The dial that decides nothing

The twist angle looks like the free parameter of a tessellation. It is fenced at both ends — turn too far and the pleats have no paper left, turn too little and something stranger happens — but between the fences it is a dial, and turning it changes the pattern visibly and continuously.

It changes everything a folder measures, and nothing a folder decides.

What the dial does change #

Take a square twist patch and turn its angle from just above the lower fence to just below the upper one.

At 0.002 radians the twist polygon is barely turned; the sectors at an interior vertex are 89.7°, 90°, 90.3°, 90°, and the pattern is a grid with a hint of a rotation in it. At 1.54 radians the same vertex reads 0.7°, 90°, 179.3°, 90°: one sector has collapsed to under a degree and its opposite has opened almost straight.

Between those, the paper the pleats consume falls from 24 per cent of the sheet to two hundredths of one per cent. The kept area rises from 0.764 to 0.9998. The folded footprint changes shape, the crease length changes, and the object a folder ends up holding is a different object at every setting.

What it does not change #

The number of mountain-and-valley assignments that satisfy every condition in the subject is sixteen, at every one of those angles.

Not approximately sixteen. Not sixteen over most of the band. Sixteen at 0.002, at 0.01, at 0.1, at 0.3, at 0.6, at 0.9, at 1.2, at 1.45 and at 1.54, on a patch with twelve free creases and four interior vertices, counted by exhaustive enumeration over all 4,096 labellings.

The reason is short once it is stated. Developability and Kawasaki are conditions on sums of angles, and the twist construction satisfies both identically at every angle in the band; they do not filter labellings at all. Maekawa counts letters and does not look at angles. That leaves the big-little-big lemma, which is the only condition in the subject that reads a comparison between angles — and what it reads is which sector is strictly smallest, never by how much.

So the lemma constrains the same pair of creases at every angle where the same sector is smallest. And on a square twist, the twist sector is smallest at every angle in the band: it starts at 89.7° against a neighbour at 90° and finishes at 0.7°, and it is under its neighbours the whole way.

Where the count does move #

If the count is decided by the ordering of the sectors, it should change exactly where the ordering changes. On a triangular twist it does.

At a triangular twist the vertex reads 60°, x, 120°, 180° − x, with x growing as the angle is turned. Below 0.216 radians the sectors sit in one order; at 0.216 exactly they tie, at 60° and 60° and 120° and 120°; above it they sit in another.

The count of assignments is 128 below and 64 above, and the change happens at that angle and nowhere else. Measured at 0.05, 0.15 and 0.21 it is 128; at 0.216, at 0.22, at 0.25, at 0.3, at 0.6 and at 0.9 it is 64.

That is a factor of two, arriving discontinuously at a tie, which is exactly the shape the same lemma produces at a single vertex — where a degree-four vertex with two smallest sectors equal admits eight assignments and four once the tie is broken by a tenth of a degree.

There is a way of putting the whole result that makes it less surprising and more useful. The four conditions are of two kinds. Three of them — developability, Kawasaki, Maekawa — are equalities: sums that must match, counts that must differ by two. An equality either holds or it does not, and a construction that satisfies it satisfies it identically across a family. The fourth is an inequality, and an inequality has a boundary; a family crosses that boundary at isolated parameter values and is otherwise on one side of it.

So a one-parameter family of patterns can only change its assignment count where it crosses an inequality’s boundary. The square twist never crosses one. The triangular twist crosses one, at 0.216 radians, and that is the only place its count moves.

Both results fall out of the sector formulas #

The invariance on the square and the single jump on the triangle are reported as measurements, and both can be read off the sector expressions the essay has already given. That is worth doing, because it turns two observations into one rule.

The lemma reads which sector is strictly smallest. So the count can only change where the identity of the smallest sector changes — where two sectors cross at the bottom of the list. Everywhere else the same pair of creases is constrained, and the same constraint on the same pair removes the same labellings.

