygon that tiles, and everything above it is squeezed into the gap between three and two.

This is a fact about the plane and has nothing to do with paper. It is the same arithmetic that limits the regular tilings, it was known to the Greeks, and it applies equally to floor tiles, honeycomb and anything else that has to fill a plane without gaps.

Reading the sector angles #

The table’s middle column is worth reading rather than trusting, because the arithmetic is one addition and the whole essay rests on it.

For the triangle the sectors are 60, 60, 120, 120. Alternating sums: 60 + 120 = 180, and 60 + 120 = 180. For the square: 90, 90, 90, 90, and both sums are 180 trivially. The pentagon: 108, 108, 72, 72, and 108 + 72 = 180 twice over. The heptagon: 128.6, 128.6, 51.4, 51.4, and the two sums are both 180 to the accuracy the numbers are printed at.

Every row is the same identity — the interior angle of any polygon and its supplement add to a straight angle — dressed up as a theorem about folding. That is why the figure prints both alternating sums instead of a tick: a condition satisfied identically looks exactly like a condition satisfied by good luck, and printing the two numbers is the difference between showing it and asserting it.

The radius is the one genuinely free parameter of a twist, and it is free within limits rather than absolutely — past about a third of the sheet the pleats have nowhere to run. That is a constraint from the paper’s boundary, which is a third kind of restriction alongside the local conditions and the tiling, and it is the one a figure notices first because it draws ink off the page.

The vertex conditions never see regularity either #

The derivation above used the interior angle of a regular K-gon, and it is worth noticing that it never needed the polygon to be regular.

At a twist vertex sitting on a corner of interior angle $A$, the four sectors are $A$, $A$, $180° - A$, $180° - A$ whatever else the polygon is doing. Kawasaki’s two alternating sums are both $A + (180° - A)$, which is a straight angle by definition of a supplement. Maekawa is a statement about letters and sees no angle at all. The smallest sector still has its equal twin beside it, so the smallest-sector lemma is still silent.

Each vertex is checked on its own, and each carries its own $A$. So the twist construction works on any convex polygon, with a different interior angle at every corner if it likes, and the identity holds corner by corner rather than by symmetry.

That widens the second constraint and leaves the third exactly where it was — which is what makes the tiling arithmetic worth redoing without the word “regular” in it.

Dropping regularity moves the boundary by one #

Ask which convex polygons tile the plane at all, in any arrangement, and the answer is a genuinely different list from the three.

Every triangle tiles. Two copies make a parallelogram. Every convex quadrilateral tiles, including entirely irregular ones, by rotating copies about the midpoints of its edges — a fact that surprises people who have only met the regular case.

Some pentagons tile, and which ones was a hundred-year question: fifteen families are now known and the classification was completed in 2017. So a pentagon twist tessellation is possible, and the regular pentagon’s failure was a failure of regularity rather than of five-sidedness.

Some hexagons tile — three families, settled in 1918. And above six, nothing: no convex polygon with seven or more sides tiles the plane, which is a theorem rather than an absence of examples.

So the honest boundary is at six rather than at the familiar three, and it falls in two different ways. A regular pentagon twist cannot tile because 360 divided by 108 is not a whole number. A heptagon twist cannot tile for a reason no adjustment of its angles can repair.

That makes the table’s last row categorically different from its middle one. The pentagon is a near miss with fifteen ways round it. The seven-sided twist is a perfectly good piece of paper that will never be a material, however it is drawn.

What a lone twist is good for #

If a pentagon twist folds but cannot tile, the natural question is whether it is worth anything, and it is — in two places.

As a local feature. A twist does not have to be repeated to be useful. A single twist in the middle of a sheet is a way of gathering paper into a rotating boss, and designers use twists of many kinds as elements within a model rather than as a global pattern. There is no requirement that the surrounding paper be twists too, and a five-sided twist in the middle of an otherwise ordinary sheet is a perfectly ordinary thing to fold.

In a mixed tiling. Regular polygons that cannot tile alone can tile in combination — an octagon with squares, a pentagon with rhombi — and a twist tessellation built on a mixed tiling is a perfectly good tessellation. The restriction is on regular tilings by a single polygon, which is narrower than it first appears.

That second point matters for reading the literature. Twist tessellations in practice include a great many patterns that are not one of the three, and every one of them is a tiling that mixes shapes or relaxes regularity. The three are the simplest cases rather than the only ones.

There is a third use and it is the one that connects this rung to the rest of the site. A twist is a way of taking paper out of a sheet — the central polygon and its pleats consume area, and the sheet contracts. Used once, that is a local gather. Used everywhere, it is a material whose properties come from its creases, and the contraction becomes a Poisson’s ratio rather than a detail. Which of those two a twist is depends entirely on whether it tiles, and that is the practical weight behind an arithmetic fact about interior angles.

