A square that turns

Fold a small square in the middle of a sheet, run a pleat out from each of its sides, and collapse the whole thing. The square rotates. It ends up turned through some angle relative to the paper around it, and nothing about the flat pattern suggests that it should.

That rotation is the whole family. Twists tile, they are the standard unit of origami tessellation, and their geometry is not a design choice — it is what the vertex conditions leave.

Deriving the shape #

Start with the object and let the conditions cut it down.

A twist has a central regular polygon with k sides. At each of its corners, four creases meet: two sides of the polygon, and two pleat lines running outward. So each corner is a degree-four vertex, and the whole local theory applies.

The interior angle of a regular k-gon is fixed: 60° for a triangle, 90° for a square, 120° for a hexagon. That is one of the four sectors at each corner, and it is not negotiable.

Kawasaki’s condition says the alternating sums must each be a straight angle. The sector opposite the polygon’s interior is therefore its supplement — 120° for a triangle, 90° for a square, 60° for a hexagon. Also not negotiable.

That leaves two sectors, which must sum to 180° and are otherwise free. One parameter, and the twist’s whole shape is in it.

The lemma picks the value #

The free parameter is not quite free, because a third condition is watching.

The big-little-big lemma says that a sector strictly smaller than both its neighbours must be flanked by creases of opposite assignment. It is a constraint on the assignment rather than on the shape, and it is easiest to satisfy by arranging that no sector is strictly smallest at all.

Setting the two free sectors to the interior angle and its supplement does exactly that. The sector list becomes two equal pairs — 60, 60, 120, 120 for a triangle; 90, 90, 90, 90 for a square; 120, 120, 60, 60 for a hexagon — and with ties everywhere the lemma has nothing to forbid.

That choice also fixes the pleat directions, and one of them turns out to be a coincidence worth noticing. For every k, one of the two pleat creases at a corner is collinear with a side of the polygon, extended past that corner. So half of the pleat lines are the polygon’s own sides continued, which is why a square twist looks like a square with its four sides run out to the edge of the paper.

Which theorem was checked, and how #

Three things are computed rather than stated in these figures.

The sectors are measured off the pattern. The generator builds the crease pattern from the corner positions and the pleat directions, and then reads the sector angles back out of it with the same routine the theorem-checker uses. The numbers printed are those measurements, not the design values — so a mistake in the construction shows up as a sector list that does not sum to 360°.

The pleats are clipped to the sheet. A pleat ray is infinite and a square of paper is not. The generator computes where each ray leaves the square and stops it there, which is a rule this site has been caught by before and now enforces.

The assignment is found, not chosen. For the triangle and square the whole space of colourings is enumerated and the valid ones counted: sixty-four of five hundred and twelve for the triangle, two hundred and fifty-six of four thousand and ninety-six for the square. For the hexagon, where the enumeration is too large, the search is restricted to colourings whose central ring takes a stated shape, and enumerated whole within that restriction.

That last point matters because the alternative would have been to report a negative on the basis of not having looked. A search that has been narrowed must say what it searched.

Four to the k, exactly #

The two enumerated counts are not arbitrary and they check the essay’s own argument about the lemma, which is worth doing because that argument is the one the shape rests on.

A $k$-gon twist has $k$ polygon sides and two pleats at each corner: $3k$ creases and $k$ interior vertices. Maekawa halves the letterings at each vertex, and the smallest-sector lemma — silenced everywhere, because the sectors come in two equal pairs — removes nothing. So

$$2^{3k} / 2^{k} = 2^{2k} = 4^{k}$$

At $k = 3$ that is 64 of 512. At $k = 4$, 256 of 4096. Both are the measured numbers, and the formula predicts 4,096 of 262,144 for the hexagon — the count the essay says is out of reach.

So the sector choice the lemma motivated is confirmed by the census: if any sector were strictly smallest anywhere, the counts would have come out halved again, at 32 and 128.

Which prices the ring #

The same arithmetic separates the ring’s letterings from the rest, and it explains where thirty-two comes from.

The ring is $k$ of the $3k$ creases. A uniform ring is one of two letterings of those $k$ — all mountain or all valley — out of $2^{k}$, so uniform-ring letterings are $2/2^{k}$ of the admissible set. At $k = 4$: $256/8 = 32$, which is the count the essay reports and which is therefore a fact about the ring’s parity rather than about anything geometric.

