One step per panel is a table size

There is a number that keeps appearing in this collection and has never been explained. A box-pleating grid at nine sizes from four panels to two hundred and fifty-six takes exactly one search step per panel. A tapered leaf at four widths: one step per panel. A Miura at four sizes: one step per panel. Six crumples of deepening severity: one step per panel, less a constant.

Forty-six patterns from five constructions, and the reading offered was that these are patterns which fill their own sheet, so their vertices are all interior, so the propagation is decisive and the search never guesses wrong.

Every clause of that is true and the conclusion does not follow, because the constant is not one. It is eight.

What a table is #

Before any lettering is chosen, the conditions at a vertex are applied to the vertex alone. Maekawa says the mountains and valleys must differ by two. The big-little-big lemma says the two creases bounding a sector strictly smaller than both its neighbours may not carry the same letter. Kawasaki reads angles and no letters at all, and refuses the vertex outright if the angles are wrong.

What survives is a table: the list of labellings of that vertex’s own creases which the conditions permit. Everything the search does afterwards is bookkeeping over those tables — propagate the ones that are forced, branch where two remain, and test the result.

A degree-four vertex has sixteen labellings of its four creases. Maekawa cuts that to eight, and the big-little-big lemma cuts it to four when it has something to say. So a degree-four vertex carries eight or four, depending on whether one of its sectors is strictly smallest, and the word doing the work is strictly.

A degree-six vertex has sixty-four labellings. Maekawa cuts that to thirty, and the lemma to eight.

The four families on the line #

The grid, the leaf, the Miura and the crumples have this in common and nothing else: every vertex admits eight labellings.

Their geometries are unrelated. A grid is orthogonal and a Miura is not; a leaf tapers and a crumple has no repeating structure at all. What they share is that each has degree-four vertices with no strictly smallest sector — or with one, in the Miura’s case, which turns out not to change the count here — so the tables come out at eight apiece. That is the same count the preliminary and waterbomb bases carry at every vertex, and for the same reason.

And all four sit at 1.00 nodes per panel: two hundred and fifty-six steps on two hundred and fifty-six panels, twenty-four on twenty-four, seventy on seventy-one.

It is worth checking that the coincidence is a coincidence of tables and not of something else, because four families agreeing is exactly the kind of agreement that invites a wrong cause.

They are not the same size: four panels to two hundred and fifty-six, a factor of sixty-four. They are not the same degree of order: a crumple is produced by folding a sheet at random and flattening it, and a grid is the most regular object here. They are not built by the same construction: the grid comes from a lattice, the leaf from a taper, the crumple from a random fold sequence. They do not have the same sector angles: a Miura’s are set by its zigzag, a crumple’s by where the folds happened to land.

What they share is the count, and it is exactly eight in every case — not approximately, not on average, but eight at every interior vertex of every pattern in all four families. That is a strong enough coincidence to be a cause.

The family below the line #

The twist tessellation patches admit four.

Every interior vertex of a clipped patch is a corner of a twist polygon where two of the polygon’s own sides meet two pleat creases, and its sectors are unequal in a way that gives the big-little-big lemma something to say at all four of them. Four labellings rather than eight.

Their cost per panel is 0.53 on the square, 0.47 on the triangular, 0.51 on the honeycomb, 0.52 on the elongated and 0.51 on the rhombille. Half the line, on all five.

That is the family this collection long called the difficult one, and per panel of paper it is the cheapest object here — because its vertices are the most decided.

The family above it, and the controlled comparison #

The Yoshimura’s vertices are degree six: a horizontal course crossed by two creases going up and two going down. At the proportion everybody draws it — a row height of $\sqrt3$ halves of a column, which makes all six sectors exactly sixty degrees — no sector is strictly smallest, because each small sector’s neighbours include another of the same size. The lemma says nothing, and the tables hold thirty.

Change the row height and the sectors change with it. At a row height greater than $\sqrt3$ the two sectors between the up-creases and the down-creases become smaller than the four around them, and each is strictly smaller than both its neighbours. The lemma bites at every vertex and the tables drop to eight.

Nothing else about the pattern moves. Same panels, same vertices, same degree, same construction, same number of creases; only the row height, and only through the sectors.

Thirty labellings: fifty-seven steps on sixty-five panels. Eight labellings: nineteen.

