A corrugation never backtracks

A corrugation is the same vertex repeated until the sheet stops being a sheet and becomes a material. Its crease pattern is as regular as anything in the subject: one template, stamped on a lattice, with every interior vertex a copy of every other and every sector angle drawn from a list of two or three.

That regularity makes it the natural place to ask how the cost of a question grows. There is no shape to a corrugation beyond its size, so anything that grows must be growing with the size, and a family that reaches from four panels to two hundred and fifty-six is a family with a genuine range in it.

The answer is that nothing grows except the size. Four panels, four steps. Nine panels, nine steps. Two hundred and fifty-six panels, two hundred and fifty-six steps. The search decides every crease it decides at the first attempt and never takes one back.

What a step is, and why one per panel is the floor #

A step here is a node of the search: a point at which the vertex conditions have been propagated to a fixed point, the arcs have been checked for a circle, and a crease has been chosen to decide.

One step per panel is not merely cheap. It is close to the least a search of this shape can cost, because the search has to visit the sheet — the propagation is a sweep over the vertices, and a pattern with two hundred and fifty-six panels has two hundred and twenty-five interior vertices whose conditions have to be read at least once. A search costing one step per panel is a search whose cost is the reading rather than the deciding.

The number that would signal difficulty is the gap between steps and decisions taken and withdrawn. Every backtrack is a step that produced nothing, and the count of them is separately available: on this ladder it is zero, at every size. Not small; zero. There is no size of orthogonal grid at which the search’s first guess is ever wrong.

There is a second floor worth naming, because it explains why the line is exactly one step per panel rather than approximately. The search branches only where the propagation has stopped deciding, and on a grid the propagation stops at exactly one crease per panel — the panel’s own commitment, from which its neighbours follow. So the count is not an empirical near-linearity that happens to fit; it is the number of genuine choices the pattern has, and the search takes each of them once.

That is a strong statement about the pattern rather than about the search, and it is the reason the same number appears on the leaf, on the crumples and on the corrugations. Any pattern whose constraints determine a letter from each of its neighbours has as many real decisions as it has panels, and no arrangement of any search can do better.

The share that falls to one per cent #

That would be unremarkable if consistent letterings were common on these patterns, and they are not.

Drawing letterings at random — letterings that pass every condition at every vertex, drawn by propagating and branching at random where the propagation stops — and testing each afterwards for a circle in the arcs gives a share that collapses as the grid grows. Two divisions: a hundred of a hundred agree. Four: ninety-four. Six: seventy-three. Eight: fifty-eight. Ten: thirty-six. Twelve: fifteen. Fourteen: two. Sixteen: one.

Sixteen divisions is not an arbitrary end point. It is roughly the scale a box-pleated design is actually drawn at, so the one per cent is the share at the size where the question matters — and at that size a randomly chosen admissible lettering is almost certainly inconsistent while a searched-for one is found without a single wrong turn.

The rarity is not a sum of local risks #

The falling share invites an obvious model, and the model is wrong in a way that says something.

An $n$-division grid has $(n-1)^2$ interior vertices, and each is a four-panel circuit at which a contradiction could sit. If each vertex were an independent hazard — a fixed chance $q$ of being fine, whatever its neighbours do — the share of drawn letterings that agree would be $q^{(n-1)^2}$.

Calibrate $q$ on the cheapest point. Four divisions has nine interior vertices and ninety-four letterings in a hundred agree, so $q = 0.94^{1/9} = 0.9931$. Every other size is then a prediction with nothing left to fit.

Six divisions: 84 predicted, 73 measured. Eight: 71 against 58. Ten: 57 against 36. Twelve: 44 against 15. Sixteen: 21 against 1.

The model is wrong everywhere, always in the same direction, and the gap widens monotonically — a factor of 1.15 at six divisions and a factor of twenty at sixteen. That is not a mis-set constant. No choice of $q$ fixes it, because the shape is wrong: an independent-hazard model is an exponential in the vertex count, and the measurements fall faster than any exponential in $n^2$.

Which means the circuits are long #

Matching the fall needs the hazard to grow roughly as $n^3$ rather than $n^2$ — as the vertex count multiplied by the linear size of the sheet.