Now look at the square twist. Its vertex reads 89.7°, 90°, 90.3°, 90° at one end of the band and 0.7°, 90°, 179.3°, 90° at the other, which is the family $90° - t$, $90°$, $90° + t$, $90°$ with $t$ running from a third of a degree to eighty-nine. The smallest is $90° - t$ throughout: it starts below the two fixed sectors and only falls further, and the two ninety-degree sectors never move. The identity of the minimum never changes, so the count cannot. Sixteen at every angle is not a coincidence of the enumeration; it is what the family’s own sector expressions say.

The triangular twist reads 60°, $x$, 120°, $180° - x$. The last two are never below sixty while $x$ stays under 120, so the minimum is whichever of 60° and $x$ is smaller — and those two cross exactly once, at $x = 60°$. One crossing, one jump, and the jump is at the angle the measurement puts it at.

Which bounds how often any dial can matter #

That gives a rule covering families this essay never examines, and it is a bound rather than a case list.

In a one-parameter family the sector sizes are continuous functions of the dial, and on these constructions they are monotone — some rising, some falling, some fixed. The minimum of $d$ monotone functions changes identity at most $d - 1$ times as the parameter is swept. So a family of degree-$d$ vertices can change its assignment count at most $d-1$ times across its whole band, whatever the construction and whatever the tiling.

At degree four that is three. The square twist uses none of them and the triangular twist uses one, so both sit comfortably inside a bound that neither approaches. A family that used all three would be one whose sector list reorders twice more on the way across, which is a thing a construction could do and none of these does.

The bound also says what to look for when a dial does change something. The jump is always at a tie, ties are where the lemma falls silent, and a tie in a one-parameter family is a single point rather than a region. So a designer sweeping an angle can cross at most a handful of such points, will not notice crossing one, and will find the combinatorics of their pattern unchanged everywhere in between — which is the practical form of the whole finding.

What that means at the paper #

A folder choosing a twist angle is choosing a shape. The pattern will be tighter or looser, the model smaller or larger, the pleats fatter or thinner. What they are not choosing is how much there is to get wrong: the number of decisions between the drawn creases and a folded model is the same at every setting, and so is the number of ways to make them.

That is worth stating because it runs against the intuition that a tighter pattern is a harder one. A twist at 1.4 radians is very much harder to fold — the sectors are under three degrees and the paper does not want to make them — but the combinatorial difficulty is identical to the same tessellation at 0.1, where the creases are comfortable. The difficulty that grows is material rather than geometric, and the two have been running together in the folklore.

Which theorem was checked, and how #

The counting is exhaustive rather than clever: every one of the 4,096 labellings of a twelve-crease patch is put past the site’s own pattern checker, which runs developability, Kawasaki, Maekawa and the big-little-big lemma at every interior vertex. Nothing is pruned, so the count is the count, and the same routine reproduces the published totals it is checked against elsewhere.

The invariance is not asserted from a formula. The sector angles are read off the built pattern at each angle and their ordering is recorded, and it is the ordering that is observed to be constant on the square and to change once on the triangular. If the ordering had moved without the count moving, or the count without the ordering, the explanation would be wrong.

The patches are small on purpose. A larger patch has hundreds of free creases and cannot be enumerated; what a larger patch would give is a count that is a product over vertices of the same per-vertex numbers, which is the same statement raised to a power.

One more measurement is worth reporting because it bounds how much of this is an artefact of a small patch. The square patch enumerated has four interior vertices and twelve free creases; its sixteen assignments are, vertex by vertex, four choices at one vertex and the rest forced by the creases they share. Doubling the patch to sixteen interior vertices does not double the count, and it does not square it either — the shared creases couple the vertices, and how tightly is precisely what the propagation experiment measures and finds to be very loosely. What can be said with confidence is that whatever the count is on a large patch, it is the same number at every angle in the band, because every ingredient of it is.

Where the model stops #

The count is of assignments that satisfy the local conditions, which is a filter and not a decision: deciding whether a whole pattern folds is NP-hard, and some of the sixteen may not fold. That does not weaken the finding — the claim is that the filter’s output is constant across the band, and a constant filter output on a family of patterns is exactly what a folder’s decision count is bounded by — but it does mean sixteen is an upper bound rather than an exact count of foldable markings.

