What the rim was doing

A crease pattern cut out of an infinite tessellation has a property no corrugation has: its boundary runs through the middle of its own structure. The construction fills the plane, a square is taken out of it, and the square’s edge falls wherever it falls — through a pleat, across a twist polygon, between two of them. Nothing about the pattern anticipates the cut.

That property was offered as the explanation for why such a patch is harder to reason about than any other pattern in this collection, and the argument for it was a good one: five families of pattern behave identically and one does not, and each of the five kills a different alternative. Size is killed by a grid four times larger than any patch. Disorder is killed by a crumple. Irregularity is killed by a mesh whose vertices are all different. The construction is killed by the corrugations, which come out of the same machinery. What is left standing is the rim.

Standing is not the same as measured. A refutation of four alternatives is a statement about four alternatives, and the comparison that would settle it — one pattern with a rim against the same pattern without one — was not available, because everything about a folded sheet here assumes a disc of paper with an edge somewhere.

Joining the edges #

A twist tessellation repeats. Take a rectangle that is exactly one period of it and a crease running off the right-hand side runs back on at the left, in the same place, at the same angle, because the drawing on one side of the rectangle is the drawing on the other side moved over.

So the two pieces the picture shows are one crease, and the sheet they belong to has no edge at all: it is the plane, and the rectangle is a bookkeeping device for describing it. Joining the edges makes that explicit. A crease divided by the cut becomes one crease again and takes one letter. A panel touching the left edge and the panel touching the right edge at the same height are one panel. And every vertex has a full turn of paper around it, because there is nowhere left for a vertex to be short of paper.

The counts come out as a torus’s counts must. On a two-period square cell the drawing shows twenty-five panels and the sheet has sixteen; forty crease pieces are thirty-two creases; there are sixteen vertices. Sixteen minus thirty-two plus sixteen is nothing, which is what Euler’s formula gives for a torus and is a check rather than a restatement, because those three counts come from three separate acts of joining.

What is held still, and what moves #

The comparison this makes possible is deliberately narrow. The rectangle is the same rectangle. The drawing inside it is the same drawing, out of the same construction at the same turn, pitch and fill. The vertices are in the same places and the four conditions asked of each of them are the same four conditions.

One thing differs, and it is the letters. A crease the cut divides is two creases on the cut sheet, free to carry different letters on the two sides of the cut, and one crease on the joined sheet. On the two-period square cell that is exactly eight creases. On the four-period cell it is sixteen. In general it is four for every period of edge, because a square cell of an n by n patch has four sides and one crease crosses each side per period.

The vertex count does not move at all. This is not luck: the rectangle’s corner is placed where its sides run furthest from every drawn point, so the cut falls between the vertices rather than through them, and a figure that found otherwise would be refusing to draw. Sixteen vertices asked before, sixteen after; four labellings admitted at each, before and after.

The measurement #

Both sheets are searched the same way: propagate the conditions at every vertex to a fixed point, branch on the crease the propagation has left least decided, and test at each step whether the letters so far have contradicted themselves. Same procedure, same order of choices, same test.

Cut out of the plane: five steps at one period, thirteen at two, thirty at three, forty-eight at four.

Joined up: three, nine, six hundred and twenty-five, and fifty-six thousand seven hundred and seventy-two.

The lower line is linear and stays linear. Forty-eight steps on eighty-one panels is a little under six-tenths of a step per panel, and the ratio holds between a half and two-thirds at every size on every tiling here. The upper line is not linear in any sense: it multiplies by three, then by seventy, then by ninety, and at the largest size measured it is more than a thousand times the lower one.

Four letters per period of edge buy that.

The arithmetic is worth stating as arithmetic. At four periods the joined sheet has a hundred and twenty-eight creases and the cut one has a hundred and forty-four, so the cut sheet is choosing among $2^{16}$ times as many letterings before any condition is applied. It searches a thousand times less. Whatever the extra sixteen letters do, they do not enlarge the problem in the sense that matters — they dismantle it.

The same shape on four tilings #

One tiling could be a peculiarity of the square. It is not.

The triangular tessellation’s period is a rectangle one tiling unit across and $\sqrt3$ up. Cut out, it costs twelve steps at one period, forty-six at two, ninety-three at three and a hundred and fifty at four, on twenty-three, sixty-nine, a hundred and thirty-nine and two hundred and thirty-three panels. Joined, it costs eight at one period, four hundred and fifty-five at two, and does not finish inside two hundred thousand at three. The honeycomb behaves the same way: twelve, thirty-six, seventy-six and a hundred and forty-eight cut, against eight, one thousand and forty-three, and unfinished.

The pattern is identical in every case and the crossover is at the same place: two periods of edge is where the joined sheet stops being a small problem, and by three periods it is out of reach of a search that reads a hundred and thirty-nine panels of cut paper in under a hundred steps.