What the number actually predicts #

It is worth being exact about how far the association goes, because it is an association and not a formula.

The direction is reliable: fewer labellings at a vertex, fewer steps. Four gives half a step per panel, eight gives one, thirty gives just under one on a family with proportionally fewer vertices.

The magnitude is not, and the grid and the tilted Yoshimura say so plainly. Both admit eight labellings; the grid costs 1.00 per panel and the tilted Yoshimura 0.29. What differs is the degree. A degree-six vertex has six creases in it, so settling one vertex settles more of the sheet than a degree-four vertex does, and the tilted Yoshimura has forty-five vertices on a hundred and nineteen panels where a grid has two hundred and twenty-five on two hundred and fifty-six.

So the honest statement has two parts, and one of them is a caution. The number of labellings a vertex admits is the quantity that moves when nothing else does — the Yoshimura’s two proportions prove that with everything held still. It is not by itself enough to predict a family’s constant, because how many creases each vertex covers matters too, and this collection has no formula that combines them.

Reading the two directions apart #

The essay’s claim can be stated in two sentences that sound the same and are not, and separating them is most of the work.

A pattern whose vertices admit fewer labellings has fewer letterings. That is trivially true and it is a statement about a set: the admissible letterings of the whole sheet are a product over vertices, more or less, so halving the count at each vertex divides the total by a great deal.

A pattern whose vertices admit fewer labellings is cheaper to letter. That is the claim here and it is a statement about a procedure, and it does not follow from the first. A smaller set is generally harder to hit by sampling, not easier — which is exactly why rarity and cost move independently on the grid ladder, with the share of drawn letterings that agree falling by two orders of magnitude while the search cost stays exactly linear.

The two go the same way here for a specific reason: the search does not sample. It propagates, and a smaller table propagates further, because a table with one entry writes its letters and a table with four does not. So constraint that would hurt a sampler helps a propagator, and the same number moves both quantities in opposite directions.

What a step actually counts #

The reason the arithmetic works at all is worth stating, because it makes the constant less mysterious.

A step of the search is not a guess. At each step the conditions are propagated to a fixed point — every vertex whose table has narrowed to one option writes its letters, those letters narrow their neighbours’ tables, and the sweep repeats — and then one crease that is still undecided is chosen and given a letter.

So the number of steps is the number of creases the propagation never decides on its own: the pattern’s genuine independent choices, counted. On none of the families here is a choice ever withdrawn. Across forty-six patterns from five constructions the number of decisions taken back is zero, so the step count is not an exploration cost at all. It is a census.

That is why the constant is stable within a family and different between families. It is not measuring how hard the search works; it is measuring how much freedom the pattern has, and the tables are where the freedom is.

The waterbomb, which has both #

One family here has vertices of both kinds and it behaves the way a mixture should.

A waterbomb tessellation has degree-four vertices where its diagonal creases meet the grid and degree-six vertices where three creases cross, and its tables run from eight to thirty across one pattern. Its cost per panel is 0.93, 0.92 and 0.89 at three sizes — between the grid’s 1.00 and the tilted Yoshimura’s 0.29, and much nearer the grid because most of its vertices are the degree-four kind.

A mixture landing between its components is weak evidence on its own. It is worth having because the alternative reading — that the constant is a property of a construction rather than of its vertices — predicts nothing in particular for a mixture, and would have to explain why a pattern built by one recipe lands between two others.

The count at a vertex, from sixteen down #

The arithmetic that produces eight and four is short enough to give in full, and it explains why those two numbers and no others.

A degree-four vertex has four creases, so sixteen labellings. Maekawa requires the mountains and valleys to differ by two, which admits three-and-one or one-and-three: four of each, so eight. Big-little-big then forbids the two creases bounding a strictly smallest sector from carrying the same letter, which removes half of what is left: four.

A degree-six vertex has sixty-four labellings. Maekawa admits four-and-two or two-and-four, which is fifteen of each: thirty. Big-little-big at two of the six sectors removes all but eight.

So the table sizes available here are sixteen, eight, four, thirty and eight again — and the two eights are different objects, one reached from sixteen and one from sixty-four. That coincidence is why the association is not a formula: a grid’s eight and a tilted Yoshimura’s eight say very different things about how much of the sheet one vertex settles.