The survivors have a ring of two letters in adjacent pairs, and at $k = 4$ there are four such rings of the sixteen — so a quarter of the admissible set, sixty-four letterings, contains the eight that fold.

One lettering in eight of that quarter folds, and none of the other three quarters does. That is a considerably sharper statement of where the survivors live than “the eight have adjacent pairs”, and it says the ring’s shape is a necessary condition doing most of the filtering before the layers are consulted at all.

It also gives the hexagon’s restricted search a justification rather than an excuse. Narrowing to a stated ring shape is not an arbitrary convenience — it is narrowing to the only quarter of the space that has ever contained a folded state, and enumerating it whole is enumerating the part that matters.

The ring that reads as one letter, and why it is not the one drawn here #

The uniform ring is worth dwelling on, because it is what makes a twist recognisable and because it is a trap.

In every case a valid assignment exists in which all k sides of the central polygon carry the same letter. It is the assignment anybody draws: the paper appears to turn the same way at every side, the polygon lifts as one piece, and the visual signature of a twist — a rigid-looking little polygon spinning against the sheet — follows from it.

None of those assignments has a folded state. The square twist has thirty-two letterings whose ring reads as one letter, every one of them satisfies every condition at every vertex, and not one of them can have its panels put in an order. The reason is a loop: each corner region between two consecutive pleats has to lie above one of them and below the other, and going round the four pleats the requirement closes on itself.

So the pattern printed here takes two letters on its ring rather than one — two mountains and two valleys, in adjacent pairs, which is the shape all eight of the survivors have. The polygon still turns. What it does not do is turn for the reason the uniform picture suggests.

What is verified, and what is not #

Here is the honest boundary, and it is the reason this essay does not draw a tessellation.

Each twist unit above is verified: every interior vertex satisfies developability, Kawasaki, Maekawa and the big-little-big lemma, and the assignment came from a search over those conditions.

A tessellation is a different object. Twists tile the plane, and in a real tessellation the pleat from one twist runs into the next, which means the two have to agree about the order of the layers in the pleat between them. Nothing in the local conditions examines that agreement, and deciding it in general is intractable.

So the site draws a unit and says so. A figure captioned “this tessellation folds flat” on the strength of checking its vertices would be exactly the claim this site refuses to make, and it would be wrong for the ordinary reason: passing every local test is necessary and not sufficient.

The compensating fact is that these particular tessellations demonstrably do fold, because people fold them. What is missing is a proof, not the phenomenon.

What tiling actually requires #

Since the essay declines to draw a tessellation, it should at least say what one would need.

A twist tiles by repeating on a lattice — triangular for the triangle twist, square for the square, triangular again for the hexagon. Adjacent twists share their pleats: the pleat leaving one twist is the pleat arriving at the next, and the two halves have to be the same crease with the same assignment.

That much is a condition on the pattern and it can be arranged. What cannot be arranged locally is the layer ordering. In the pleat between two twists the paper from one twist and the paper from the other overlap, and their order has to be consistent — and consistency is a constraint linking twists that share no vertex.

Chase that around a loop of four twists and the constraints can close on themselves. Whether they close consistently is exactly the question that is intractable in general, and for twist tessellations specifically it is answered by folding one rather than by proving anything.

So a tessellation figure would be asserting a global property from local checks, which is the one thing this site refuses to do. The compensating fact is that the tessellations demonstrably fold, because people fold them by the hundred; what is missing is a proof, not the phenomenon.

The family is larger than three #

Regular polygons are the tidy case and the family does not stop there.

The derivation above used a regular k-gon because its interior angle is the same at every corner, which makes every vertex identical and the sector arithmetic uniform. Nothing in Kawasaki requires that. An irregular polygon works too, as long as each corner’s four sectors satisfy the alternating condition — which now gives a different pleat geometry at every corner.

The lemma then starts to bite. With unequal sectors there generally is a strictly smallest one, and the two creases flanking it must differ, which constrains the assignment rather than the shape. Some irregular twists have no valid assignment at all.

Beyond irregular polygons there are the patterns built by repeating twists at more than one scale, the ones with twists of different orders in the same sheet, and Fujimoto’s families generated by systematic operations on a base pattern. The tessellation literature is largely a catalogue of these, and it is much larger than the derivation here suggests.

