The designer's grid is the dearest thing here

A box-pleated design is drawn on a square grid, and the grid is the plainest object in this collection. Right angles everywhere, one vertex per cell, no free parameters, nothing to choose. It is the pattern anybody would name if asked for the simplest thing here.

Per panel of paper, it is the most expensive.

The measurement #

An $n$-division grid has $n^2$ panels and $(n-1)^2$ interior vertices. Searched for a consistent lettering under a fixed letter order, it costs:

Two divisions, four panels, four steps. Three, nine, nine. Four, sixteen, sixteen. Six, thirty-six, thirty-six. Eight, sixty-four, sixty-four. Ten, a hundred, a hundred. Twelve, a hundred and forty-four, a hundred and forty-four. Sixteen, two hundred and fifty-six, two hundred and fifty-six.

Exactly equal at every size, with no decision ever taken back. That is one of the cleanest arithmetics in the collection and it has been known here for a while.

What has not been said is that it is the largest such constant. Nothing here costs more than one step per panel, and almost everything costs less.

What else costs #

A tapered leaf at four widths: one step per panel exactly. A Miura at four sizes: exactly. Six crumples: one per panel less the last one.

So four families sit on the line together, and none of them is above it.

A twist tessellation patch: 0.53, 0.47, 0.51, 0.52 and 0.51 steps per panel on the five tilings, at every size measured.

A Yoshimura at the proportion everybody draws it: 0.88. The same Yoshimura at any row height past $\sqrt3$: 0.23.

The grid is at the top of that list, and the object at the bottom is a corrugation drawn at a different proportion — a quarter of the grid’s cost per panel of paper.

Why the grid costs exactly one #

The equality is exact rather than approximate and the reason is worth having, because it explains the ceiling.

A step of the search is a crease the propagation does not settle. The propagation writes every letter that a vertex’s conditions force and repeats until nothing more follows, and on a grid it stops at exactly one crease per panel: each panel’s own commitment, from which its neighbours follow.

So the step count is not an empirical near-linearity. It is the number of genuine choices the pattern has, and a grid has one per panel because its vertices are degree four, admit eight labellings apiece, and each of them narrows to two once one crease is known. Two options is a choice; a choice is a branch; a branch is a step.

Two, not one, and that is where the ceiling comes from. Eight labellings at a degree-four vertex is what Maekawa alone leaves. No condition here can leave more at a degree-four vertex, so no degree-four pattern can have more genuine choices per vertex than a grid does, and the vertices tile the sheet one per panel. Which makes one step per panel a ceiling rather than a typical value, and the four families that reach it the ones whose conditions leave the most.

There is a second way of saying the same thing that some readers will prefer. A grid’s vertex is the least constrained degree-four vertex there is: Maekawa is the only condition that bites, and Maekawa is a counting condition rather than a geometric one. Every other degree-four vertex here has at least as much said about it, and most have more.

So the grid’s position at the top of the list is not an accident of the family. It is where a pattern lands when the geometry contributes nothing, and geometry contributing nothing is exactly what four equal sectors means.

What the grid is missing #

The condition that everything cheaper here has and the grid does not is the big-little-big lemma.

It says the two creases bounding a sector strictly smaller than both its neighbours may not carry the same letter, and where it applies it halves the table from eight to four. A grid’s sectors are four right angles: nothing is strictly smaller than anything, and the lemma has no opinion.

A twist polygon’s corner has sectors of $46.15°$, $90°$, $133.85°$ and $90°$, so the smallest is isolated between two right angles and the lemma applies at every vertex of the pattern — which is why a twist patch is the most decided object here.

So the grid is expensive per panel for exactly the reason it is simple to draw. Its regularity is a tie among its sectors, and a tie is what silences the one condition that would have decided more.

What “expensive” means here, and what it does not #

The claim is per panel and per panel only, and it is worth guarding against three readings it does not support.

It does not mean a grid is hard. Two hundred and fifty-six steps on two hundred and fifty-six panels, with no backtracking, is a search that reads the pattern once. A grid is trivially easy in absolute terms and remains so at every size a designer would draw.

It does not mean a grid is expensive to fold. The cost measured here is decisions taken by a machine; the cost a folder feels is two hundred and fifty-six panels of paper and two hands.

