The pieces without the list

The letterings a crease pattern folds in are a set, and a set has a shape. Walking between two foldings asked what that shape is at a single vertex and found one piece; the creases that cannot move asked it of a whole pattern and found many, with a rule attached — the number of pieces is two to the power of the number of buried creases, meaning creases with an interior vertex at each end.

That rule was found by writing every lettering of a pattern down and joining the ones a single change relates. Writing them down is the problem. A pattern with n creases has 2^n letterings, the walk in this repository refuses at eighteen, and the patterns anybody actually folds are far past that: the Miura fold printed here has thirty-eight creases and the Yoshimura eighty-six.

So the rule is a conjecture about every pattern worth folding, found on the ones nobody would bother with. This essay is the instrument that checks it on the others, and its whole design is that it never enumerates anything.

Three measurements, and none of them is a list #

The rule connects two things: a count of creases, which is cheap, and a count of pieces, which is not. Checking it means finding evidence about the second that does not go through enumeration, and there are exactly three kinds available.

The buried creases are read off the graph. A crease is buried when both its ends are interior vertices, and that is a question about the pattern’s own edge list — one pass, no folding, no search. It is what the rule is stated in terms of, and on the Yoshimura it comes to forty-eight, which predicts 2^48 pieces of a set with 2^86 members. Neither of those numbers is reachable by counting to it.

The freeze is checked move by move. The rule’s mechanism is that a buried crease cannot be re-lettered: flipping it disturbs two vertices at once, and a move has only one other crease to spend. That is a claim about the moves a pattern admits — a few hundred of them — rather than about its letterings, and it is decidable in milliseconds. Every pair of creases meeting at an interior vertex is flipped together and the whole pattern re-checked, and the rule requires that no surviving move touch a buried crease.

The pieces are counted by landing in them. Independent letterings can be drawn without enumerating anything, by the same constraint propagation that decides whether a pattern has an assignment at all with the branch order randomised. Each draw’s buried letters name the piece it landed in. On a pattern with four buried creases forty draws should collide constantly; on one with twenty-two they should almost never collide. Both are observable, and neither requires knowing 2^b.

What a move costs, and how rarely one exists #

The move census produces a second fact before it produces the first, and it is the one that makes the rest of the essay feel different at the table.

Across the eight printed patterns there are 862 pairs of creases that meet at an interior vertex. Of those, 107 survive — that is, flipping both letters leaves a pattern that still satisfies every condition at every vertex. And none of the 107 changes a buried crease, which is the rule’s mechanism confirmed on 862 tries rather than argued from a picture of one vertex.

The share varies enormously and it varies in a way that is worth noticing. On the preliminary base, twenty-five of twenty-eight pairs can be flipped: the pattern has one interior vertex, nothing is buried, and the folding set is one large connected piece. On the Miura, four of ninety. The tapered corrugation is the same, four of a hundred and eight.

So a Miura is not merely in one of four million pieces. The piece it is in is a place where almost nothing can be changed at all — which is exactly what a folder finds when they try to re-crimp one and discover that the sheet will not have it.

The walk that goes nowhere #

A census tries every move once from one lettering. A walk does something a folder would recognise better: take a move, then look for another from wherever that landed, and keep going.

A hundred and twenty steps from a lettering of the Miura visit sixteen distinct letterings. Not sixteen pieces — sixteen letterings, total, out of the millions the pattern admits. The same walk on the Yoshimura visits a hundred and fifteen, on the waterbomb tessellation a hundred and eleven, on the preliminary base seventy-three.

And on every one of them, the letters on the buried creases at the end of the walk are the letters they had at the start. Nine hundred and sixty steps across the eight patterns, and not one signature moved.

That is the claim as a folder would state it. A pattern’s letters divide into the ones a hand can still argue with and the ones the pattern settled when it was drawn, and the second kind are exactly the creases that never reach the edge of the paper.

Landing in different pieces #

The third measurement is the one that could most easily have come out the other way, because it does not test the mechanism at all — it tests the count.

Forty independent letterings are drawn from each pattern. The draws are independent because the search is randomised, not because the letterings are: propagate the vertex conditions to a fixed point, and where propagation stalls, branch on a crease with the two letters tried in whichever order the stream decides. Every draw is a lettering that satisfies every condition, and two draws differ because the decisions differ.

The two patterns with no buried crease produce one piece and forty draws that agree on it. The square twist produces fifteen distinct pieces out of a possible sixteen, which is what forty draws on sixteen boxes look like. And the four large patterns produce forty out of forty, every single time, which is what forty draws on four million boxes look like.

