Where a sector crosses sixty

A search that runs out of options has proved something. Not no lettering was found, which is what a sample reports, but no lettering exists — every branch tried, every branch refused. Nothing in this collection had ever made a search do that, because every pattern it had been pointed at turned out to have an answer.

Sweeping the twist angle made it happen, and what it found was not what the sweep was looking for.

The two sides of the line #

At a turn of 0.2155 radians, the triangular patch has no labelling at all that satisfies the four conditions at all sixty of its interior vertices. The search visits fifteen nodes, dead-ends eight of them at vertices whose options have run out, and finishes with the space exhausted.

At 0.216 — five ten-thousandths of a radian further, about three hundredths of a degree — the same search finds a labelling in about forty-five nodes and never takes a letter back.

Everything else is the same. Eighty-three panels on both sides. A hundred and forty-two creases. Sixty interior vertices, all of degree four, each admitting exactly four labellings of its own creases. Sixty independent closed chains of panels. Even the arcs are the same arcs: the same pairs of panels joined, the same panels turned over, the same graph in every respect a graph can be described by.

What does change #

One thing, and it is a number rather than a structure.

At a typical interior vertex of this patch the four sectors are 120°, s, 60°, and 180° − s, where s falls as the twist angle grows. At a turn of 0.18 radians s is 64.41°; at 0.20 it is 61.92°; at 0.2155 it is 60.06°; at 0.216 it is 59.997°; at 0.35 it is 46.15°.

So s crosses 60° at a turn of about 0.2155 radians — and which sector is the smallest one changes there. Below the crossing the smallest sector is the fixed 60° one; above it, the shrinking one.

That matters because of the fourth condition, the one this subject calls the big-little-big lemma: the two creases bounding a strictly smallest sector must carry different letters. Which two creases those are is decided by which sector is smallest. So the crossing swaps the pair of creases the lemma forces apart at every one of the sixty vertices at once, and the two forcings are not equally satisfiable.

Above the crossing, they are. Below, they are not — not at any one vertex, but jointly, across the patch.

Every vertex satisfiable, and no pattern #

The part worth dwelling on is that nothing is locally wrong below the crossing.

Every one of the sixty vertices has exactly four labellings of its own four creases that satisfy developability, the angle condition, the count and the lemma. Not one vertex is short of options. A folder examining any single point of this pattern would find nothing to object to, and so would any check that examines one point at a time.

The pattern nevertheless has no labelling, because the choices cannot be made consistently across the sheet. Each crease belongs to two vertices, each vertex constrains its own four, and the constraints chain around the patch until they contradict.

This is the difference between local and global, and it is the first time this collection has an instance of it in this exact form: not a pattern that passes every vertex and then fails on layers, but a pattern that passes every vertex individually and has no assignment at all.

That is a stronger statement than the usual one, and it needed a search to make. A sample cannot distinguish no labelling exists from none was found, and the collection has one case where the difference mattered enormously. Here it matters in the other direction: the search says the space is empty, and it says so by exhausting it in fifteen steps.

What an exhaustion is worth #

This is the first proof of absence in the collection that came from a search rather than from an enumeration, and it is worth being precise about what kind of object it is.

An enumeration proves absence by writing every candidate down and rejecting each. That is available on a pattern with twelve creases, where there are four thousand and ninety-six labellings to write down, and it is what the square twist’s exact counts are. It is not available on a patch with a hundred and forty-two creases, where the list has more members than anything has a name for.

An exhausted search proves absence differently. It never writes the candidates down; it explores a tree whose branches are partial labellings, and it prunes a branch as soon as some vertex has no option left. Fifteen nodes were enough here because the contradiction arrives early: the first few letters chosen force enough of their neighbours that a vertex somewhere runs out, and it does so whichever letters were chosen first.

The two proofs are equally conclusive and enormously different in cost. That difference is the whole practical case for the instrument, and this is where it pays: a claim about a hundred and forty-two creases, settled in fifteen steps, that no amount of sampling could have made.