Why freedom is cheap #

The mechanism that was offered for the rim making things hard is right about where the work happens and wrong about what the work is.

A vertex in the interior of a pattern has all of its creases present, so its conditions constrain fully and a single known letter usually settles the rest. A vertex whose paper runs out has fewer creases in its conditions and constrains less, and a crease with one end on the rim answers to one vertex instead of two. All of that is true, and it does mean the propagation stops earlier near an edge than in the middle.

But a search’s cost is not the number of decisions it takes. It is the number it takes and withdraws. A crease that nothing constrains can be given either letter and neither choice can ever be wrong, so a decision about it costs exactly one step and is never revisited. The rim is full of such creases. It is not a region where the search struggles; it is a region where the search cannot fail.

Take the freedom away and every one of those creases acquires a partner on the far side of the sheet that it must now agree with. A letter written at the left edge is a letter written at the right edge, and the consequences of the two propagate towards each other across the whole pattern before anything discovers whether they are compatible. That is a constraint of the worst possible shape for a propagating search: long-range, invisible locally, and discovered only after a great deal of work has been done on the assumption that it holds.

There is a second reason, and it is about when the constraint is discovered. The propagation is a local sweep: it reads a vertex, writes what the vertex forces, and moves on. A constraint between two creases at opposite edges of the sheet is invisible to every vertex individually — no vertex has both creases in it — so nothing in the propagation can report it. It is discovered only when a chain of forced letters has crossed the whole pattern and arrived at the far edge disagreeing with what is written there, at which point every decision taken along the way is suspect and the search has to undo them one at a time.

That is precisely the shape of constraint that makes a lettering rare while leaving it cheap to find, read in the other direction. On a cut sheet the only global condition is the one about circles in the layer arcs, and the propagation is decisive everywhere else. Joining the edges adds a second family of global conditions — one per severed crease — and the propagation is decisive nowhere near an edge, because there is no longer any such thing as near an edge.

The four-period cell, in detail #

The largest joined cell measured here has sixty-four panels, sixty-four vertices and a hundred and twenty-eight creases, and its search visits fifty-six thousand seven hundred and seventy-two nodes. The cut version of the identical rectangle has eighty-one panels — the extra seventeen are pieces of panels the cut divided — a hundred and forty-four creases and the same sixty-four vertices, and its search visits forty-eight.

The cut sheet has more panels, more creases and exactly as many conditions, and it is a thousand times cheaper. Any account in which difficulty tracks size is refuted by that one line.

The reading that has to be withdrawn #

Five families sit at one step per panel. The clipped tessellation patches were said not to. Measured under a fixed letter order rather than a randomised one, they sit at about six-tenths of a step per panel, at every tiling and every size — which is not on the line, and is below it rather than above.

So the family that was called the hard one is the cheapest object in the collection per panel of paper, and the variability that made it look hard was manufactured by one line of the search’s own arrangement and was removed as soon as it was found. The rim was blamed for a cost that, once the coin was taken out, is not there.

What the rim does is the opposite. It makes a patch cheap. Cutting a sheet out of a tessellation is not damage to the pattern’s structure from the search’s point of view; it is the removal of every long-range constraint the pattern had, replaced by a ring of creases nobody has to get right. That is the same observation the corner of a sheet makes about paper turned into a statement about search: an edge is where the constraints run out.

What the counts say about the two objects #

It is worth being exact about what joining the edges does to the pattern as a graph, because the direction of every change is the opposite of the direction difficulty is usually supposed to run in.

Panels fall: twenty-five to sixteen at two periods, eighty-one to sixty-four at four, because the pieces the cut divided are reunited. Creases fall: forty to thirty-two, a hundred and forty-four to a hundred and twenty-eight. Vertices stay: sixteen and sixty-four either way. And every vertex that was interior stays interior, while the vertices the rim had left short of paper — the ones a patch’s conditions never reach — cease to exist, because there is no rim for them to sit on.

So the joined sheet is smaller in every count and asks every question the cut one asks. It is a strictly smaller, strictly more constrained problem, and it is a thousand times dearer.

What this does not say #

It does not say a rim is always slack. A rim is slack here because the construction is periodic and the cut severs constraints that ran between one period and the next. A pattern whose creases end at the paper’s edge because the construction put them there — a grid, a corrugation, a leaf — has no severed constraints at its rim, because it had none to sever. That is why those families sit at exactly one step per panel and a patch sits below: the patch has had something taken away.

It does not say the joined sheet is harder for a folder. Nobody folds a torus. The joined sheet is what the pattern is, considered as an object rather than as a piece of paper, and it is the object every claim about a tessellation ought to be about — the tessellation, rather than one square of it. A patch is a specimen; the sheet with no edge is the thing the specimen was taken from, and the counts a patch reports depend on where its edge fell in ways the tessellation’s own properties do not.