What the derivation does give is a reason the regular cases are the ones everybody folds: they are the ones where the lemma is vacuous, so the assignment is free and the folding is forgiving.

The idealisation this rests on #

Twists are the patterns where zero thickness stops being a convenient fiction and starts being conspicuous.

At the centre of a twist, k layers of pleat converge on a small polygon, and each pleat is itself two or three layers. A hexagon twist has something like a dozen thicknesses of paper meeting in a region a few millimetres across, and in a tessellation of a hundred of them the accumulated bulk is what actually limits how small the twists can be.

The mathematics does not see any of it. Every layer is a plane of no depth, the twist closes exactly, and the pattern tiles indefinitely. A real sheet of kami reaches its limit at a twist of about a centimetre; tissue foil goes smaller; nothing goes arbitrarily small.

That is the first of the four idealisations doing its usual work, and it is worth naming here because tessellations are the one place in the subject where a reader can feel it — the finished piece has a thickness gradient that the pattern does not predict.

The surprise: it is bistable #

There is a mechanical property of the square twist that has nothing to do with the flat-folding theory and was noticed only recently.

A Miura fold has one degree of freedom and moves smoothly from flat to folded. A square twist does not. It has two stable states — open and closed — with an energy barrier between them, and it snaps from one to the other rather than moving continuously.

The reason is that the twist is not rigid-foldable. Getting from one state to the other requires the panels to bend on the way, and the bending stores energy that is released once the pattern passes its midpoint. In paper that shows up as an audible click.

That makes the square twist a mechanical switch made of geometry, and it is why the pattern turns up in the metamaterials literature. A sheet tiled with square twists is a surface with a large number of independently switchable cells, and the switching is a property of the crease pattern rather than of anything the material is doing.

It also means the pattern is a counterexample worth keeping. Flat-foldable and rigid-foldable are different properties; the square twist satisfies the first and fails the second, and the failure is what makes it useful.

Folding one #

The unit above is printable at true scale, which is worth doing once, because a twist is the pattern where the difference between reading and folding is largest.

The creases are quick to make: a small polygon in the middle, two rays from each corner. The collapse is not. Every pleat has to be persuaded to lie the same way round the ring at the same time, and until they all do, nothing holds — the pattern gives no partial credit.

What a folder notices doing it is the rotation, and what is surprising about the rotation is that it is not gradual. The polygon sits flat, resists, and then turns through its whole angle as the sheet closes. That is the bistability described above, met by hand rather than described.

It is also the clearest demonstration this site can offer of why a local checker is not enough. The pattern’s vertices all pass; the sheet still refuses to close until the layers are arranged, and arranging them is something the fingers find and the theorems do not.

Who found it, and when #

The twist has two independent origins and they did not meet for decades.

Ron Resch, an American artist and computer scientist, was folding triangular and hexagonal twist tessellations by the early 1960s and patented several in 1968. His interest was in surfaces that could be made rigid and self-supporting, and he worked at architectural scale as well as in paper. He was designing structures rather than studying theorems, and his patterns were published as engineering.

Shuzo Fujimoto, a Japanese schoolteacher, developed twist tessellations independently in the 1970s and published them privately. His work is the ancestor of most of the tessellation folding done today, and it reached the wider community slowly.

The mathematical account came much later still, with the flat-folding conditions of the 1990s and the metamaterials work of the 2010s. So the pattern was an artwork, then a structure, then a mechanism, and only lately a thing with a derivation — which is a sequence this subject repeats often enough to be worth noticing.

Because Resch’s and Fujimoto’s patterns are published as mathematics and as structure rather than as models, they are reproduced freely; a living designer’s crease pattern is their work, and this site does not print one.

The ladder from here #

Later rungs against this anchor: the tessellation proper, and what it would take to verify one. Fujimoto’s family and the operations that generate it. The Resch pattern as a rigid structure, which is what he was actually after. Bistability, measured rather than described. Twists with irregular polygons, where the sectors are no longer equal and the lemma bites. The relationship between twist tessellations and the periodic patterns of crystal symmetry. And the practical question of how small a twist can be folded, which is a material limit and has been measured by nobody.

The pattern looks designed and is derived. What is left over after two conditions and a lemma have taken their share turns out to be the most-folded family in the subject.