What a grid costs in circuits

Box-pleating is the technique that made complex origami design tractable. Put every crease on a square grid, build each flap from a rectangle of grid cells, and the awkward question of where the creases go becomes an arrangement of blocks — a design problem on a lattice rather than a construction problem on a continuum.

What the grid does to the letters has never been part of that account, and it turns out to do a great deal.

The ladder #

A grid of n divisions has n² panels, 2n(n−1) creases and (n−1)² interior vertices, and every one of those vertices has degree four with four right angles.

Drawing a hundred labellings from each, under the four conditions at every vertex, and asking how many have no circle in the arcs they force:

Two divisions: a hundred of a hundred. Three: ninety-seven. Four: ninety-four. Five: eighty-six. Six: seventy-three. Eight: fifty-eight. Ten: thirty-six. Twelve: fifteen. Fourteen: two. Sixteen: one.

That is not a gentle decline. It is flat to about six divisions, falls steadily to twelve, and then collapses.

The collapse is where the practical consequence sits. A reader trying this on squared paper will work at four or five divisions, where nine labellings in ten are fine and the whole question looks like a technicality. A designer works at sixteen or more, where it is one in a hundred, and nothing about the small case gives any warning that the large one is different in kind.

Why a grid is the worst shape for this #

A circle in the arcs needs a closed chain of panels to run round, and the number of independent closed chains is what decides how many places a contradiction could sit. On a grid that number is exactly the interior vertex count — one, four, nine, sixteen, up to two hundred and twenty-five at sixteen divisions.

The grid is the densest arrangement of those this collection draws. Every interior vertex of a grid is a four-panel ring, and the rings share every edge with their neighbours, so the sheet is circuits all the way across with no slack between them.

Compare the alternatives. A fold-and-cut outline has one to three chains, because a straight skeleton is a tree. A Miura has one per interior vertex too, but a Miura of realistic size has fifteen; a Yoshimura has twenty-two. A tessellation patch has thirty-six to a hundred and twenty-six, and that is the family this collection calls difficult.

A sixteen-by-sixteen grid has two hundred and twenty-five, which is more than any tessellation patch here.

One rate per chain, up to twelve divisions #

The ladder is not merely a decline; it is a decline with a shape, and the shape has one parameter.

Model each of a pattern’s $V$ chains as closing with probability $\varepsilon$, independently, so the share of labellings with no circle anywhere is $(1-\varepsilon)^{V}$. On a grid of $n$ divisions $V$ is $(n-1)^{2}$.

Fit $\varepsilon$ to the middle of the ladder and it comes out at 1.25 per cent. That single number then predicts: 89 per cent at four divisions against a measured 94, 82 against 86 at five, 73 against 73 at six, 54 against 58 at eight, 36 against 36 at ten. Two exact hits and three within a standard error of a hundred draws.

So up to twelve divisions a grid is one number, and the number is a per-chain probability of the letters closing.

Where the collapse actually is #

Past twelve the model breaks, and the break is the finding the ladder was pointing at.

At fourteen divisions the independent model predicts 12 per cent and the measurement is 2. At sixteen it predicts 6 and the measurement is 1. Those gaps are five and seven standard errors — far outside what a hundred draws can produce by chance in the other direction.

So the collapse is not the exponential finally biting. It is the chains ceasing to be independent, somewhere between twelve and fourteen divisions, and beginning to conspire: a closure on one chain making a closure on its neighbours more likely rather than less. Below that size a grid’s chains are effectively separate problems; above it they are one problem.

That is a sharper account than “it collapses”, and it is testable — a sweep at thirteen, and at fourteen with a thousand draws rather than a hundred, would locate the onset rather than bracket it.

Which also settles the comparison with a twist patch #

The same parameter disposes of the factor of ten at equal chain count, and disposes of it exactly.

A twist tessellation patch’s per-chain rate, fitted the same way from its own ladder, is 5.5 per cent — four and a half times a grid’s. At thirty-six chains the two models give $(1-0.0125)^{36} = 64$ per cent and $(1-0.055)^{36} = 13$ per cent.

The measurements are seventy in a hundred and thirteen in two hundred.

One number per family reproduces both. So the grid’s chains are looser than a twist patch’s in a way that is now quantified rather than merely plausible: not looser in some unmeasured sense, but by a factor of four and a half in the chance any one of them closes. What produces that factor is still unmeasured, and it is the right next question — but the quantity to explain is a single ratio rather than a pair of curves.