And it does not say that the search on the joined sheet is the right search. It is not, quite, and the reason is the subject of the loop that turns out not to be one: the test for a contradiction that this collection has used since it began is a test for a disc, and on a sheet with no edge it rejects letterings that are perfectly good. The numbers above are from the test that reads a sheet with no edge correctly. Under the collection’s own test the joined cell is not merely expensive; it comes back with a proof that no lettering of it exists.

Where the cut falls #

If the cut is what makes the pattern cheap, a natural question is whether it matters where the cut falls, and the answer is that it barely does.

Sliding the same rectangle across one whole period of the same drawing gives twelve different patches: different creases divided, different part-panels round the edge, panel counts running from forty-nine to sixty-one. Their search costs run from twenty-five to thirty-three steps. Every position of the cut is within a factor of one and a third of every other, against a factor of more than a thousand for whether there is a cut at all.

That is a useful shape for the claim. If the rim’s contribution were about what it happens to sever — which creases, which corners of which twists — the spread across cut positions would be large. It is not. What the rim contributes is that it severs at all.

The instrument, and what it had to be checked against #

A comparison like this is only as good as the guarantee that both sides are the same drawing, and the guarantee here is that both are cut out of one set of segments computed once. The construction’s geometry — the tiling, the side distance propagated across it, the polygons — is computed before anything is cut, and the two sheets are two different things done to the same output of it.

The joining itself is checked in four ways that can each fail. Two routes to one panel must agree about which period it belongs to, and they agree to within a part in a hundred million million. Every crease meeting a side must have exactly one partner across it, and every panel touching a side must have a partner. Every folded crease must lie on exactly one crease of the drawing. And the three counts must satisfy Euler’s formula for a torus, which they do on fifteen cells over five tilings.

What a designer takes from it #

Very little directly, and one thing indirectly.

Directly: nothing. A designer’s sheet has an edge, always, and the patterns in this essay that have no edge are objects of study rather than objects to fold.

Indirectly: a claim about a pattern’s difficulty should say which sheet it is about. The tessellation is an infinite object and a patch is a finite one, and they are not merely different in size — they are different in kind, and the finite one is systematically easier because a cut destroys constraints. Any statement of the form this family of tessellations is hard to letter that was established on patches has been established on the easy case.

The cost of the other question #

There is a second measurement on the glued sheet worth setting beside the first, because it says how far the two objects have come apart.

Asking the wrong question of the joined rectangle — applying the collection’s own rule that any loop in the layer relations is a contradiction — closes a search tree and reports that no lettering exists. Doing that costs three steps at one period, thirty-five at four and three thousand four hundred and fifty-five at nine, and does not finish inside two hundred thousand at sixteen.

So on the same object there are three numbers rather than two: forty-eight steps to letter it cut out of the plane, fifty-six thousand seven hundred and seventy-two glued, and an unbounded amount to prove the false thing about it. The rim is worth the gap between the first two; the missing hypothesis is worth the third.

Where the cut falls, on three more tilings #

The square tessellation’s twelve cuts move its cost by a third, and the other tilings are tighter still.

The triangular grid’s twelve positions of the same rectangle give panel counts of a hundred and thirty-three, a hundred and thirty-nine or a hundred and forty-five, a hundred and eight vertices at every one, and costs from seventy-four to ninety-one steps. The honeycomb’s are seventy-five to ninety-two on the same counts. The elongated triangular tiling’s run from a hundred and twenty-nine to a hundred and thirty-seven — a spread of six per cent.

Four tilings, forty-eight patches, and the whole variation across every way of cutting each of them is between six and thirty-three per cent, against a factor of more than a thousand for taking the cut away.

What the picture cannot show #

The upper line stops. At four periods the joined square cell costs fifty-six thousand steps and at three periods on the triangular tiling the search does not finish inside two hundred thousand — so the shape of the growth beyond that is not measured, and no claim is made about it. What is measured is a gap of three orders of magnitude at a size where both sides can still be finished, on a pair of objects that differ in one thing.

Nor does a step count say how the two searches differ in kind. Both propagate and both branch; what changes is how often a branch turns out to be wrong. On the cut sheet, across every size measured, the answer is never. On the joined sheet it is often, and the withdrawn decisions are where the whole difference sits.

And a step count says nothing about seconds. Each step propagates every vertex condition to a fixed point and then tests the letters so far, and both cost more on a larger pattern, so the flat lower line is flat in decisions rather than in time. What it says is that the cut sheet’s search is not growing in the way that matters — it is reading the pattern rather than exploring it, which is exactly the distinction a corrugation makes most cleanly and the one a patch turns out to share.