One solution of a search nobody ran

Open any crease pattern in this collection and the letters are already there. Every crease is drawn as a mountain or a valley, in a colour and a dash pattern chosen so that both survive a monochrome printer, and they look exactly as much a part of the drawing as the lines they are drawn on.

They are not part of the drawing. A crease pattern is a set of lines; the letters are a second object, chosen after the lines and constrained by them, and on the patterns here they are chosen by a machine that stops as soon as it has an answer.

The question this raises is how much of the pattern that arbitrary choice accounts for. The answer is: most of it.

Where a pattern’s letters come from #

The constructions in this collection all end the same way, and the ending is easy to overlook because it is short.

A twist tessellation is built from a tiling: place a polygon at every vertex of the tiling, turn each one by the same angle, join them with pleats. That is the geometry, and it is completely determined — the polygon sizes are solved from a matching condition and the pleat widths follow. At the end of it there is a set of lines with no letters on them.

The letters are then found by propagation: hold, for each interior vertex, the labellings of its own creases that satisfy every condition; strike the ones the decided letters contradict; where a vertex has run out of choices about a crease, write that crease’s letter in; and where nothing is being decided any more, guess, and carry on.

That procedure returns the first complete lettering it reaches. It is not choosing among the solutions and it is not looking for a good one. It stops.

The result is correct, in the strict sense that it satisfies everything asked of it. The mistake is in reading it as though it were derived.

Two correct answers that share nothing #

The measurement is straightforward. Take a patch, keep its own lettering, run a search for a lettering with no circle in its arcs, and count the creases the two disagree about.

Forty-five of eighty-four on the square patch. Sixty-six of a hundred and six on the elongated. Sixty-seven of a hundred and forty-two on the hexagonal, eighty-seven of a hundred and forty-two on the triangular, and a hundred and fifty-five of two hundred and eighty-two on the rhombille.

As shares of each patch that is 54%, 62%, 47%, 61% and 55% — four of the five above half and the fifth a little under.

It would be less surprising if the search had been forced away from the drawn lettering — if the drawn one were defective and the search had to repair it. On the largest patch that is the case: its own construction hands it a circle of sixteen panels, so a consistent lettering has to be somewhere else. But on the other four the drawn lettering is perfectly consistent. The search was not escaping anything. It found a different answer because there was no reason for it to find the same one.

The five shares average to a half, which is the whole finding #

Those five percentages are worth looking at as a group, because their average is the result and no individual one is.

Two letterings drawn independently, with each crease equally likely to go either way, would disagree on half their creases. The five measured shares are 54%, 62%, 47%, 61% and 55%, and their mean is 55.8% — a handful of points from chance, with the scatter running to either side of it.

So the construction’s lettering and the searched one do not merely differ a lot. They agree at about the rate two unrelated letterings would, which is the strongest available statement of the essay’s claim: the drawn letters carry essentially no information about the found ones, and the construction’s answer is statistically indistinguishable from an independent draw out of the admissible set.

That is a different assertion from more than half differ, and a better one. A pair of letterings could differ on two-thirds of their creases and still be strongly related — anticorrelated is as informative as correlated. Landing at chance is what carries no signal at all.

And the pieces are not few #

The partition makes the same point in a currency that cannot be argued with.

The rhombille patch has 222 buried creases and no admissible move touches one, so the letterings fall into $2^{222}$ pieces that nothing walks between. That is more pieces than there are atoms anywhere anybody has counted, and the two letterings this essay compares are two of them.

Which is why the chance-level agreement is not a curiosity about these five patches. Two draws from a space partitioned that finely have no reason to resemble one another, and the measurement says they do not.

The number is not a distance #

There is a natural way to read a hundred and fifty-five creases differ which is wrong, and it is worth blocking because the whole of this collection’s earlier work on repair was built on it.

The reading is: the two letterings are a hundred and fifty-five steps apart, so a walk of a hundred and fifty-five steps gets from one to the other. That would make the difference a distance, and a large distance is at least a thing one could cross.