A useful coincidence falls out of it too. The Miura’s fitted rate is 1.27 per cent, against the grid’s 1.25. Two patterns of degree-four quadrilateral vertices, arrived at independently, with the same per-chain rate to two figures — which is what one would expect if the rate is a property of the vertex arrangement rather than of the pattern, and is the first evidence here that it is.

And why it is not the worst in practice #

Chain count is not the whole story, and the grid is the clearest demonstration in this collection that it is not.

A seven-by-seven grid has thirty-six independent chains, which is exactly what the square tessellation patch has. Their consistency shares are seventy in a hundred and thirteen in two hundred — a factor of ten apart, at the same chain count.

So a grid’s chains are, in some sense not measured here, looser than a twist patch’s. A plausible account is that a grid’s chains are all four panels long and each shares an edge with at most four neighbours, while a twist patch’s chains are the polygons and pleats and each interlocks with several others at different lengths; but nothing in this collection measures chain overlap, and the account is offered as consistent with the numbers rather than established by them.

What is established is the numbers. Chain count orders patterns within a family and does not compare across families, which was already found in a corrugation and is found again here from the other end.

What this means for a designer #

The practical consequence is narrow and worth stating carefully, because a box-pleated design is not an unlettered grid.

A designer working in box pleats does not choose letters freely. The pattern comes with a structure — flaps, rivers, hinges, the axial creases that separate them — and much of the assignment follows from what each region is doing. A grid’s full labelling space is not the space a designer is choosing from.

But some of it is. Where a design has a region whose letters are not forced by its function, a designer fills them in, and the measurement above says what filling them in blind is worth at design scale: about one attempt in a hundred, on the whole grid, with the failure invisible until the model is nearly closed.

That is not a reason to distrust box-pleating, which plainly works. It is a reason to notice that what makes it work is the structure — the flap layout constraining most of the letters — rather than any looseness in the grid itself.

What the fall looks like as a curve #

The shape of the decline is worth reading rather than only its endpoints, because it says which quantity is driving it.

If each independent chain were an independent chance of failure, the consistent share would fall geometrically in the chain count: some fixed probability per chain, multiplied together. The chain counts here are one, four, nine, sixteen, twenty-five, forty-nine, eighty-one, a hundred and twenty-one, a hundred and sixty-nine and two hundred and twenty-five, so a geometric fall would show as a straight line against those numbers.

It does not quite. The share is ninety-four per cent at nine chains and fifty-eight at forty-nine, which is a per-chain survival of about ninety-nine per cent; from forty-nine to two hundred and twenty-five it goes fifty-eight down to one, which is about ninety-eight per cent per chain. Close enough to geometric that the model is not embarrassed, and not so close that anything should be read off it.

What the near-geometry does say is that the chains are behaving roughly independently, which is the property a grid has and a twist patch conspicuously does not. On a twist patch the same arithmetic would predict a far higher share than the thirteen per cent measured, and the gap is where the interlocking sits.

What a search costs on the same object #

The other half of the table is the part that does not move.

At sixteen divisions the search finds a consistent labelling in two hundred and sixty-one steps, on a pattern with two hundred and fifty-six panels, having taken back five letters. That is one node per panel plus five, which is what a search looks like when it barely searches.

At every smaller size it is the same: four nodes for four panels, sixteen for sixteen, a hundred for a hundred, with at most one backtrack anywhere below twelve divisions.

So the grid at design scale is a pattern where a random labelling almost never works and a directed one always does, immediately. Rarity and difficulty are different quantities and this is the sharpest instance of it in the collection: a share of one per cent, and a search that never really has to search.

The reason is the same as everywhere else. The search tests the arcs while choosing the letters, so it never enters the bad region; the draw chooses under the vertex conditions alone and finds out at the end.

What the grid shares with a map #

There is a second family in this collection built on exactly the same graph, and putting the two measurements side by side sharpens both.

A map is a grid of paper scored on its lines, and its panel graph is the grid’s panel graph. On the three-by-three map — nine panels, twelve creases, four interior vertices — every labelling can be enumerated rather than sampled: two hundred and fifty-six satisfy every vertex condition, and four of them close a circle.

Four in two hundred and fifty-six is one and a half per cent, and the sample at the same size gives three per cent. Those agree about as well as a hundred draws can be expected to, which is the check that makes the sampled ladder above worth reading: at the one size where an exact answer exists, the sampler is not far off it.

