The other grid

Box pleating puts every crease on a square grid and every fold at a multiple of forty-five degrees, which turns a design into a drawing on squared paper and is why it is the method most complex work is done in. The whole alphabet of that grid is short enough to print.

Forty-five degrees is a choice inside that method rather than a consequence of it. Half a turn divides by four at forty-five degrees, by six at thirty and by eight at twenty-two and a half, and each of those is a grid a designer can work on. The twenty-two-and-a-half-degree grid in particular is a real school of design with real models in it, and it is almost always described as box pleating at higher resolution.

It is not higher resolution. It is a different alphabet, and counting the alphabets shows how different.

Counting the alphabets #

A vertex on a grid has every sector a whole multiple of the grid’s step, and the sectors must close and satisfy Kawasaki — the alternating sums equal. That is a finite condition on a finite set, so the vertices a grid admits can be enumerated completely rather than sampled: there is a catalogue, it is short, and it is a complete list rather than a draw.

Up to degree six:

Up to degree eight the counts are six, thirty and a hundred and forty-nine — and every number here is a complete enumeration rather than an estimate, because a grid’s vertices are a finite set.

The growth is not a factor of two per halving of the step. From forty-five to twenty-two and a half the vocabulary multiplies by eleven, and almost all of the growth is at degree six: two kinds of degree-six vertex become forty-six.

Why the coarse grid has so few #

The reason is the pigeonhole principle, applied to sector sizes.

At forty-five degrees a sector is 45°, 90° or 135° — three values. A degree-six vertex has six sectors, so six values are drawn from three and at least two must coincide. Every degree-six vertex on a forty-five-degree grid has a tie, necessarily, and it is not a coincidence in any meaningful sense: there is no room for it not to happen.

At twenty-two and a half degrees a sector is one of seven values, and six of them still cannot all be different — six distinct whole numbers sum to at least twenty-one and a full turn is only sixteen units. So ties are forced on the finer grid too, and the pigeonhole is not where the difference lies.

That single fact runs through everything below, because a tie is exactly where the big-little-big lemma falls silent — a sector that ties for smallest is not strictly smallest, so the lemma forbids nothing — and where the crimp reduction loses its forced move.

Where the difference actually is #

The pigeonhole forces a tie on both grids, so the mechanism has to be something narrower, and the arithmetic says exactly what.

The lemma needs a sector strictly smaller than both its neighbours. A tie among large sectors costs nothing; only a tie at the minimum silences it. So the question is not whether a vertex has equal sectors but whether its smallest sector is unique.

On the forty-five-degree grid a degree-six vertex has six sectors summing to eight units with each at least one, so the excess over the smallest possible total is two — which can raise at most two of the six above a single unit. At least four sectors are one unit each, the minimum is one, and it is achieved four times or more. The smallest sector is never unique, and the lemma is silent at every degree-six vertex the coarse grid admits, without exception and by arithmetic.

On the twenty-two-and-a-half-degree grid a degree-six vertex has six sectors summing to sixteen units, and the excess is ten. There is room: one, two, three, four, three, three sums to sixteen and has a unique minimum. So the finer grid admits degree-six vertices at which the lemma bites, and the coarse grid admits none.

That is the whole of it. Both grids force coincidences; only one of them forces the coincidence at the smallest sector, which is the only place a coincidence costs anything.

Which the lettering totals confirm #

The totals in the table are a check on that account and they are worth reading as one, because they can be reconstructed from the catalogues without any measurement.

A degree-four vertex has sixteen letterings and a degree-six vertex sixty-four. The forty-five-degree catalogue is three of the first and two of the second: three sixteens and two sixty-fours is a hundred and seventy-six, which is the figure the table reports. The twenty-two-and-a-half-degree catalogue is ten and forty-six: a hundred and sixty plus two thousand nine hundred and forty-four is three thousand one hundred and four, again the reported figure.

So the letterings are dominated by the degree-six entries on both grids — a hundred and twenty-eight of a hundred and seventy-six on the coarse one, and ninety-five per cent on the fine one. And degree six is precisely where the two grids differ about whether the smallest sector can be unique.