The letters do not work like that. A single crease cannot be flipped on its own: flipping one changes the counts at both its vertices, and the count has to differ by two at every one of them. The smallest change that keeps a lettering admissible is a pair of creases meeting at an interior vertex, flipped together — the change a folder makes by pushing a point through.

And that move set is far more restricted than it sounds. No move that survives the conditions ever changes a buried crease — a crease with an interior vertex at each end — which was measured across every move every printed pattern admits, and holds without exception.

So the two letterings are not a hundred and fifty-five steps apart. They are in different pieces of the space, and the number of pieces is two to the power of the buried crease count. The rhombille has two hundred and twenty-two buried creases. Nothing walks between them.

That is the real content of the number, and it is why every attempt to repair a pattern’s lettering in this collection has come to nothing. The attempts were looking for a short path in a space with no paths.

Twenty seeds, twenty answers #

If the construction’s lettering were special, a search would tend to rediscover it. It does not, and the failure to rediscover it is systematic rather than a single accident.

Run the search from twenty different seeds on the rhombille patch — same pattern, same conditions, different order of guesses — and twenty consistent letterings come back, no two of them the same. On the four smaller patches the same thing happens for the same reason.

Twenty is a count of what was seen and not an estimate of what is there; nothing here divides it by anything. What it does show is that the search is not converging on some distinguished answer either. Both procedures are picking a member of an enormous set, and neither has any principle for picking.

What the propagation is actually doing when it guesses #

It is worth being precise about where the arbitrariness enters, because it does not enter everywhere and the places it does not are the places the letters mean something.

The propagation alternates between two activities. The first is deduction: a vertex whose surviving labellings all agree about some crease has decided that crease, and writing it in is not a choice. The second is a guess, taken when every vertex has at least two labellings left and no crease is forced by any of them.

On a tessellation patch the second activity dominates. The deductions run two or three creases and stop, because the conditions do not chain — fixing one letter settles about three others out of a hundred and fifty-eight, and then the implications run out. So a patch of two hundred and eighty-two creases is decided by something on the order of a hundred guesses, each of which could have gone the other way.

A hundred guesses is two to the hundred orders in which the same procedure could have run. It is not surprising that two runs share little; it is surprising only if one has been thinking of the procedure as a derivation.

The printed patterns are different in degree and not in kind. A fold-and-cut outline’s pattern has one to three interior vertices, so the deductions cover nearly everything and the guesses are few; two runs on it will often agree. The Miura’s fifteen chains sit in between. The arbitrariness is proportional to how much of the pattern the conditions fail to pin, and on a corrugation that is most of it.

The habit this corrects in the drawings themselves #

There is a small, concrete consequence for how the figures in this collection should be read, and it applies to several dozen of them.

A figure that draws a pattern and says the twist tessellation is drawing one lettering of the twist tessellation. If the caption goes on to say something about the letters — how many mountains there are, where the mountains cluster, which creases carry the same letter as their neighbours — that statement is about the drawn lettering and not about the pattern, and a second run of the same construction could contradict it.

Statements about the lines are safe: how many creases there are, what the sector angles are, how the pleats meet, how much the sheet shrinks. Statements about the letters are safe only when they are statements about every admissible lettering, and those have to be established by a sweep rather than by looking at the picture.

This collection has mostly got that right, and the place it most nearly got it wrong is instructive: a figure that shaded the panels lying on a circle, and then drew the pattern at its own letters rather than at the letters the shading was measured from. Two pictures pretending to be one, and it was caught by the assertion that the drawn lettering must have a circle in it — which the pattern’s own lettering did not.

What is a fact about the pattern, then #

Something has to be, or the letters would carry no information at all, and they plainly do: a folder handed a mountain where a valley belongs cannot fold the model.

Three things are facts about the pattern rather than about the choice.

The first is which letterings are admissible at all. That set is determined by the lines, and it is what every condition in the subject is about.

The second is the buried signature. Which piece of the space a lettering lives in is decided by its buried creases, and no move changes those — so the buried letters are not a free choice once the piece is chosen, and two letterings agreeing about them are genuinely related in a way that two letterings agreeing about the rest are not.

