Origami & Folding
what one sheet can do
Designing a base

Ninety-nine in a hundred pass

A designer checks a box-pleated pattern the way every text teaches: vertex by vertex, counting mountains and valleys, watching the smallest sector. At sixteen divisions that check passes a hundred letterings in a hundred, and one of them folds. The check that separates them costs a single sweep over the crease list and is in no recipe anywhere.

Assumes What a grid costs in circuits and Designing on a grid.

21 min read 9 figures Local rules, global behaviourFlat is rare

A box-pleated design lives on a square grid, and by the time it is finished the grid carries a great many creases. At sixteen divisions there are two hundred and fifty-six panels, four hundred and eighty creases and two hundred and twenty-five interior vertices, and every one of those vertices has to be given letters that satisfy the conditions at a point.

That is the check every account of the technique teaches, and it is the check a careful designer performs: go round the pattern, count the mountains and valleys at each vertex, make sure they differ by two, make sure the two creases either side of the smallest sector are not the same letter. It is a real check, it catches real errors, and it is what a schoolteacher’s theorem is for.

At the scale a box-pleated design is actually drawn at, it passes essentially every lettering that cannot be folded.

Box pleatingDesigning on a grid, with every crease running along a grid line or at forty-five degrees to it. It gives up the efficiency of a free circle packing and gains something worth more for complex work — the creases meet where they are supposed to, and the errors do not accumulate.16 × 16 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley
Fig. 1 Sixteen divisions: two hundred and fifty-six panels and two hundred and twenty-five interior vertices, each of which has to satisfy the conditions at a point, and every one of which can be made to.

The measurement

Draw a hundred letterings of the sixteen-division grid, each one satisfying every condition at every one of the two hundred and twenty-five vertices — not random letters, but letterings produced by propagating the conditions and choosing at random only where they stop deciding, so that every draw passes the designer’s check everywhere by construction.

Then ask a different question of each: do the statements those letters make about which panel lies above which contradict one another?

One of the hundred survives. The other ninety-nine carry a circle — a chain of panels each of which the letters place above the next, closing back on itself, which is a proof that no stacking exists.

What a grid costs in circuitsThe orthogonal grid a box-pleated design is laid out on, at five sizes, with a hundred letterings drawn at random from each. The share that agrees with itself falls steadily as the grid grows, because every interior vertex of a grid carries the shortest circuit a panel graph can have.the bar is how many of a hundred random letterings agree with themselveson the orthogonal grid a box-pleated design is drawn on, at five sizes2 by 21001 interior vertices · 4 panels · found in 4 nodes4 by 4949 interior vertices · 16 panels · found in 16 nodes6 by 67325 interior vertices · 36 panels · found in 37 nodes8 by 85849 interior vertices · 64 panels · found in 65 nodes12 by 1215121 interior vertices · 144 panels · found in 145 nodes16 by 161225 interior vertices · 256 panels · found in 261 nodesevery interior vertex is a four-panel circuit, so the number of places a contradiction could sit is the number of vertices
Fig. 2 The same measurement at six grid sizes. At two divisions every lettering that passes the vertex check also folds; by sixteen, one in a hundred does.

The share falls the whole way: a hundred of a hundred at two divisions, ninety-four at four, seventy-three at six, fifty-eight at eight, thirty-six at ten, fifteen at twelve, two at fourteen, one at sixteen. A designer working at four divisions is nearly always right; one working at sixteen is nearly always wrong; and nothing in the check they are performing changes between the two.

It is worth pausing on what those hundred letterings are, because the number would mean something quite different if they were random letters.

Writing letters on a sixteen-division grid at random would produce something admissible about once in every four million attempts — the conditions at two hundred and twenty-five vertices are a great deal to satisfy by luck. So a random-letters measurement would say nothing about the designer’s check, because a random lettering fails it immediately.

What is drawn instead is a solution: propagate the conditions until they stop deciding anything, branch where they have stopped, and take the branch at random. Every draw is a lettering the designer’s check passes everywhere, and two draws differ because the branches differed. That is the population the ninety-nine per cent is measured over, and it is exactly the population a careful designer’s own work lives in.