Three restrictions, in three different places #

Setting the constraints side by side is the clearest summary this essay has, and it is worth doing explicitly because they are usually run together under the word “the twist tessellations”.

The vertex conditions. Kawasaki, Maekawa, big-little-big. These constrain nothing here — every polygon passes, identically, and the passing is arithmetic rather than luck.

The paper’s boundary. The pleats have to fit on the sheet, which limits the twist radius to about a third of the side. This is a constraint on the drawing and no theorem mentions it.

The tiling. Only 3, 4 and 6 have interior angles dividing 360°. This is a constraint on the plane, it is two thousand years old, and it is the one that produces the three familiar patterns.

The interesting one is that the constraint everybody attributes to the folding is in the third row, where folding does not appear. A tessellation is one vertex repeated and the repeating is a separate question from the vertex — a separation this site keeps returning to and which is easiest to lose exactly where the two halves happen to have the same answer.

Where the computation runs out #

There is a limit on this essay’s own table and it is worth stating, because it is a limit of the machinery rather than of the subject.

The assignment for each twist is found rather than stated — as it is everywhere on this site — by enumerating mountain-and-valley strings and testing them against the local conditions. A K-gon twist has 3K free creases: K in the ring and 2K in the pleats. At seven sides that is twenty-one, and two to the twenty-first is manageable. At eight it is twenty-four, and the enumerator refuses anything above twenty-two.

So the table stops at seven, and it stops for a reason that has nothing to do with twists. A search over letters doubles with every crease, and a pattern with two dozen creases is already past what a build can enumerate.

The refusal is a thrown error rather than a truncated search, which is the right behaviour: a partial enumeration reporting “no assignment found” would be reporting that nobody looked. Past the enumerator’s limit the search is narrowed to assignments whose central ring takes a stated shape — which cuts the space by a factor of 2ᴷ⁻¹ and is why the hexagon is answerable at all.

That narrowing deserves a word, because it is a choice with a consequence. It used to require one letter throughout the ring, which is a design preference — the thing that makes a twist look like a twist, with the central polygon turning as a rigid piece — and not a theorem. It turned out to be worse than arbitrary: not one lettering with a uniform ring has a folded state, on any twist small enough to check. The narrowing now prefers a ring running as one block of each letter, and the preference is broken only after the folded state has decided.

The honest summary is that each row of the table means: this polygon admits at least one flat-foldable assignment, and where the panels are few enough to search, at least one of those assignments has an ordering. Neither statement is a count, and a count is what the enumerator would give if it could afford to run.

What no figure here can show #

The figures draw single twists, and a tessellation is not a single twist repeated — it is a single twist repeated and the pleats between neighbours reconciled. Two adjacent twists share their pleats, and whether the shared pleats can carry consistent letters is a question about the pair rather than about either one.

That question is exactly the local-versus-global gap this site returns to, and none of these figures addresses it. Every twist here is verified as an isolated pattern on its own sheet. A tiling of them is a different object and this site does not check it.

The second absence is the folded state. Every figure on this page is a crease pattern, which is the artefact that is verified; the twisted, contracted sheet is a projection that no figure here draws. A pattern flat on the page is not the folded object, and where both appear elsewhere on this site the caption says which is which.

The idealisation, named #

A twist gathers paper into a small central polygon, and the paper it gathers has to go somewhere: at the ring, three or more layers meet along each edge.

For a single twist that is trivial. For a tessellation it is the binding constraint, because layer count is what real material runs out of and a twist tessellation at any useful density has layers accumulating across the whole sheet. The classical patterns are folded from very thin paper for exactly this reason, and the twist radius in these figures is a geometric parameter with no thickness attached to it.

There is a second and subtler one. The pleats are drawn as running to the edge of the sheet, which is what the construction says. In a tessellation they run into the next twist, and the length available is fixed by the tiling rather than by the paper. A twist whose radius is too large for its cell does not fail a local condition; it simply has no room, and no vertex test sees that either.

Who found these, and when #

Twist tessellations as a systematic subject belong to Ron Resch, who was folding and patenting them in the 1960s, and to Shuzo Fujimoto, whose independent work in Japan produced much of the classical repertoire. The mathematics of which regular polygons tile the plane is far older and belongs to nobody in particular; Kepler wrote it down carefully in 1619.

What this essay adds is the separation, and the separation is easy to lose. The twist is a local construction with no restriction on it; the tiling is a global constraint with a two-thousand-year-old answer; and the three familiar tessellations are the intersection. Reading the intersection as a property of twists is the mistake, and it is an easy one because the three are the only ones anybody draws.

Where the ladder goes next #

A construction with no local restriction, limited by something global — that shape recurs. The universal molecule fills any convex polygon and refuses a reflex corner, which is the same story with the constraint moved from the tiling to the shape, and with the consequence that a design’s difficulty gets handed backward to whatever chose the polygons.