The whole alphabet of a grid

Box pleating is usually defended as a trade. A free circle packing is more efficient; a grid gives that up and buys creases that land exactly where they should, which for a design with hundreds of folds is not a compromise but the only thing that makes it foldable. What the grid settles adds a second purchase: on a lattice the best packing is a finite question with an answer, and off it nobody knows the best packing of six circles in a square.

There is a third, it is stronger than either, and it is not usually stated at all.

Six vertices #

Box pleating puts every crease along a grid line or along a diagonal of the grid. There are four such directions, so every sector at every interior vertex is a whole multiple of forty-five degrees.

That is a much stronger restriction than a restriction on positions. A vertex has to be developable — its sectors sum to a full turn — and it has to satisfy Kawasaki — its alternating sums are equal. Ask for both, with every sector a multiple of forty-five, and enumerate: at degree four there are three vertices, at degree six two, at degree eight one, and that is the complete list.

Six kinds of vertex, and a box-pleated design of any size is built entirely out of them.

The alphabet is short because the constraints compound. Developability is one equation, Kawasaki is another, the entries are positive integers in units of forty-five, and the compositions of eight units into four, six or eight positive parts with equal alternating sums are very few. There is no cleverness in the count; it is an exhaustion, and it terminates immediately.

There is a way of feeling how strong the restriction is without doing the arithmetic. A general flat-foldable four-crease vertex is two free numbers — a continuum of vertices, no two of them alike. On a forty-five degree grid that continuum collapses to three points. Every degree-four vertex in every box-pleated design ever drawn is one of those three, and which three they are was fixed the moment somebody decided to work on a grid.

What each letter costs #

The catalogue is worth more than its length, because the six are not interchangeable.

Three of the six are clean. The 45–90–135–90 vertex admits four markings, folds in all four, and each marking names one object. The square vertex — four right angles — admits eight, folds in all eight, one object each. The eight-crease rosette admits 112, folds in all 112, and there the cleanliness stops: only sixteen of those markings name a single object, and thirty-two of them name four.

Three are not clean. The 45–45–135–135 vertex admits eight markings and folds in six — two of the eight satisfy every condition in the subject and have no folded state, because with two smallest sectors equal there is no forced crimp. Both degree-six letters are worse: each admits thirty markings and folds in twenty and eighteen respectively.

And the multiplicity runs the other way. The 45–45–90–45–45–90 vertex — the waterbomb’s — has markings that fold six different ways each, so a design containing one has six objects where its pattern shows one.

The two columns therefore measure different failures and a letter can be good at one and bad at the other. A vertex with a gap accepts markings that cannot be folded, which is a checker’s problem: a design built from it can pass every condition and be unmakeable. A vertex with multiplicity accepts markings that can be folded several ways, which is a folder’s problem: the design is makeable and the pattern does not say which model it is. The square vertex has neither. The rosette has only the second. The waterbomb’s letter has a great deal of both.

Another grid, and how much worse it gets #

The restriction is a property of the angle rather than of the idea of a grid. Change the quantum and the alphabet changes size dramatically.

At sixty degrees there are two letters: the tied four-crease vertex and the six-fold vertex with every sector equal. At ninety there is one, the square vertex. At thirty there are thirty, and the catalogue stops being an alphabet and becomes a list.

So forty-five degrees is a compromise in a sense nobody usually names: it is large enough to keep the vocabulary to six and small enough that six is enough to design with. Thirty degrees buys expressiveness and pays for it with a vocabulary five times larger, half of whose letters carry a gap between what the conditions accept and what folds — including one whose worst marking reaches seventy-one distinct folded states.

Which theorem was checked, and how #

The enumeration is over compositions: every sequence of positive integers summing to the number of quanta in a full turn, of even length, with equal alternating sums. Each sequence is reduced to a canonical form under rotation and reflection, because a vertex turned round is the same vertex and a vertex mirrored is the same vertex.