Which condition the search actually backtracks onFor each of five patches, how many times the search took a letter back and what refused it. Every backtrack is a loop in the arcs the letters force. The four conditions at a vertex — developability, Kawasaki, Maekawa and the big-little-big lemma — refuse nothing at all, because they are propagated into the choice rather than tested after it.the bar is how many times the search took a letter backand every one of those was the arcs closing a loop, never a vertex running out of labellingsthe square patch126 nodes · 1 refused by the arcs · 0 by the vertex conditionsthe elongated patch335 nodes · 3 refused by the arcs · 0 by the vertex conditionsthe hexagonal patch241 nodes · 2 refused by the arcs · 0 by the vertex conditionsthe triangular patch747 nodes · 7 refused by the arcs · 0 by the vertex conditionsthe rhombille patch246561 nodes · 246 refused by the arcs · 0 by the vertex conditionsthe vertex conditions are propagated rather than tested, so they narrow the choice instead of refusing it
Fig. 3 Which of the four conditions does the refusing, over the same patterns. The gap the check leaves is not spread evenly across the conditions — nearly all of it is one of them, which is why a designer can pass ninety-nine in a hundred and still be wrong about the hundredth.

Why the check gets worse as the design gets bigger

The vertex conditions are local, and a circle is not.

Every interior vertex of a grid is a four-panel circuit — four panels meeting round a point — so each vertex is a place a contradiction could sit. A two-division grid has one such place; a sixteen-division grid has two hundred and twenty-five. A lettering has to avoid all of them at once, and the chance of managing it by satisfying local conditions alone falls roughly geometrically.

That is local rules and global behaviour at the scale a designer works at rather than in principle. The conditions at a point are necessary and they were never sufficient; what changes with size is how much of the difference they miss.

The unwelcome part is that the check gets less informative exactly as a design gets more ambitious. Small patterns, where a designer could fold a test model in five minutes, are where the check is reliable. Large ones, where folding a test model is an evening’s work, are where it is not.

What a witness costs, against how rare one is5 tessellation patches. The bar is how many nodes a search visits before returning a lettering whose arcs have no loop in them; the note beside it is how many of 200 letterings drawn at random from the same pattern turn out to agree with themselves. 2 of them were never given one by the draws.the bar is how many nodes the search visited before it found onethe note is how many of the same pattern's random letterings agree with themselvesthe square patch2649 panels · 26 of 200 drawn letterings agreethe elongated patch3562 panels · 5 of 200 drawn letterings agreethe hexagonal patch4177 panels · 2 of 200 drawn letterings agreethe triangular patch4783 panels · 0 of 200 drawn letterings agreethe rhombille patch561157 panels · 0 of 200 drawn letterings agreea search that stops at the first witness; nothing here counts how many there are
Fig. 4 What the missing check costs to satisfy rather than to run: the work of finding a consistent lettering against how rare one is. The rarity is what moves as a design grows, and the work follows it.

The check that is missing

The test that separates the ninety-nine from the one is short enough to state in a sentence and it is in no recipe anywhere.

Each crease, given its letter, says which of the two panels it separates ends up on top. Collect those statements — one per crease — and look for a cycle among them: a set of panels each placed above the next, closing back on itself. That is one pass over the crease list, it needs no geometry beyond what the pattern already carries, and a cycle in it is a proof that no ordering of the panels works.

It is not a proof of foldability. A lettering with no cycle can still fail the two non-crossing rules, and consistency among the letters is necessary and not sufficient. But it is the cheapest test that catches ninety-nine per cent of what the vertex check misses on a grid, and its cost is a single sweep.

A lettering of the square patch that agrees with itselfThe square tessellation patch, lettered by a search that tests the arcs the letters force at every step rather than after every letter is chosen. Mountain and valley are distinguished by colour and by dash. Every panel of the folded sheet can be ordered consistently with these letters, which is not true of the lettering the construction itself produces.a lettering of the patch that agrees with itselffound by testing the arcs while the letters were chosen, not after26 nodes · 1 backtracks · verified against a rebuilt folded sheet49 panels · 84 creasesits own lettering has no loop in it26 of 200 random letterings agree with themselvesthis one was found in 26 nodes and 1 backtracksit differs from the drawn lettering on 45 of 84 creasesthe drawing is the pattern; nothing here is a picture of the folded object
Fig. 5 The check that is missing, done once: a lettering of the square patch found by search and then put past the two tests that did not find it. The arrows it forces carry no circle, which is the statement a grid of this size is never asked for.

There is a second reason the check degrades, and it is about what a circle is rather than about counting places.

A cycle among the panels can be long. On a sixteen-division grid the shortest possible one is four panels round a single vertex, and those a designer might spot; but a cycle can run through twenty panels scattered across the sheet, each of which is placed correctly with respect to its own neighbours. Nothing local is wrong anywhere along it. The contradiction only exists as a property of the whole chain, and a chain of twenty is invisible to any inspection that looks at one vertex at a time — which is every inspection a person performs.

