The rim is four letters a cell

The usual account of what a sheet’s edge does to a crease pattern is about vertices. A vertex in the middle of the paper has a full turn of creases around it and answers to four conditions; a vertex near the edge has paper on one side only, so the conditions do not apply and nothing checks it. That is true, it has been measured here, and the vertices nobody checks are a real feature of every patch this collection draws.

It is also not what the edge of a tessellation patch does, and the difference matters because it is the difference between a pattern being under-checked and a pattern being under-constrained.

The comparison #

A twist tessellation repeats, so a rectangle exactly one period across and one period up is a description of the whole pattern rather than a piece of it. The same rectangle can be read two ways: as a piece of paper with an edge, which is what every patch here has always been, or with its opposite sides joined so that no crease is divided, which is what the pattern actually is.

Two readings of one rectangle. Same drawing, same coordinates, same construction at the same turn, pitch and fill. Whatever differs between them is the edge and nothing else.

The vertices do not move #

The first thing to check is the account above, and it fails.

The rectangle’s corner is placed where its sides run furthest from every drawn point, which puts them through the middle of pleats rather than through polygon corners. A vertex is therefore either wholly inside the rectangle or wholly outside it, never cut. So the vertices inside are interior on both readings, with all their creases present and all four conditions applying.

Sixteen vertices on the two-period square cell, cut or joined. Sixty-four on the four-period cell, cut or joined. Forty-eight on the two-period triangular and honeycomb cells; a hundred and eighty on the three-period elongated one. Fifteen comparisons across five tilings, and the vertex count is identical in every one.

Each of those vertices admits four labellings, before and after. The conditions are not merely present in the same number; they are the same conditions, admitting the same answers.

What does move #

Creases. A crease that the rectangle’s side divides is drawn as two pieces, and on the cut reading those two pieces are two creases, each free to carry whichever letter it likes. On the joined reading they are one crease with one letter.

On the two-period square cell the drawing has forty crease pieces and the pattern has thirty-two creases. Eight creases are divided. On the three-period cell it is twelve; on the four-period cell, sixteen.

The rule is four per period of edge. A square cell n periods on a side has four sides, each n periods long, and the drawing crosses each period of each side once — so $4n$ creases are divided, and $4n$ letters exist on the cut sheet that do not exist on the pattern.

The other tilings give the same shape with different constants, because their repeating rectangles are not squares and their pleats meet the sides at different rates. The triangular and honeycomb cells divide ten creases at one period and twenty at two, thirty at three and forty at four; the elongated cell and the rhombille divide twelve, twenty-four, thirty-six and forty-eight. Ten per period on two tilings, twelve on two others, four on the square — one number per tiling, and it is the number of times the drawing crosses one period of one side, multiplied by four sides.

Why the constant is what it is #

Four on the square, ten on the triangular and the honeycomb, twelve on the elongated and the rhombille. The differences are not arbitrary and they are not about how complicated the tiling looks.

The number is how many creases cross one period of one side, times the four sides. A side of length one period cuts whatever the drawing puts across a line of that length, and a twist tessellation puts across it the pleats it happens to run through — two creases per pleat, plus any polygon side the line clips.

So the constant is a measure of how densely creased a period is along a straight line, which is a very different quantity from how many creases a period holds. The square cell holds twelve crease pieces and divides four of them; the rhombille cell holds sixty and divides twelve. A third and a fifth respectively — the rhombille’s period is denser in creases and proportionally less exposed to a cut, because a period of it is larger.

Panels, and why they run the other way #

There is a third count and it moves in the direction nobody expects.

The cut rectangle has more panels than the pattern, because the sides slice panels in two and each half is counted separately. Twenty-five against sixteen at two periods on the square; eighty-one against sixty-four at four; two hundred and thirty-three against a hundred and ninety-two on the four-period honeycomb cell.

So the cut object is larger in the panel count, larger in the crease count, and identical in the vertex count. Every quantity that would make a problem look big is bigger on the cut sheet, and every quantity that would make it look constrained is the same or smaller.

That arrangement is the reason the accounting is worth doing carefully. A reader told only that a patch has twenty-five panels and a tessellation sixteen would conclude that the patch is the more complicated object, and in the sense that matters for lettering it is the simpler one by a factor of a thousand.