That is exactly the signature of a constraint whose violations are not local. A circuit in the arcs is a cycle, and a cycle can be as long as the sheet is wide. A sixteen-division grid does not have two hundred and twenty-five small chances to fail; it has that many places a cycle may pass through, and cycles that run across the whole pattern. The number of ways to fail therefore carries a factor of the sheet’s extent that a per-vertex count cannot see.

And that is the mechanism the cost-and-rarity split was missing. The two conditions on a lettering have different reach. The vertex conditions are local — four creases at a point — which is why the propagation is decisive and the search never guesses wrong. The circuit condition is global, which is why the failure rate outruns the vertex count and a sampler that is fine at four divisions is hopeless at sixteen.

Both facts have one cause: locality. It makes the search flat and it is precisely what the expensive condition does not have.

Why rarity and cost are unrelated here #

The two numbers measure different things and the collection has made this point once before on a crumple. The grid ladder makes it much more sharply, because both quantities are measured on the same objects across the same range and they move by two orders of magnitude in one case and not at all in the other.

Rarity is a statement about a set: what fraction of the admissible letterings are consistent. It falls because the number of ways to be inconsistent grows faster than the number of ways to be consistent — every interior vertex is a four-panel circuit, so a sixteen-division grid has two hundred and twenty-five places a contradiction could sit against a two-division grid’s one, and a lettering has to avoid all of them at once.

Cost is a statement about a procedure: how much work reaching one member takes. It stays flat because the procedure never has to sample. It propagates, and the propagation on a grid is decisive — a letter written on one crease forces its neighbours, those force theirs, and the forcing sweeps across the sheet without ever leaving a genuine choice that could be got wrong.

So the sampler’s difficulty and the search’s difficulty are not two readings of one quantity. The sampler is trying to hit a shrinking target by throwing at it; the search is walking to the target along a path that the constraints lay out for it, and the target’s size is irrelevant to the walk.

It is worth pushing the contrast one step further, because there is a version of the rarity claim that sounds paradoxical and is not.

At sixteen divisions, one lettering in a hundred agrees with itself. The number of admissible letterings of that pattern is astronomically large, so one per cent of it is still astronomically large — there is no shortage of consistent letterings, only a shortage of them among randomly drawn ones. A sampler struggles because it is sampling from the wrong distribution, not because the target is small in any absolute sense.

That is the same distinction that settled a question two thousand draws could not: a sampler measures a density and existence is not a density. Here the density is falling and existence is never in doubt, so the two instruments diverge steadily as the grid grows, with the sampler getting worse and the search staying exactly where it was.

The same reading on four other families #

The grid is the cleanest case and it is not a special one.

The tapered leaf pattern costs exactly one step per panel at every geometry. The crumples cost one step per panel less a constant, the constant being the last panel that needs no decision. The eight fold-and-cut patterns cost five steps on seven panels apiece. The four corrugation families — the Miura, the leaf, the Yoshimura and the waterbomb — cost twenty-four, twenty-eight, fifty-seven and forty-eight steps on twenty-four, twenty-eight, sixty-five and fifty-two panels.

Not one of them backtracks. Across forty-six patterns from five constructions the search’s first guess is never wrong, and the total number of withdrawn decisions in the whole collection of corrugations is zero.

Where a corrugation’s difficulty actually lives #

If none of this is difficult, the natural question is what a corrugation is hard about, and it has an answer.

It is hard about which rule to use. A corrugation is generated by a repeating instruction — a few bits saying which letter each row and column carries — and sweeping every such rule shows that most of them do not fold: sixteen of sixty-four for the Miura and its relatives, thirty-two of five hundred and twelve for the waterbomb. Choosing a rule is the difficulty, and it is a difficulty of design rather than of search.

It is hard about layers. The letters agreeing with themselves is a necessary condition and no more; whether the panels can actually be stacked is a separate question, and that one has no cheap answer at all.

And it is hard about scale, in the sense that matters to somebody folding one. A sixteen-division grid is two hundred and fifty-six panels of paper, and the difficulty of putting them all in the right place with two hands is not a difficulty any search here measures.

The one place the line bends #

Across five constructions and forty-six patterns the line is straight, and there is exactly one family in this collection where it is not: the clipped tessellation patches.

A patch cut out of an infinite tessellation is not a corrugation in the relevant sense. Its interior is regular and its boundary is not — most of a patch is edge, the vertices near the rim have creases running off the paper, and the propagation that sweeps cleanly across a corrugation stalls where the pattern stops. So the search has genuine choices to make in places a corrugation has none, and on one patch of five those choices produce a cost that varies by three orders of magnitude with the search’s own arrangement.