And it does not mean the grid is where a box-pleated design’s difficulty lives. The grid settles a great deal and what it does not settle is the design, which is a question about where the flaps go and not about which letters the creases take.

The two ways to be below the line #

Everything cheaper than a grid gets there in one of two ways, and both are worth naming because a designer can influence one of them.

More decided vertices. A twist patch’s corners admit four labellings rather than eight, so each vertex leaves half as much to choose. That halves the constant, and it is bought by having a strictly smallest sector — which is a matter of the pattern’s angles.

Fewer vertices per panel. A degree-six vertex covers six creases instead of four, so a pattern built from them needs fewer vertices to cover the same paper. A 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.

The tilted Yoshimura has both and is at 0.23. The twist patch has the first and is at 0.5. The equilateral Yoshimura has the second and not the first, and is at 0.88 — nearly the grid’s, because its vertices are less decided than a grid’s and there are fewer of them, and the two nearly cancel.

Four families, three combinations of the two effects, and the arithmetic comes out in the order the mechanism predicts.

Where the grid’s difficulty actually is #

Two places, and neither is the search.

Rarity. The share of drawn letterings that agree with themselves falls from a hundred in a hundred at two divisions to one in a hundred at sixteen — so a designer assigning letters by eye at working scale will produce an arrangement that satisfies every condition at every vertex and fails globally, ninety-nine times in a hundred. The circuits are long and the failures are not local.

Layers. Whether the letters can actually be stacked is a separate question with no cheap answer, and a lettering whose relations contain no circle is still not a lettering that folds.

Both of those are about the sheet rather than about the vertices, and both are untouched by anything in this essay. The second is the one a designer is most likely to be caught by, since ninety-nine drawn letterings in a hundred pass every local test at working scale. The search cost being at the ceiling is a statement about how much the local conditions decide, and on a grid they decide the least of anywhere here.

A designer’s grid, and a designer’s proportion #

There is a practical shape to this that generalises past the grid, and it is worth pulling out because it applies to every pattern a person draws by preference.

Regular patterns have ties among their sectors. Right angles, sixty-degree wedges, equal divisions — these are the proportions people choose, because they are the ones that can be drawn with a folded reference and the ones that look correct. And a tie is precisely what stops the big-little-big lemma from saying anything.

So the patterns a designer naturally draws are the patterns whose local conditions decide the least, and therefore the ones on which checking a lettering vertex by vertex is worth the least. The Yoshimura is the sharpest instance of this — its natural proportion sits exactly on the boundary, on the silent side — and the grid is the most common one.

Nothing about that argues for drawing at irregular proportions. It argues for knowing that the local check is weakest where the drawing is most regular.

The exception that has not been built #

The ceiling argument above is about degree-four vertices and it has a gap in it, which is worth stating because it is the obvious next thing to measure.

Maekawa leaves eight at degree four and thirty at degree six. So a degree-six pattern’s vertices leave more, not less, and if such a pattern had as many vertices per panel as a grid it would cost more than one step per panel and the ceiling would not be a ceiling.

The Yoshimura is the only degree-six family here and it does not test it, because its vertices are sparse: forty-five of them carry a hundred and nineteen panels, where a grid’s two hundred and twenty-five carry two hundred and fifty-six. Fewer vertices, more permissive each, and the two nearly cancel at 0.88.

A degree-six pattern with a dense arrangement of vertices — one per panel, say, as a corrugation on the triangular lattice would have — would settle whether one step per panel is a ceiling or merely the top of what has been drawn. It is not built here, and the claim in this essay is therefore about the patterns that exist rather than about all patterns.

The other grid #

There is a second orthogonal grid in box-pleating and it behaves the same way.

A box-pleated design uses a square grid for the pleats and a diagonal grid at forty-five degrees for the corners that turn, and the two together are what the alphabet of a grid is built from. The diagonal vertices are also degree four, and their sectors are also equal — four right angles, rotated — so they admit eight labellings apiece and contribute one choice each.

So a box-pleated pattern’s cost per panel is the grid’s cost per panel wherever it is, which is a tidy result: the search cost of a design drawn on this system does not depend on the design. It depends on how many panels the design uses.

What a designer could do with the freedom #

There is a use for a pattern’s genuine choices that has nothing to do with searching, and it is the reason the count is worth having.