The band itself is a model. The upper fence is where the pleats run out of paper and the lower one is where the sector ordering changes, and both are computed on an infinite tiling; a finite sheet clips the pattern and its edge vertices are subject to no conditions at all, so a real patch has slightly more freedom than this account gives it.

And zero thickness is doing its usual work. At 1.5 radians a twist polygon’s sector is under a degree, which on a 150 mm sheet is a wedge of paper a fraction of a millimetre wide at the scale a crease is thick. The pattern exists; the paper does not.

A second limit is about what “the same count” is a count of. Sixteen labellings of a twelve-crease patch is sixteen ways to mark the sheet, and two of those may fold to the same object or to different ones — the count of objects is a different number, and it is not a function of the count of markings. Nothing here says the folded states are invariant across the band; only the markings are, and the two questions come apart at exactly the vertices whose sectors tie.

What the picture cannot show #

An invariant is not a picture. Two tessellations at two angles look completely different and their assignment counts are the same number, so any figure showing the two patterns is showing the thing that changes rather than the thing that does not — and the only honest representation of the finding is the number written under each.

Nor can anything show what the lemma is reading. It reads an ordering, and an ordering is not visible in a drawing of angles: the reader has to be told which sector is smallest, and at 0.002 radians the difference between smallest and next is six tenths of a degree.

The generalisation #

The statement worth carrying out of this is about what kind of information a condition consumes.

Every condition in flat-folding is scale-free in the crease lengths — not one of them mentions how long a crease is — and that has been said here before. What this essay adds is a second and stronger blindness: the conditions are almost scale-free in the angles as well. Three of them see only sums, which the construction fixes; the fourth sees only an order.

So the whole apparatus reads a crease pattern as a combinatorial object with a few equalities attached, and the continuous geometry a designer works in — how sharp, how wide, how deep — passes straight through it. That is why a tessellation can be redrawn at a completely different angle and be, from the theory’s point of view, the same pattern.

It also explains something about how the subject is taught. A twist is always drawn at a comfortable angle, and the drawing is treated as the twist rather than as a member of a family, because nothing in the theory distinguishes the members. The family only becomes visible when the pattern is generated from a rule with the angle as an argument, which is what makes it a fact this site can find and a hand-drawn tradition could not.

Who found it, and when #

Twist tessellations are Ron Resch’s and Shuzo Fujimoto’s independently; the twist angle as a continuous parameter, and the two fences on it, belong to the computational treatments of the last twenty years. The big-little-big lemma is Jacques Justin’s, from 1986, and it is the only one of the four conditions that compares angles at all.

The invariance appears not to have been stated anywhere, and the reason is probably that it is only visible from a construction that takes the angle as an input. A tessellation drawn by hand has one angle; a tessellation generated has all of them, and the question of what varies across the family is one only the second can ask.

There is a design reading of all this that is worth one paragraph. A tessellation is usually specified by its tiling and its angle, and the angle is chosen for appearance or for how much the sheet shrinks. This says the angle is purely a shape parameter: it can be tuned to hit a target footprint, a target pleat width or a target crease length without any risk of changing what has to be decided when the sheet is folded. That is an unusually clean separation for this subject, where most parameters change several things at once, and it is worth having stated rather than assumed.

Where the ladder goes next #

The obvious continuation is the one this essay’s method cannot reach: the count on a large patch, where a product over vertices is not the whole answer because vertices share creases. Fixing one crease and propagating the conditions settles almost nothing, so the true count is somewhere between the product and a number nobody has computed.

The other direction is the fence itself. The lower fence on the triangular twist is a tie, and ties are where several of this subject’s conditions go quiet at once — the lemma, the reduction that decides a single vertex, and now the assignment count. Whether every fence in the subject is a tie is a question worth asking of the whole library rather than of one pattern.