The third is the pattern’s circuit structure, which decides how many places a contradiction could sit. The independent closed chains of panels number the interior vertices, and that count predicts the share of letterings that agree with themselves across every family measured. It is a property of the drawing alone.

What is not a fact about the pattern is the particular lettering it arrived with. That is a fact about the order the propagation happened to take.

Why this matters for what gets printed #

This collection prints eight patterns at true scale for a reader to fold, and every one of them carries a lettering that was found this way. It is fair to ask whether that is good enough.

It is, and for a reason worth stating rather than assuming. The printed patterns are verified — every one is put past all four conditions at every interior vertex, and past the circle test, and the ones small enough are put past a search over the orderings of their panels as well. A lettering that survives all of that is foldable, which is the whole of what the printed sheet promises. The reader is not promised the canonical lettering of the waterbomb tessellation, because there is no such thing.

What would be wrong is to describe the letters as though they were derived from the geometry, and this collection has occasionally slipped into doing so. The waterbomb tessellation’s essay says its assignment was searched for rather than remembered, which is exactly right. Elsewhere the language runs the other way, and a pattern is said to have a lettering, as though the drawing determined it.

The one case where the letters are derived #

There is an exception, and it is instructive because it is the case where the language everyone uses is correct.

A crumpled sheet’s pattern is read off a folding that has already happened. The creases are where the paper was folded, and each one’s letter is which way it was folded, so the lettering is not chosen at all — it is a record. A crumple’s own letters never contradict themselves, at any depth, because a lettering derived from an actual folding is a lettering that has a folded state by construction.

Reletter the same crumple and the guarantee evaporates immediately: at eight folds, only eleven of forty redrawn letterings agree with themselves. The pattern is the same object. The lettering is a different one, and it was never the pattern’s to begin with.

That is the distinction this essay is about, in the one place where the two halves of it can be seen side by side.

The same holds, less obviously, for a pattern found rather than designed. The Yoshimura is what a thin cylinder does when it is crushed, and the buckled shell arrives with its letters on it in the same sense a crumple does: the ridges are ridges and the valleys are valleys because that is which way the material went. What this collection prints is a drawing of that pattern, and the drawing’s letters were found by the same propagation as everything else — so the printed Yoshimura’s lettering is a chosen one even though the object it depicts has a derived one. The two happen to agree here, and nothing guarantees it.

Whether any lettering deserves to be called the pattern’s #

One could try to distinguish a lettering by some property other than being first — the one with the most mountains, the most symmetric one, the one whose buried signature is all valleys — and call that the canonical choice.

Nothing recommends any of them. Symmetry is the closest to a real criterion, since a symmetric pattern’s letterings need not be symmetric and the ones that are form a much smaller set worth naming. But a tessellation patch clipped out of an infinite pattern has had its symmetry cut away at the rim, and the sixteen repeating rules of a grid corrugation are already the symmetric letterings of that family, so on the two cases where the idea would do work it has either been destroyed or already used.

The position this collection takes is the plainer one. A pattern has a set of letterings; the printed sheets carry one member of it, verified; and any statement about the letters that is not a statement about the whole set says which lettering it is talking about.

Where the ladder goes next #

If a construction’s lettering is one arbitrary solution, the obvious next question is what the whole family of solutions looks like when it can be written down. On a tessellation that is hopeless — the count has tens of digits — but a repeating pattern has a much smaller family hiding inside it: the letterings that repeat with the pattern. There are sixty-four of those for a grid corrugation, and sweeping all sixty-four says exactly which bit of the rule is forced and which is free.

And there is a question about the printed shelf this leaves open, which nothing here settles. Of the many letterings each printed pattern admits, is the one it ships with the easiest to fold — the one whose creases can be set in the fewest passes, or whose layers seat with the least fighting? That is a question about hands rather than about graphs, this collection has no instrument pointed at it, and the honest position is that the printed letterings were chosen by a propagation and have never been compared to their alternatives on any criterion a folder would recognise.