Each surviving kind is then run through the two machines this site keeps for the purpose. The first is the site’s own vertex checker, unaltered — the same four conditions every figure here is gated on. The second is an exhaustive search for a stacking of the sectors, put past the three non-crossing rules, sharing no code with the first. The gap column is the difference, the stacking column is the count of legal orderings, and both come from the same run.

The check that matters is negative. If the enumeration had produced a seventh forty-five degree vertex, or had missed the square one, the catalogue would be wrong in a way nothing else here would notice — so the count is verified at three quanta, at two maximum degrees, and against the vertices this site’s own patterns actually contain.

One more count is worth reporting because it makes the thirty-degree comparison concrete rather than rhetorical. Of the thirty letters a thirty-degree grid admits, twenty-two carry a gap, and the gaps run from two markings to seventy-six. Of the six a forty-five degree grid admits, three carry a gap and the largest is twelve. So the finer grid is not merely five times as large a vocabulary: it is a vocabulary in which nearly three quarters of the letters are ones a checker built from the four conditions gets wrong, and gets wrong by more.

The multiplicities behave the same way. The worst forty-five degree letter reaches six folded objects for a single marking; the worst thirty-degree letter reaches seventy-one. A design containing a handful of those has, on paper, one pattern and a number of possible models that runs into the thousands.

One further reading is worth setting down, because it changes what the catalogue is for. A designer does not choose vertices; they choose a tree, a packing and a grid, and the vertices arrive as a consequence. So the alphabet is not a menu — it is a list of everything that can happen, which is a different and more useful kind of object: it means that anything a box-pleated design does at an interior point has already been enumerated, checked and given its two numbers. There is nothing left to discover about box pleating’s vertices, only about how they are put together.

That is a rare position to be in. Almost every other question in this subject is about an infinite family whose members have to be checked one at a time.

Where the model stops #

The catalogue is of vertices, not of designs, and a design is a great deal more than a bag of vertices. Which letters can sit next to which, how many of each a sheet of a given size can hold, and whether a given sequence of letters along a row is realisable are all questions this does not touch. The alphabet bounds the vocabulary and says nothing about the grammar.

The enumeration stops at degree eight. Box-pleated designs do contain higher-degree vertices — a point where many flaps meet can carry twelve or sixteen creases — and the count above degree eight is not made here, because the stacking search is factorial in the degree and eight is where it stops being instant. What can be said is that the number of higher letters grows and the enumeration remains finite, which is the property that matters.

And the whole thing rests on the sectors being exact multiples. Real designs are drawn on a grid and then adjusted; a crease a degree off a diagonal makes a vertex that is not in the catalogue and is not flat-foldable either, which is why the grid is worth the efficiency it costs.

What the picture cannot show #

The catalogue is a table because that is what a finite list of things with numbers attached is. Drawing all six vertices would show six fans of lines that differ in ways a reader cannot count by eye — the difference between 45·45·135·135 and 45·90·135·90 is where one crease sits, and it is the difference between a vertex the conditions decide and one they do not.

Nor can any figure show the absence of a seventh. An exhaustion’s result is that nothing else exists, and nothing else existing has no picture.

The generalisation #

The pattern here is one that turns up wherever a continuous design space is put on a lattice, and it is worth stating in the general form because the trade-off it names is not obvious.

Restricting to a grid does two things at once. It shrinks the vocabulary, which is what makes designs on a grid tractable — a finite alphabet can be catalogued, checked, taught and automated, and a continuum cannot. And it concentrates the degeneracy, because a lattice is exactly where coincidences live: equal angles, equal lengths, ties. Everything that goes wrong at a tie in this subject therefore goes wrong at a grid vertex, systematically rather than by accident.

So the grid is not a neutral restriction that happens to be convenient. It buys finiteness and it pays for it in exactly the currency the subject is most sensitive to. Half the letters of the forty-five degree alphabet are tied vertices, and every one of those is a place where the local conditions over-count, the reduction has to search, and the marking may not name the object.