Every step of the search is a crease the conditions do not decide, which means it is a crease the designer may decide. On a sixteen-division grid that is two hundred and fifty-six independent decisions, each of which changes the folded object without breaking any local condition — and most of which, taken carelessly, break a global one.

That is the same number from two directions. As a search cost it is what a machine must work through; as a design space it is how much room the pattern leaves. Sixty-four rules generate a corrugation and sixteen of them fold, which is the same accounting on a much smaller object: the freedom is real, most of it does not work, and knowing how much there is is the first step to searching it.

So the grid being at the ceiling reads better as the grid is the most permissive pattern here than as the grid is the dearest. Both are the same measurement, and the first is the one a designer wants.

Which theorem was checked, and how #

The equality between steps and panels is asserted rather than read off a chart. Every pattern in the ladder is required to cost no more than one step per panel, and the requirement fails on the first one that does — which is a stronger statement than the counts coming out linear, because a search could visit exactly $n$ nodes while taking a decision and giving it back.

The table sizes are enumerated: all sixteen labellings of a degree-four vertex, filtered by Maekawa’s difference and by the lemma at each sector, counted. Eight is what survives at a grid vertex and four at a twist corner, and both are counts rather than formulas.

And the comparison across families is measured under one letter order on all of them, so that the grid’s 1.00 and the patch’s 0.51 are two numbers from the same instrument rather than two runs of a randomised one.

What a grid’s freedom is spent on #

There is one more reading of two hundred and fifty-six independent choices that is worth having, because it connects the number to something a designer already knows.

A box-pleated design is specified by a tree — where the flaps go, how long each is — and the grid is where that specification is written down. The letters on the grid’s creases are what turn a layout into a folded object, and the count of genuinely free letters is the count of ways a given layout could be realised.

Most of those ways do not fold. At sixteen divisions the share of drawn letterings that agree with themselves is one in a hundred, so of the enormous space the two hundred and fifty-six choices open up, the overwhelming majority is unreachable — and every one of the failures satisfies every condition a designer can check at a vertex.

So the grid’s position at the top of the range is the same fact as the difficulty of box-pleating by hand, seen from the other end. The pattern leaves the most room and the room is mostly empty.

The reversal, stated plainly #

Two sentences that would both have been accepted here until recently, and which turn out to be the wrong way round.

The grid is the simplest pattern here, so it is the least work. It is the most work per panel, because simple means regular and regular means a tie among its sectors and a tie means the geometric condition says nothing.

The tessellation patches are the intricate ones, so they are the most work. They are the least work per panel, because their corners have four different angles and four different angles is what gives the lemma something to say.

The reversal is not deep and it is worth having explicitly, because the intuition it corrects is very strong. A drawing’s visual complexity and the amount its conditions decide are close to unrelated, and where they are related here they run opposite ways: the more even the angles, the less is settled.

The pattern at the other extreme #

The clearest contrast to a grid is not a tessellation patch but a corrugation at a proportion nobody draws, and it is worth putting the two side by side.

The Yoshimura at its natural proportion has six equal sectors, so the big-little-big lemma is silent and its vertices keep thirty labellings — more than anything else here. It still costs less per panel than a grid, at about nine-tenths, because a degree-six vertex covers six creases and there are fewer of them per panel of paper.

Two effects, pulling opposite ways, nearly cancelling. The grid has neither: four creases a vertex, one vertex a panel, and the least said about each of them by any condition.

What the picture cannot show #

A ladder of nine points with every point on the line is a picture of an arithmetic identity, and identities are the hardest thing to draw interestingly. What it cannot show is the ceiling: that no pattern can be above the line, which is an argument about what Maekawa leaves at a degree-four vertex rather than a measurement of anything.

Nor does the per-panel measure say anything about what a step costs. Each step propagates every vertex condition to a fixed point, and a sixteen-division grid’s step is a great deal more work than a two-division grid’s. The line is flat in decisions and steeply rising in seconds.

And a per-panel figure gives no sense of the absolute scale, which for a designer is the number that matters. Two hundred and fifty-six steps is a fraction of a second; the difference between the grid and the cheapest pattern here is a fraction of that fraction. Everything in this essay is a statement about which patterns have more genuine freedom in their letters, and freedom is the interesting quantity for reasons that have nothing to do with how long a computer takes.