There is one condition on believing it, and it is worth stating because it is the failure mode such a search has. An exhaustion is only a proof if the search is complete — if every branch it declined to explore was genuinely refused rather than skipped. The search’s completeness rests on the propagation being sound: a crease is only written in when every surviving labelling at some vertex agrees about it, and a branch is only cut when some vertex has nothing left. Both are checked by the same machinery that produces the positive answers on the other side of the threshold, which is the strongest guarantee available short of a second implementation.

At the crossing itself the lemma says nothing #

The sectors go round as 120°, $s$, 60°, 180° − $s$, and reading the smallest-sector condition off that arrangement says more than which pair it forces.

The 60° sector’s neighbours are $s$ and 180° − $s$, so it is strictly smallest exactly when $60 < s < 120$. The sector $s$ has neighbours 120° and 60°, so it is strictly smallest exactly when $s < 60$.

At $s = 60$ neither is strictly smallest. They are equal, the lemma has no strictly smallest sector to act on, and it says nothing at all — so at exactly the crossing every vertex admits eight labellings instead of four.

That is a constraint vanishing rather than switching, and it has a consequence for the shape of the transition.

Which makes the transition one-sided #

Below the crossing the lemma forces one pair apart and a labelling exists. At the crossing it forces nothing, so every labelling that worked below still works and the solution set can only be larger. Immediately above, it forces a different pair and the set is empty.

So the solution set does not shrink smoothly to nothing. It grows at the crossing and then collapses, which is a discontinuity of a particular kind: the empty side is $s > 60$, and the crossing itself belongs to the non-empty side.

That answers one of the questions the essay leaves open. The verdict cannot change a shade before or after the sector crossing, because the constraint set itself changes at exactly $s = 60$ and nowhere else — every condition in the subject is continuous in the angles except the lemma, and the lemma is a strict inequality that flips at one point.

So the threshold is exactly the turn at which $s = 60°$, and the bracket found by bisection is a numerical statement about where that turn is rather than about where the verdict changes. Solving $s(\theta) = 60°$ from the pleat geometry is the closed form the essay says ought to be available, and the sector arithmetic says it is the whole of it.

It also sharpens the search’s own report. Fifteen nodes to exhaust at 0.2155 and forty-five to find an answer at 0.216 is not two searches in similar conditions; it is a search under a constraint that has just been imposed and one under a constraint that has just been lifted, and the fifteen is short because the new forcing contradicts almost immediately.

The same threshold on three tilings #

The transition is not a peculiarity of one tiling. The elongated patch and the hexagonal patch have it too, at the same place: no labelling at 0.21, a labelling at 0.22.

Three different tilings, three different panel counts, one threshold. That is a strong hint that the crossing is a property of the twist construction rather than of any tiling, and the sector arithmetic says why it should be: the shrinking sector’s size depends on the turn angle and the tiling’s own angles, and 60° appears in all three because the pleat geometry puts it there.

The square patch is the exception, and it has a labelling at every turn tried, from 0.1 radians to 1.3. Its vertices have no 60° sector to cross: the four sectors there are two supplementary pairs that move together as the turn changes, so whichever is smallest at one angle is smallest at all of them and the lemma forces the same pair apart throughout. The rhombille is the awkward case in the other direction — below the threshold its search does not exhaust inside any budget tried, so it is not known whether it has a labelling or merely a very expensive proof that it has none, and it is the one patch here whose verdict at a shallow turn is genuinely open.

What this corrects #

This collection has said twice that the twist angle changes nothing, and both statements were about a measurement that really does not move: the share of a patch’s letterings that agree with themselves is thirteen in a hundred and twenty at every turn from 0.15 radians to 1.0, and the essay that reported it drew the conclusion that the question is combinatorial and the dial is geometric.

That measurement was made on the square patch, which is exactly the one patch that has no threshold. It is correct as far as it goes, and the conclusion drawn from it is too broad.

The corrected statement is more interesting than the original. The angle does not move the share of letterings that agree — that is a fact about the combinatorics, and the earlier essay is right about why. What the angle moves is whether the set of admissible letterings is non-empty at all, which is not a share and which no proportion could have shown.

A dial that changes nothing about a proportion and everything about existence is a good reason to be careful about what a flat curve licenses. The lemma that says nothing at equal sectors is the same warning from the other side.