So the check misses more as the pattern grows both because there are more places to fail and because the failures get longer, and neither is something a designer can compensate for by being more careful.

The check carries no evidence, which is worse than being unreliable

There is a way of stating the sixteen-division result that is sharper than “one in a hundred”, and it is the one that says why care cannot help.

A test is worth something when passing it is more likely for a good pattern than for a bad one. The ratio of those two likelihoods is what the test is worth. Here, every draw passes the vertex check by construction — the letterings were produced by propagating those very conditions — so the ratio is exactly one, and a likelihood ratio of one is zero bits of evidence.

Not a weak test. No test. A designer who inspects all two hundred and twenty-five vertices twice knows exactly what a designer who inspects none of them knows, because the letters were written to pass and they do.

Which is why repetition does not rescue it

That distinction matters because it defeats the obvious workaround. If the check merely had a high false-pass rate, drawing several letterings and keeping the ones that pass would improve the odds. It cannot: all of them pass, so the designer chooses among a hundred indistinguishable candidates and is right one time.

It also sizes what is needed. A one per cent prior has to be moved to something a person would bet a weekend on, and reaching even even-odds takes log⁡299=6.6\log_2 99 = 6.6 bits of evidence — while reaching the 95% a designer actually wants takes about 11.

The cycle test supplies essentially all of them in a single sweep, because it fails ninety-nine of the hundred and passes the one. That is the useful way to compare the two: not that one check is local and the other global, but that one carries no information about the question and the other carries nearly all of it, at the same cost as reading the pattern once.

What a designer should actually do

Three things, in increasing order of what they cost.

Do not letter a large grid by hand. At sixteen divisions the chance of a hand-drawn admissible lettering also being consistent is about one in a hundred, and every one of the ninety-nine failures looks correct at every vertex a designer inspects. This is not a matter of care.

Search for the letters instead. A consistent lettering of any grid a designer is likely to draw is found at one step per panel with no wrong guesses anywhere: the grid ladder costs exactly its own size, four steps at two divisions and two hundred and fifty-six at sixteen. The letters were never the expensive part of box-pleating.

Fold a test model anyway, because the cycle test is necessary and not sufficient, and the remaining gap — the non-crossing rules, which is where the two hundred and fifty-six panels actually have to fit past one another — has no cheap test at all.

The same gap, one level up

The cycle test is the missing check for the letters. It is worth saying plainly that there is a further missing check, and that this collection has no cheap version of it either.

A lettering with no cycle says the statements about which panel lies above which are mutually consistent. It does not say the panels can actually be arranged: two more rules — one forbidding a panel from sitting between two panels folded round a common crease, one forbidding two such pairs from interleaving — have to hold as well, and testing them requires searching over orderings rather than reading the crease list. On a pattern of thirteen panels that search visits over a million nodes; at two hundred and fifty-six panels it is not available at all.

So the honest ladder of checks for a box-pleated pattern is three rungs long and the third is missing at scale. The vertex check catches gross errors and passes ninety-nine in a hundred failures. The cycle test catches almost all of what is left, in one sweep. And whether the panels fit past one another is answered by folding a test model, because nothing else can answer it.

Refused at one lettering is not refusedSix developable quadrilateral meshes, each with every labelling of its creases enumerated and every consistent one put to a search over orderings of its nine panels. Two of the meshes fold at no labelling whatever. Two others were refused at the labelling they were built with and fold at others.the bar is how many letterings of the mesh can have their panels stackedout of every labelling of its twelve creases, enumeratedmesh 3016 pass every vertex · 16 agree with themselves · arrived refusedmesh 5032 pass every vertex · 32 agree with themselves · arrived refusedmesh 8832 pass every vertex · 32 agree with themselves · arrived refusedmesh 11832 pass every vertex · 32 agree with themselves · arrived foldablemesh 141416 pass every vertex · 14 agree with themselves · arrived foldablemesh 19416 pass every vertex · 16 agree with themselves · arrived refusedtwo of the meshes have none at all, and two more were refused only at the lettering they came with
Fig. 6 The third rung, on patterns small enough for it to run: six meshes whose letters are all consistent, four of which have no stacking. The cheap test passes all six.

Why the technique works despite all this

A reasonable objection: box-pleating is a working technique, designs get folded, and if ninety-nine per cent of letterings failed nobody would get anywhere.