That closes the account. The coarse grid’s search share is high because most of its letterings live at degree-six vertices where the lemma has nothing to say; the fine grid’s is low because most of its letterings live at degree-six vertices where it usually does.

The consequence that runs the wrong way #

The obvious expectation is that a bigger vocabulary is a harder subject. More kinds of vertex, more cases, more to check.

Measured over each catalogue in full — every kind of vertex, every lettering of each — the share of decisions that need a search rather than a forced move goes the other way:

A designer working at twenty-two and a half degrees is drawing from a vocabulary eleven times larger in which two decisions in nine need a search, against a vocabulary of five in which three decisions in five do.

The mechanism is the tie again, and it is the same one four populations of vertex disagreed about. A vertex whose sectors coincide has more letterings admitted by the conditions and fewer of them forced, because the conditions constrain by distinguishing — the lemma says something exactly when one sector is strictly smaller than both its neighbours. Give the grid more sizes to draw from and coincidences become rare, so the conditions have more to say and the search has less to do.

What the extra vertices are for #

A catalogue is a vocabulary and a vocabulary is only worth something if the words are usable. The forty-five-degree catalogue’s five entries include the two that every box-pleated design is made of: the degree-four vertex with sectors 45°, 135°, 135°, 45° and the degree-six one with three sizes.

The twenty-two-and-a-half-degree catalogue contains those and fifty-one more. Its extra degree-four vertices — 22.5°, 157.5°, 22.5°, 157.5° and its relatives — are the ones that make a flap point in a direction the coarse grid cannot reach, which is the practical reason the school exists: a limb at 22.5° to the sheet’s edge is a limb a box pleater has to approximate.

That is a design argument rather than a geometric one and it is worth marking as such. What the counting establishes is that the vertices are available and how many there are; whether a designer wants them is a question about models.

What the alphabet does to a whole pattern #

A vertex catalogue is a local object, and the practical question is what a design made of those vertices inherits.

Two things, both measurable and both already on this site. The first is that a pattern’s letterings are constrained at every vertex at once, so a vocabulary in which each vertex admits fewer letterings is a vocabulary in which the pattern admits fewer — and what a local change can reach is decided by the pattern’s graph rather than by its grid, so the two effects are independent.

The second is subtler and cuts the other way. On the coarse grid the ties that make each vertex ambiguous also make its foldings easier to walk: a gridded degree-six vertex stays in one piece under the neighbour-only move where a generic one comes apart. So the coarse grid gives a designer a small vocabulary, many letterings per word, hard decisions and an easy walk; the fine grid gives a large vocabulary, few letterings per word, easy decisions and a harder walk.

Neither is a recommendation. What the pair of measurements does establish is that ambiguity and navigability are not the same axis, and that a grid choice moves them in opposite directions.

There is a third consequence and it is the one a folder feels. A vertex whose decision is forced can be folded without thinking: the paper only goes one way, and a mistake announces itself immediately. A vertex whose decision needs a search can be folded two ways at the moment the fingers reach it, and only one of them lets the rest of the model close — so the error appears somewhere else, several steps later, as a sheet that will not lie flat. Sixty per cent of decisions on the coarse grid are of the second kind and twenty-two per cent on the fine one, which is a statement about where mistakes get discovered rather than about how many are made.

Which theorem was checked, and how #

The catalogues are complete enumerations, not samples, which is the property that makes them worth counting at all — a number about a sampler is a number about the sampler. Every composition of a full turn into whole multiples of the step is generated, filtered by Kawasaki, and reduced modulo rotation and reflection — so a catalogue of fifty-six is a list of fifty-six things rather than an estimate of how many there are.

A step that does not divide a full turn is refused. Thirty-five degrees is not a grid and the enumerator says so rather than rounding.

Every lettering of every catalogue entry is decided by the reduction and the answer is checked against an exhaustive stacking search. The share needing a search is a count over that reduction’s own branching, and a vertex whose reduction branches is one where two smallest sectors are offered at once.