That is a useful boundary for the claim of this essay. A corrugation never backtracks because it is uniform, and uniformity is exactly what a clip destroys. The relevant property is not being a tessellation; it is not having an edge.

The number that is the same in two places #

There is a coincidence in the measurements above that is not one, and it is worth spending a paragraph on because it makes the ladder’s flatness less mysterious.

The grid’s step count equals its panel count. The leaf’s step count equals its panel count. The corrugations’ step counts equal their panel counts less a handful, and the handful is exactly the panels whose letters the boundary has already settled. Four different constructions, four different geometries, one arithmetic.

The reason is that all four produce patterns in which every interior vertex has degree four and the same local structure. A degree-four vertex with a strictly smallest sector has exactly two admissible labellings once one of its creases is known, which means one decision per vertex and no more — and the vertices tile the sheet one per panel. The arithmetic is not a fact about grids or leaves; it is a fact about degree-four vertices laid out regularly, and every construction here produces those.

The prediction that follows is testable and has been tested in one direction: the Yoshimura, whose vertices are degree six, breaks the equality — fifty-seven steps on sixty-five panels rather than sixty-five. A degree-six vertex has more admissible labellings and the propagation settles more of the sheet without branching, so the count comes out below the panel count rather than above it. The exception confirms the mechanism rather than the rule.

Which theorem was checked, and how #

Every lettering counted here is verified twice over. The search returns one, and it is then written back onto the pattern and put past the four conditions at every interior vertex by the pattern’s own reader, and past a folded sheet rebuilt from the coordinates and walked for a circle. A lettering the search believes and those two do not is a defect rather than a result.

The claim that the search never backtracks is asserted directly rather than inferred from the step counts: every pattern in every ladder is required to cost no more than one step per panel, and the assertion fails on the first pattern that does. That is a stronger statement than “the counts came out linear”, because a search could visit exactly n nodes while withdrawing decisions and taking others, and the requirement rules that out.

The share of drawn letterings that agree is measured by the sampler rather than by the search, and the two share no code — one draws letterings and tests them afterwards, the other tests while it chooses. Their agreement about which patterns have consistent letterings at all is what makes the pair of numbers a comparison rather than two readings of one thing.

What the picture cannot show #

A step count is a count of decisions, not of seconds. Each node propagates the vertex conditions to a fixed point and then tests the arcs, and both cost more on a large pattern — so a sixteen-division grid’s two hundred and fifty-six steps are a great deal more work than a two-division grid’s four, and the flat line in the figure is flat in decisions rather than in time. What it says is that the search is not growing, and the arithmetic underneath it is.

Nor does the ladder say a corrugation can never be difficult. It says these corrugations, generated by these constructions, are not — and the constructions all produce patterns whose vertices are regular and whose propagation is decisive. A corrugation with irregular sectors, or one built from a rule that nearly fails, might well behave differently, and the family that would test it has not been built.

What a designer takes from it #

A box-pleated design is drawn on this grid, and the practical reading is worth separating from the theoretical one.

The theoretical reading is that a consistent lettering always exists and is cheap to obtain. The practical reading is narrower and more useful: do not draw the letters by hand and hope. At sixteen divisions the chance that a hand-drawn assignment satisfying every vertex condition also has no circle in its arcs is about one in a hundred, and every one of the ninety-nine failures looks correct at every vertex. A designer checking their work vertex by vertex will pass a pattern that cannot fold, and will do so almost every time.

The corresponding good news is that the repair is not laborious. Since the search costs one step per panel and never backtracks, a lettering that works can be produced for any grid a designer is likely to draw in the time it takes to read the pattern once. The expensive part of box-pleating was never the letters.

There is a caution attached, and it is the one that runs through the whole subject. A lettering whose arcs close no circle is not a lettering that folds. Consistency among the letters is necessary and not sufficient, and a designer who takes a searched-for lettering as a guarantee has taken the wrong guarantee.

Where the ladder goes next #

The same measurement on a family that is not regular at all is the obvious next reading: a crumple has no tail either, which is a stronger statement than it looks, since a crumple is the least structured pattern this collection can produce and it costs the same one step per panel as the most structured one.