The answer is that a designer does not draw a lettering at random from the admissible ones. A box-pleated pattern is built from pieces with known letters — a flap here, a river there, a hinge somewhere — each of which has been folded before and each of which brings its assignment with it. The whole is assembled from parts that already fold, and assembly is far more likely to preserve consistency than a fresh assignment is to achieve it.

So the measurement above is not a statement about how designers work. It is a statement about what their check would catch if they were wrong, and the answer is almost nothing. The technique survives on the reliability of its components rather than on the check applied to the result.

Rounding a design is not rounding its limbsThe worst limb error against how fine the grid is, for the obvious rounding and for the best whole-number version there is. The obvious one is not monotone — a finer grid can round worse — and the best one is flat over a wide range, because the same coarse set of whole numbers goes on being the best answer.42:3:4:363:4:6:583:4:6:5125:7:11:8165:7:11:8248:11:17:13each limb roundedthe best whole numbershow wrong the worst limb isgrid units across the longest limbthe numbers under the axis are the best whole-number limbs at that resolution
Fig. 7 How a box-pleated pattern is really assembled: from components whose letters are known, rather than by lettering a blank grid. That is why the technique works and why the check that would catch a mistake almost never fires.

That has a consequence worth stating. The moment a designer does something genuinely new — grafts two patterns that have not been combined before, or modifies a component’s letters to fit — they leave the regime where the components carry the guarantee, and the check they fall back on is the one that passes ninety-nine unfoldable patterns in a hundred.

What this says about the recipes

Every published account of box-pleating this collection has looked at teaches the vertex check and none teaches the cycle test, and it is worth being fair about why.

The vertex check is teachable. It is local, it is visual, it can be demonstrated on a diagram of one point, and a reader can apply it with a pencil. The cycle test is a sweep over a data structure — collect one statement per crease, look for a cycle — and while it is not difficult, it is not something a person does comfortably on paper at two hundred and fifty-six panels.

The check that is taught is therefore the check that can be taught, and its inadequacy at scale is invisible from within the teaching, because the examples in a book are small and at small sizes the check is fine. Half the recipe is decoration is the same shape found in a corrugation’s two-instruction rule: a recipe survives because it is applied to cases where its gaps do not bite.

What a grid costs in circuitsThe orthogonal grid a box-pleated design is laid out on, at five sizes, with a hundred letterings drawn at random from each. The share that agrees with itself falls steadily as the grid grows, because every interior vertex of a grid carries the shortest circuit a panel graph can have.the bar is how many of a hundred random letterings agree with themselveson the orthogonal grid a box-pleated design is drawn on, at five sizes2 by 21001 interior vertices · 4 panels · found in 4 nodes3 by 3974 interior vertices · 9 panels · found in 9 nodes4 by 4949 interior vertices · 16 panels · found in 16 nodes5 by 58616 interior vertices · 25 panels · found in 26 nodes6 by 67325 interior vertices · 36 panels · found in 37 nodesevery interior vertex is a four-panel circuit, so the number of places a contradiction could sit is the number of vertices
Fig. 8 The sizes a textbook uses. At two to six divisions the vertex check is right between seventy-three and a hundred times in a hundred, which is why nothing in the literature records that it fails.

Which theorem was checked, and how

The letterings counted here are drawn by propagating the four conditions and branching at random where they stop deciding, so every one of them passes Kawasaki, Maekawa and the big-little-big lemma at every interior vertex by construction rather than by inspection. That is a stronger version of the designer’s check than any designer performs, and it is the one that ninety-nine per cent of the failures pass — the same distinction between drawing a lettering and searching for one that separates a density from an existence.

The consistency test is run separately, on a folded sheet rebuilt from the coordinates and walked for a cycle, by machinery that shares no code with the drawing. And the one lettering in a hundred that survives is verified again the other way round: written back onto the pattern and put past both instruments.

The share is asserted to be falling rather than merely reported, because a figure claiming that a check gets worse with scale is worthless if the underlying numbers ever stop doing so.

Four patches the search walks through, and one it does notThe same search run from a hundred and twenty different seeds on each of five patches, and the middle result. Four of the patches cost between twenty-five and fifty-six nodes whatever the seed. The fifth runs from eighty-four nodes to past the budget, on the same pattern and the same code.the bar is the middle run of a hundred and twentysame pattern, same code — only the order the letters are tried in differsthe square patch2725 at best · 27 at the middle · 36 at worstthe elongated patch3432 at best · 34 at the middle · 39 at worstthe hexagonal patch4339 at best · 43 at the middle · 51 at worstthe triangular patch4539 at best · 45 at the middle · 53 at worstthe rhombille patch16684 at best · 166 at the middle · 48 of 120 unfinished at 20000an unfinished run is left out of the middle rather than counted as its budget
Fig. 9 The same falling share read across patches rather than across grid sizes, a hundred and twenty searches each. What falls is what a drawn lettering is worth; what does not move is the work of finding one that is worth something.