What four letters actually buy #

A free letter is a degree of freedom, and the useful way to think about what the rim contributes is to ask what the freedom is a freedom from.

On the pattern, a crease running out of the right-hand side of a rectangle and back in at the left is one crease. Whatever the conditions at its right-hand end demand of it, the conditions at its left-hand end must agree with. Those two ends are as far apart as the sheet is wide, and nothing local sees both of them.

Cut the sheet and that requirement disappears. The right-hand end answers to the vertices near it, the left-hand end to the vertices near it, and the two ends never have to be reconciled because they are no longer the same crease.

So a cut converts a long-range constraint into two local ones. Four per period of edge, and each one is a constraint that spanned the whole sheet.

That is the whole content of the rim, stated as an accounting rather than as a story, and it explains a measurement that is otherwise very hard to believe: the same rectangle costs forty-eight steps to letter cut and fifty-six thousand seven hundred and seventy-two glued, with fewer panels and fewer creases on the expensive side.

The letters and the questions, side by side #

It is worth setting the whole accounting out in one place, because the direction of each entry is what the essay is about.

Those are the two-period square cell’s numbers. Every row but the last two says the objects are the same problem; the crease row says the cut one has eight more answers to give; and the panel row says the cut one is, as a drawing, half again as large.

At four periods the same table reads 64, 4, 144 against 128, 81 against 64 — and 48 steps against 56,772.

The vertices the rim really does excuse #

None of this contradicts the earlier account; it applies to a different set of patterns.

A patch drawn the way the collection draws most of them — a square of a fixed size, cut wherever the sheet falls — does have vertices near its rim that are short of paper and are asked nothing. Those exist. The rectangle in this essay does not have them, because it was placed to avoid them, and that placement is what makes it a controlled comparison rather than two objects differing in several ways at once.

So the two accounts sit side by side. On an arbitrary patch the rim both excuses vertices and frees letters, and the two effects are tangled. On a period placed to miss the vertices only the second happens, and it can be counted.

The same accounting on a corrugation #

The natural next question is whether this is a fact about tessellations or a fact about edges, and a corrugation answers it.

A box-pleating grid, a Miura, a tapered leaf: each fills its own sheet. Its creases end at the paper’s edge because the construction put them there, not because a rectangle was drawn across a larger drawing. Cut such a pattern out of a bigger copy of itself and the same accounting would apply — but nobody does, because the pattern is the sheet.

So a corrugation has no divided creases at all, and correspondingly no free letters at its rim beyond the ones its own construction leaves. That is exactly why a corrugation costs one step per panel with nothing to spare, while a patch of a tessellation costs about half of that: the corrugation has no severed constraints to be relieved of.

The comparison is worth stating the other way round, because it is the more useful direction. The one-step-per-panel line is not a floor that a well-behaved pattern sits on. It is what a pattern costs when nothing has been taken away from it, and a patch is below the line because something has.

How the count is established #

Not by inspecting the drawing. Both counts come from union-find structures built by matching.

Crease pieces are matched by their endpoints: a piece ending on the left side at a given height is joined to the piece ending on the right side at the same height, and to the piece ending on the top at the same horizontal position if it also reaches the top. The joins are transitive, so a crease crossing two sides at once — which happens near a corner — ends up in one class with all its pieces.

A piece whose endpoint lies on a side and finds no partner is counted rather than ignored. That count is zero on every cell here, which is the arithmetic saying the rectangle really does repeat: an unmatched piece would mean the drawing on one side is not the drawing on the other.

The panel count is a separate union-find over faces, matched by probing across the sides rather than by endpoints, and each join records how many periods apart the two halves are. Two routes to one panel must agree about that displacement; the disagreements are counted and are zero, with the underlying arithmetic agreeing to about one part in a hundred million million.

And the three counts are checked against each other by Euler’s formula, which for a torus gives $V - E + F = 0$ and does so on every cell here.

The number that does not scale #

One more shape in the arithmetic is worth pulling out, because it says what happens to a patch as it grows.