The degree ceiling is stated. The counts above run to degree six for the alphabets and to degree eight for the totals; deciding a vertex means enumerating stackings, and the enumerator refuses above degree nine rather than sampling.

Where the model stops #

A catalogue is not a design. Fifty-six kinds of vertex says what can appear where creases meet; it says nothing about whether a pattern made of them folds, which is the global question and is intractable.

Degree six is where the counting stops being complete. At degree eight the twenty-two-and-a-half-degree catalogue has a hundred and forty-nine entries and the enumeration is still exact; at degree ten it would not be, and no claim is made past the ceiling.

The comparison holds the sheet fixed and varies the step, which is one of two ways to make it. A designer choosing a finer grid usually also chooses a smaller unit, so the practical comparison involves the paper as well — and how well a design can be spelled in whole units is a separate measurement with a different answer.

The comparison is about vertices and not about flaps. A flap costs a circle whatever grid it is drawn on, and the packing stage is where a design’s paper is spent; the alphabet decides what the creases can do once the packing is fixed.

Nothing here says the finer grid is better. It says the alphabets are different sizes and that the finer one is less ambiguous per lettering. Whether that is worth the extra vocabulary is a judgement about how designs get made.

What the picture cannot show #

A catalogue drawn as a grid of small vertices shows the entries and cannot show that the list is complete. Completeness is a statement about everything the grid admits, and a picture is a picture of what was drawn; the only evidence for it is that the enumeration ran over every composition and the count is what it is.

Nor can the drawings show which entries a designer would use. Every vertex in the catalogue is legitimate and a handful are the ones that appear in actual patterns, and the difference between those two sets is invisible in a figure whose subject is the enumeration.

The idealisation, named #

A grid vertex has its sectors at exact multiples of the step, and exactness is doing all the work here. A tie is an equality between two angles; on paper folded by hand the two angles are equal to within whatever the hand achieved, and the lemma’s verdict at a near-tie is the verdict at a strict inequality.

So the catalogues describe patterns as drawn rather than models as folded. That is the right object for a design method — box pleating is a way of drawing — and it is worth being explicit that the interesting behaviour of a grid comes from exact coincidences, which is precisely the property paper does not have.

The generalisation #

A finer discretisation is not the same discretisation with smaller steps. The forty-five-degree grid’s vertices are not a subset of the twenty-two-and-a-half-degree grid’s behaviour scaled down; they are the vertices of a system with three sector sizes, and a system with seven behaves differently in kind. Coincidences are the mechanism, and coincidences are a property of how many values are available rather than of how small they are.

The second half is the one worth carrying out of the subject: ambiguity comes from coarseness, not from richness. A vocabulary with few symbols produces many collisions between them, and collisions are where a decision procedure stops deciding. That is the opposite of the usual expectation, which is that a bigger space of possibilities is a harder space to work in, and the reason the expectation fails is that the difficulty being measured is per decision rather than per space.

Who folds on which grid, and why the question is old #

The three grids are not equally used and the reasons are practical rather than mathematical.

Forty-five degrees dominates because it is what squared paper gives without construction: fold the sheet in half repeatedly and the creases and their diagonals are the grid, so a designer needs no measurement at all. Twenty-two and a half degrees costs one more bisection per unit and buys the extra directions; thirty degrees is not reachable by halving at all, and a designer working on it is folding a sheet into thirds first — which is exact and is a construction rather than a fold in half.

So the alphabets differ in size and the grids differ in what it costs to draw them, and the two are not aligned: the richest of the three catalogues sits on the grid that is second-cheapest to construct, and the poorest sits on the cheapest. Whatever the history of the schools that use each, the geometry does not explain the popularity — the construction cost does.

Where the ladder goes next #

Two rungs are visible.

The fifteen-degree grid divides half a turn by twelve and would extend the table; the enumeration is the same and the counts would be larger, which makes it an exercise rather than an argument unless something changes shape.

The more interesting one is the design question this rung deliberately stops short of: given a tree to spell, which grid spells it best when both the unit cost and the vertex vocabulary are counted? Spelling a tree on a grid answers half of it — the whole-number cost — and the vertex half is the measurement above. Nobody here has put the two together.