Where the count comes from at a twist corner #

The patches’ four is worth deriving rather than asserting, because it is the one table size in this essay that is not about a symmetric vertex.

At a corner of a twist polygon four creases meet: two sides of the polygon itself, and the two creases of the pleats that run away from that corner. The sectors between them are the polygon’s own interior angle, the tile’s angle across the corner, and the two pleat sectors. On the square tessellation at the proportions this collection draws, they are 46.15°, 90°, 133.85° and 90°.

The smallest of those is the pleat sector, and its two neighbours are both ninety degrees, so it is strictly smaller than both. The lemma applies, and the table falls from eight to four.

That is why every twist patch on every tiling gives four rather than eight, and it is also why the count would change if the construction’s turn were opened far enough to push the pleat sector past the tile’s own angle — which is the transition an earlier measurement found at 0.2155 radians, arriving here as a fact about how many labellings a vertex keeps.

Why this was invisible #

The reason the reading held for so long is that every family originally measured had the same table size.

The grid, the leaf, the Miura and the crumples all admit eight, so the constant was the same in every case and there was nothing to vary. The one family that admits something else was the tessellation patches, and their number was hidden by a coin in the search that made their cost swing over three orders of magnitude — so what was visible about them was their variance rather than their level, and their level is the interesting part.

Remove the coin and the patches’ cost is not merely stable; it is stable at half the line. The table size was the explanation the whole time and there was no comparison in which it moved.

What it says about the twist patches #

The correction this makes to the collection’s reading of the tessellation patches is worth stating on its own, because it is a reversal rather than a refinement.

The patches were the exception in the essay that concluded a rim is what makes a search hard, and the exception was real: their measured cost was variable where every other family’s was flat. What was not established was the direction. A family whose cost swings between eighty-six steps and fifteen thousand has an interesting distribution and no obvious level, and the level is what this measurement gives.

The level is half a step per panel, and it is the lowest here. So the patches are exceptional in exactly the way their tables predict — four labellings against everybody else’s eight — and the variance that made them look hard came from somewhere else entirely, which is the coin the search used to choose which letter to try first.

Two exceptional things about one family, one of them about the pattern and one about the instrument, and until the instrument was fixed there was no way to see the first.

What a designer takes from it #

Something usable, and it is about proportion rather than about search.

A pattern whose vertices have a strictly smallest sector is more constrained than one whose vertices do not, and the difference is a factor between two and four in how many letterings are admissible at each point. That matters to a designer for a reason that has nothing to do with cost: a more constrained vertex is a vertex where a wrong letter is caught sooner.

A pattern drawn at a proportion where the lemma says nothing has more letterings that pass every local check and fail globally. That is the setting in which ninety-nine drawn letterings in a hundred pass and one folds, and the proportion is a lever on it — one which nobody has any reason to think about, because it is chosen for how the model looks.

The same step at a second size #

A discontinuity measured at one size is a discontinuity that might be about that size, so the Yoshimura’s step is checked at more than one.

At four columns by four rows the pattern has thirty-six panels and eleven interior vertices. Below and at a row height of $\sqrt3$ its vertices admit thirty labellings and it costs thirty-two steps; past $\sqrt3$ they admit eight and it costs fourteen. At eight columns by seven rows — a hundred and nineteen panels, forty-five vertices — the two costs are a hundred and eight and twenty-seven.

Same threshold, same two table sizes, three sizes of pattern. What moves with size is the cost and not the place the cost changes.

What the picture cannot show #

A scatter of families against table size is a picture of an association, and an association across families with different geometries is the weakest kind of evidence there is. The two things in this essay that are not weak are the pairs where one thing moves: the Yoshimura at two proportions, and the patches against the four families on the line.

Nor does the table size say anything about whether a lettering exists. A vertex admitting four labellings is more constrained than one admitting eight, and a pattern whose vertices admit four could easily have no consistent lettering at all — nine patches over a grid of parameters have none, and every vertex of every one of them admits four. Constraint makes a search cheap and makes an answer rare, and those are not the same direction.

And a table is a count of possibilities at one point, so no figure of it can show the thing that actually determines a pattern’s cost, which is how the possibilities at neighbouring points interlock. Two patterns with identical tables everywhere can differ in how much one vertex’s decision propagates into the next, and nothing in a table records that. It is the reason the association here is offered with its magnitude unexplained rather than fitted to a curve.