The dial that has to be watched #

There is a practical consequence for how this collection draws these patterns, and it is small but real.

Every figure here that draws a twist tessellation takes a turn angle, and most of them take it as a slider a reader can move. A slider that can be dragged below the threshold on a non-square tiling would be a slider that produces a pattern with no labelling — and the generator would then either refuse to draw, which is correct but abrupt, or draw the pattern with whatever letters it was last given, which would be a picture of something that does not fold.

The construction already refuses turns outside the range where the pleats have room, and that limit is computed in closed form from the polygon sizes. This is a second limit, of a different kind: not the paper runs out but the letters run out, and it sits at a different angle on every tiling.

Nothing in the collection currently draws a non-square patch below a fifth of a radian, so nothing is wrong today. What the threshold means is that the safe range for those figures is narrower than the construction’s own refusal suggests, and the two limits should be quoted together.

Why the lemma is doing this #

It is worth naming the condition responsible, because it is the one this subject treats as an afterthought.

The angle condition does not read letters at all. The count reads them but knows nothing about angles. Developability is automatic on a flat sheet. The big-little-big lemma is the only one of the four that reads both the angles and the letters, and it is therefore the only one whose verdict can change when an angle moves while the letters stay available.

That makes it the sole mechanism by which a geometric dial can decide a combinatorial question, and this transition is that mechanism firing. Below the crossing it forces one pair apart at every vertex; above it, another; and the difference is a pattern and no pattern.

What the patch looks like on either side #

It is worth looking at the two patterns, because the visual difference is almost nothing and that is the point.

At a turn of a fifth of a radian the twist polygons are barely turned: each one sits nearly square to its neighbours, the pleats between them are wide, and the sheet is mostly polygon. A hundredth of a radian further the polygons have rotated a fraction of a degree more, the pleats have narrowed imperceptibly, and nothing about the drawing announces that one of the two has no labelling.

There is a lesson in that for how these figures should be read, and it is the reason this site verifies patterns rather than drawing them. A crease pattern that cannot fold looks exactly like one that can; the difference is not visible at any magnification, and the collection has met that fact from several directions now. Here it is at its sharpest: two patterns a hundredth of a radian apart, one foldable and one not, and no reader could tell which was which.

What is left open #

Two things, and the second is more interesting than the first.

Whether the transition is sharp or gradual is not established either. The bracket says the verdict changes between 0.2155 and 0.216, and the sector crossing sits inside it, but a bracket is not a demonstration that the two coincide exactly. It is possible that the change happens a shade before or after the crossing for a reason involving several vertices at once, and the resolution of the search — one verdict per angle tried — cannot see the difference.

The exact threshold is not solved in closed form. It is bracketed between 0.2155 and 0.216 radians by search, and the sector crossing is at 60° by construction, so the closed form ought to be available from the pleat geometry. It has not been derived here, and a bracket found by bisection is a weaker object than an equation.

Whether the patch is genuinely unfoldable, or merely unlettered, is not settled either. A pattern with no admissible labelling cannot fold flat, so the verdict is safe. But the reason offered above — that the lemma’s forcing chains round the patch until it contradicts — is a description of what the search did rather than a proof of why. There is presumably a short argument: some cycle of vertices around which the forced differences accumulate to a parity contradiction. Nothing here has found it, and a search that exhausts in fifteen nodes is exactly the kind of small object such an argument could be extracted from.

Where the ladder goes next #

The obvious sweep is the one this rung stumbled into rather than planned: run the threshold hunt over every tiling, at fine resolution, and see whether the transition is always a sector crossing. Any tiling makes a twist, so there is no shortage of cases, and the construction’s own lower limit — the turn below which the pleats have no room — is a separate boundary that would then have a companion.

And there is a question about the clipped patches specifically. These patterns are cut out of an unbounded tiling, so their rim vertices are whatever the cut produced, and one of them turns out to carry twelve creases a micrometre long. Whether the threshold survives on an unclipped patch — the same tiling assembled unit by unit, with no rim — is a different measurement, and it would say whether this is a fact about twists or a fact about cutting them out.