What the picture cannot show

Which ninety-nine. The failures are not clustered anywhere a designer could learn to watch — the circle can be anywhere in the pattern, it involves panels that need not be adjacent in the drawing, and there is no visual signature. That is precisely why a sweep over the crease list is the right instrument and inspection is not.

Nor does one in a hundred mean a designer’s pattern has a one per cent chance of folding. It means a lettering drawn from the admissible ones does, and a designer’s lettering is not drawn from them — it is assembled, for the reasons above. The measurement bounds what the check is worth, not what the designer’s work is worth.

The number a designer should carry

If one thing survives from all of this into practice, it should be a ratio rather than a rule.

Two hundred and twenty-five to one. That is the number of independent places a sixteen-division grid can contradict itself against the number of vertices a designer inspects at a time, and it is the whole reason a local check cannot scale. It is also computable for any grid before a single crease is drawn: an n by n grid has (n−1)² interior vertices and therefore that many four-panel circuits, and the share of admissible letterings that fold falls roughly as that count grows.

A designer who knows the number knows when the check they were taught stops being worth performing, and the answer is somewhere around eight divisions — where the share is already down to fifty-eight in a hundred, so the check is closer to a coin toss than to a verification.

Below that, inspect. Above it, sweep.

What the one in a hundred looks like

It is worth asking what distinguishes the lettering that survives, because a designer would reasonably want to know.

Nothing visible. The consistent lettering of a sixteen-division grid is not more symmetric than the ninety-nine that fail, does not have a different mountain-to-valley ratio, and does not concentrate its letters anywhere a reader could point to. It is one member of a set defined by a global property, and global properties do not leave local signatures.

That is worth saying because the natural response to “one in a hundred” is to look for the pattern in the one. There is not one, and the reason is the same reason the vertex check fails: the property that separates them is a statement about chains of panels rather than about any place on the sheet. A rule of thumb for drawing the right lettering would be a local rule for a global property, and no such rule can exist.

The practical form of that is a small relief. A designer does not have to learn anything new about how to letter a grid — they have to stop lettering it by inspection and sweep the crease list instead, which is a change of procedure rather than of skill.

Where the ladder goes next

The grid is a sheet with an edge, and the edge turns out to matter more than anything on the inside: what makes a search hard is not structure but a boundary, which is the one property separating every pattern in this collection that searches easily from the handful that do not.

Where this goes next

the bar is how many of a hundred random letterings agree with themselveson the orthogonal grid a box-pleated design is drawn on, at five sizes4 by 4949 interior vertices · 16 panels · found in 16 nodes6 by 67325 interior vertices · 36 panels · found in 37 nodes8 by 85849 interior vertices · 64 panels · found in 65 nodes10 by 103681 interior vertices · 100 panels · found in 100 nodes12 by 1215121 interior vertices · 144 panels · found in 145 nodes16 by 161225 interior vertices · 256 panels · found in 261 nodesevery interior vertex is a four-panel circuit, so the number of places a contradiction could sit is the number of vertices What a grid costs in circuits Box-pleating puts every crease on a square grid, and a square grid is the shape with the most short circuits per panel that this collection draws. On the sixteen-by-sixteen grid a designer actually works on, one mountain-valley labelling in a hundred agrees with itself. A search still finds one in two hundred and sixty-one steps. 32 × 32 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley Designing on a grid Box pleating gives up the efficiency of a free packing and buys creases that land where they are supposed to. For a design with hundreds of folds that is not a compromise — it is the only thing that makes it foldable. each point is one pattern: panels across, nodes up00100100200200one node per panelnodes visitedpanels2 by 2 to 16 by 16, and not one backtrack anywhere in the family A corrugation never backtracks As a box-pleating grid goes from two divisions to sixteen, the share of random letterings that agree with themselves falls from a hundred in a hundred to one. The cost of finding one that does stays at exactly one step per panel — four, nine, sixteen, twenty-five, and two hundred and fifty-six — with not a single wrong guess anywhere in the family.

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AssignmentBox pleatingDesign techniqueGridInterior vertexLayer orderingMaekawa's theoremNecessary condition