That is the whole warrant for the numbers at twelve and sixteen divisions, where no exact answer is available and none will be — a sixteen-by-sixteen grid has four hundred and eighty creases and its labelling space is not a thing that gets enumerated.

The grid as an idealisation #

Three things about the object measured here are not true of a real box-pleated design, and each of them matters for how far the numbers travel.

Every crease is present. A design uses a subset of the grid’s lines; a full grid has creases everywhere, so it has the maximum number of interior vertices for its size. A design with half the creases has fewer chains and would sit higher on the consistency ladder.

No crease is a boundary. A real design has flaps whose edges are the sheet’s edge, and a vertex on the rim carries no closed chain of panels at all. The grid measured here has its rim vertices excluded already, but a design’s interior is more broken up than a grid’s.

The diagonals are missing. Box-pleating at forty-five degrees uses the grid’s diagonals as well as its lines, which changes the vertex degrees from four to six or eight in places, and a vertex of higher degree admits more labellings. Whether that raises or lowers the consistent share is not measured here and is not obvious.

So the ladder is a bound of a particular kind: it is what the bare grid does, and a design is a grid with pieces taken out and pieces added.

The corner cases at the ends of the ladder #

Both ends of this ladder are worth a sentence, because both are degenerate in instructive ways.

At two divisions the grid has four panels, four creases and one interior vertex, and every one of a hundred labellings is consistent. That is not luck: no admissible labelling of a single vertex closes a circle at any degree, so a pattern with one interior vertex cannot produce a contradiction at all. The hundred per cent is a theorem rather than a measurement.

At sixteen divisions the one surviving labelling in a hundred is close to the point where the sampler stops being able to say anything. A share of one per cent means a hundred draws produce one clean labelling on average, and a share of a tenth of a per cent would produce none — at which point a nought would report an empty set that is nothing of the kind, which is the failure this collection has learnt to watch for.

So the honest statement about the twenty-four-by-twenty-four grid, which is a perfectly ordinary size for a design, is that its share is below what a hundred draws can measure and its search still takes about one node per panel. The first half of that is a limitation of the instrument and the second is a fact about the object.

Where the two divisions meet #

There is one point where this measurement touches something the design tradition already knows, and it is worth naming because the agreement is not trivial.

Box-pleated designs are conventionally worked out on grids of sixteen, thirty-two or sixty-four divisions, and every account of the technique stresses that the assignment is decided by the structure — the flap and river layout — rather than crease by crease. A designer who fills in letters locally, working outward from a region, is doing something the tradition warns against.

The measurement says why the warning is right, in a language the tradition does not use. At sixteen divisions the labelling space is so thin that local choices made without regard to the whole will not compose; at four divisions they will, which is why the technique feels easy on a small example and is not.

The other grid this collection draws #

There is a second grid in the design field and it behaves differently enough to be worth a paragraph, because it is the one a designer would reach for when the square grid runs out.

A triangular grid puts six creases at every interior vertex instead of four, which changes two things at once. Its shortest closed chain is six panels rather than four, so a contradiction has further to run; and each of its vertices admits thirty labellings of its own creases rather than eight, so the labelling space is much larger per vertex.

Neither effect is measured on it here, and the two pull in opposite directions — more room per vertex, longer chains to close. What the corrugation families suggest is that the longer chain matters more: a family whose vertices are all of degree six closes no four-panel circle under any repeating rule, because the shortest circle is simply unavailable to it.

If that carries over, a triangular grid should be markedly more forgiving of a hand-chosen labelling than a square one at the same number of divisions. It is a one-line measurement with the machinery already built, and it is the first thing this rung leaves undone.

Where the ladder goes next #

The obvious extension is to measure a real base rather than a bare grid: take a box-pleated pattern with its flaps and rivers, count its chains, and draw labellings from it. The prediction from above is that it sits well above the bare grid’s line, because a design’s structure removes creases and adds boundary.

The more interesting extension is the diagonals. A forty-five-degree box-pleated pattern has vertices of degree four, six and eight, and the degree decides how many labellings a vertex admits — eight of sixteen at degree four, thirty of sixty-four at six, a hundred and twelve of two hundred and fifty-six at eight. More options per vertex and longer chains pull in opposite directions, and nothing here says which wins.