That is the strongest available statement of what box pleating is, and it is not the one the tradition makes.

Who found it, and when #

Box pleating as a design discipline is Neal Elias’s, from the 1960s, and its modern form belongs to the designers who took it to very high crease counts in the 1990s and after. That the vertex vocabulary is finite is implicit in every account of it and, as far as this site can find, has never been enumerated — probably because a designer’s question is which vertices to use rather than how many exist, and because the count is only interesting once the letters can be given properties.

The properties are what this site can add. The gap column needs a stacking search; the multiplicity column needs another; and neither is a thing a designer working at a drawing board could have measured.

There is a practical consequence worth writing down separately, because it is the kind of thing a piece of design software could act on. Given that a box-pleated design’s vertices come from a list of six, a checker for such designs does not need to run the conditions at all: it can look up each vertex, and the lookup carries more information than the conditions do — whether this marking is one of the ones that fails, and how many objects it names. That is a strictly better check than the one this site currently runs, and it is available only for designs on a grid, which is one more thing the grid buys.

What a lookup table would still not know #

The suggestion above — that a checker for box-pleated designs could look each vertex up instead of evaluating the conditions at it — is right and needs one sentence of restraint attached, because without it the suggestion promises something no local table can deliver.

A lookup is still local. It reads one vertex at a time and returns what is true of that vertex in isolation, which is strictly more than the four conditions return and is still a statement about a point. A pattern every one of whose vertices is individually fine can fail as a sheet, because the layers have to be ordered across the whole design and the ordering is not a vertex’s business. So a design that passes the lookup at every point is a candidate, exactly as a design that passes the conditions at every point is a candidate; what the lookup adds is that the candidates it admits are no longer contaminated by markings that fail at a single vertex.

That is a real improvement and it is worth being precise about its size. Three of the six letters accept markings that do not fold, so a checker running the conditions alone lets those through and a lookup does not. The remaining three accept only markings that fold, so on a design built from those the lookup and the conditions return the same verdict on every marking, and the whole gain is zero.

The gap compounds, and by how much is unknown #

The closing section calls the design-level count arithmetic nobody has done. Part of it can be done here, and the part that cannot is worth naming precisely, because the two are easy to run together.

Each letter has a share of its condition-passing markings that fold: all of them at the 45–90–135–90 vertex, at the square vertex and at the rosette; six in eight at the tied four-crease letter; twenty in thirty and eighteen in thirty at the two degree-six letters. So the three clean letters contribute a factor of one, and the three others contribute 0.75, 0.67 and 0.60.

If a design’s vertices were independent, the share of its condition-passing markings that survive a vertex-by-vertex check would be the product of those factors over its vertices — and a product of numbers below one over a design with a hundred tied vertices is not a small correction. At 0.75 apiece it is about three parts in ten million million million. On a design of that size essentially every marking the four conditions accept would be one no folder could make, and the conditions would be admitting a population that is almost entirely error.

They are not independent, and that is the part which is not arithmetic. Two vertices sharing a crease share a variable, so their markings are correlated, and the correlation could run either way: a shared crease might force the pair into the good markings or into the bad ones. The product is therefore an estimate under an assumption the design violates, and its value is that it fixes the scale of the question. Whether the true figure is near the product or nowhere near it is a search over a real design’s markings, and it is the same search the alphabet’s own gap column was computed with, run on something a hundred times larger.

Where the ladder goes next #

The grammar is the obvious continuation and it is a substantial one: which pairs of letters can share a crease, and what that makes of a row. It is a question about a graph whose nodes are the six letters, and it is answerable by the same kind of exhaustion.

The other direction is the one the gap column points at. Three of the six letters admit markings that no folder can make, so a design assembled from those letters has a count of unfoldable markings that is a product over its vertices — and how large that number gets on a real design is arithmetic nobody has done.