The divided creases go as the perimeter: $4n$ on an $n$-period square cell. The vertices, creases and panels go as the area: $4n^2$, $8n^2$, $4n^2$. So the fraction of a patch’s letters that are free — the fraction that is rim rather than pattern — falls as $1/n$.

A one-period square cell has four free letters out of twelve, which is a third of the whole crease list. A four-period cell has sixteen out of a hundred and forty-four, which is a ninth. A patch big enough to be interesting is, proportionally, almost all pattern.

And yet the cost of lettering it is dominated by the free letters at every size measured, in the sense that removing them changes the cost by three orders of magnitude while the rest of the pattern grows steadily. A vanishing fraction of the crease list is doing all the work — which is a familiar shape in this subject, and is the same shape as rarity falling while cost stays flat. It is also why enlarging a patch never converges to the tessellation in the way one might hope: the free letters thin out and never leave, and what a patch reports depends on where its edge fell at every size.

What a folder takes from it #

One thing, and it is about reading rather than folding.

A crease pattern printed on paper always has an edge, and the letters near that edge are the least constrained on the sheet. If a lettering is being checked by hand — vertex by vertex, as a designer checks — the edge is where the checking is least informative, because a crease with one end on the paper’s rim answers to one vertex instead of two and can very often be either letter without any vertex complaining.

The reader who wants to know whether a lettering is right therefore learns least by looking at the rim, which is the opposite of where the eye goes. Ninety-nine drawn letterings in a hundred pass every vertex condition at the scale a designer works at, and the ones that fail do so for reasons no local inspection reaches — the same reason a lettering agreeing at every vertex is not a lettering that folds.

What the accounting does not settle #

Two things, and both are worth flagging because the arithmetic above is tidy enough to be over-read.

It does not say how much each free letter is worth. Four per period of edge is a count, and the cost difference it produces is not four times anything — it is three orders of magnitude at four periods and a factor of one and a half at two. Whatever the relationship between the number of severed constraints and the difficulty of the remainder, it is not linear, and nothing here measures it.

And it does not extend beyond patterns that repeat. A crease pattern with no symmetry has no periods, so there is no second reading of any rectangle drawn on it and no way to ask what its rim contributes. Every pattern in this essay is a tessellation for that reason and not because tessellations are especially interesting here; they are simply the only patterns for which the question can be put.

The nearest thing to a general statement is negative and worth having: a rim’s contribution to a pattern’s difficulty cannot be read off the rim. It depends on what the rim severed, which is a fact about the pattern beyond the paper’s edge, and on a sheet that was never cut out of anything there is nothing to sever. The square a designer starts from is a choice about paper, and on a pattern drawn to fill it, the choice costs nothing at all.

What the four letters are worth #

The accounting says how many letters a cut adds and not what they buy, and the second number is the reason the first is interesting.

On the same rectangle at four periods — sixteen divided creases, sixty-four vertices either way — the cut sheet gives up a consistent lettering in forty-eight steps and the joined one in fifty-six thousand seven hundred and seventy-two. Sixteen letters, a factor of more than a thousand.

At one period it is four letters and five steps against three. At two, eight letters and thirteen against nine. The gap opens somewhere between two periods and three, which is where the joined sheet acquires more than one period of long-range constraint in each direction and the propagation stops reaching across it.

What the picture cannot show #

A figure of a rectangle with rings on its sides cannot show that the rings come in pairs. Which crease on the left is which crease on the right is a fact about the drawing’s periodicity, and the picture shows the drawing rather than its periodicity; a reader who wants to check the pairing has to slide one side of the figure onto the other in their head.

And no figure here shows the freedom itself, because a freedom is an absence. The eight letters a two-period square cell has and the pattern does not are not marked on the paper anywhere; they are eight questions the cut sheet is allowed to answer twice, and the only visible consequence is a search that finishes sooner.

There is one more thing outside a picture’s reach, and it is the reason this accounting exists rather than an argument. Whether a constraint is long-range or local is not visible anywhere in a crease pattern: two creases at opposite edges of a sheet look exactly like two creases, and the fact that they are the same crease of the tessellation is a fact about the drawing’s symmetry rather than about its ink. A reader looking at a patch cannot tell which of its letters are answerable freely and which are answerable only in agreement with something across the sheet — and neither, until the two readings could be built side by side, could this collection.