That count is a floor and is not divided by anything. The sampler draws from the solutions of a constraint problem and it does not draw from them uniformly; what it can establish is that at least this many pieces exist, and what it cannot establish is how many there are. The prediction supplies the second number and the draws are the check that the prediction is not absurdly high.

How weak the floor from forty draws is #

The draws are described as a floor rather than an estimate, and it is worth saying how low a floor forty draws can establish, because the answer is a long way below the prediction.

Suppose the sampler reaches $N$ pieces, all equally likely. The chance that forty draws are all distinct is the birthday calculation, and it stays above a half only when $N$ is bigger than about eleven hundred. So forty draws coming back all distinct establishes something over a thousand pieces and nothing more.

Against a prediction of four million on the Miura, that is three and a half orders of magnitude short. Against the Yoshimura’s two hundred and eighty million million, it is eleven orders short.

The essay is right that the count is a floor and not an estimate, and this is the size of the gap. The draws rule out the prediction being absurdly high in the way a prediction of ten pieces would have been ruled out; they do not distinguish four million from four thousand.

What would narrow it #

The bound improves as the square root of the draws, which is the discouraging direction. A thousand draws all distinct would establish about seven hundred thousand pieces; a million draws would establish about seven hundred million. Reaching a floor comparable with the Yoshimura’s prediction means drawing more letterings than there are pieces in the small patterns anybody can enumerate.

So the collision statistic is the wrong instrument for the large patterns, and it is worth being explicit that the evidence for them rests elsewhere. What actually supports the prediction on a Miura is the mechanism — the move census, which shows on eight hundred and sixty-two tries that no legal change touches a buried crease — together with the rule’s exact agreement on the two patterns small enough to enumerate.

That is a perfectly ordinary evidential arrangement and it is worth naming as one. A rule verified exactly on small cases, with a mechanism checked directly on the large ones, and a sampling check that rules out gross error and nothing finer. The sampling is the weakest of the three and is the only one that looks like a measurement of the quantity being predicted — which is exactly why it should be quoted with its own limit attached rather than as the number that confirms the count.

Two patterns with nothing buried #

The rule has a degenerate case and it is worth dwelling on, because a rule whose interesting behaviour is universal is usually a rule about the construction rather than about the object.

The preliminary base is both diagonals and both midlines of a square: eight creases, one interior vertex in the middle, and every crease running from that vertex to the edge of the paper. Nothing is buried, so the prediction is one piece, and the measurement agrees — forty draws, one signature, twenty-five of twenty-eight moves legal.

The fold-and-cut triangle is the same shape of answer for a completely different reason: it has one interior vertex too, and its six creases all reach an edge.

So the two patterns on the shelf that a beginner meets first are the two whose letterings are entirely negotiable. Every other pattern here has decisions in it that were taken when somebody drew the lines.

The arithmetic that ties the two counts together #

There is a third number in the shelf table that the rule predicts without being asked, and checking it is free.

If the folding set has 2^b pieces and each piece contains the same number of letterings, then the total number of letterings is 2^b times the size of one piece. On the square twist that is sixteen pieces of sixteen letterings each, and the pattern admits 256 — which is exactly what an exhaustive count of its twelve creases returns. On the hexagon twist it is sixty-four pieces of sixty-four, and the exhaustive count is 4,096.

Both of those can be checked by enumeration because both patterns are small. Neither can be checked on the Miura, and the essay says so rather than implying that a rule verified twice is a rule verified everywhere.

Why the propagation and not a coin #

The obvious sampler is to write random letters and test them. It does not work, and the reason is worth stating because it is the same reason almost every pattern fails.

A pattern with V interior vertices admits roughly 2^−V of its letterings, so a coin-flipping sampler on a Miura accepts about one draw in thirty thousand and on a Yoshimura about one in four million. That is not merely slow; it means the sampler’s output is dominated by whatever the acceptance test is easiest to pass, and nothing about the resulting distribution is under control.

What is sampled here is therefore the search rather than the lettering. Propagation does all the work the conditions can do; the stream only decides what happens where the conditions stop deciding. Every draw is a solution, the cost is a few milliseconds, and forty draws on a pattern cost what one costs because the per-vertex tables are built once.

Where the instrument refuses #

Three refusals are built in, and each of them exists because the corresponding silent failure is available.

A propagation that runs out of budget throws rather than returning “no lettering exists”. That distinction has bitten this repository before: the assignment solver once reported a ten-fold crumple as unsolvable after two seconds of accumulated propagation, and a negative answer from a search looks exactly like a result.

A pattern with no crease at all is answered from the prediction — one lettering, one piece — rather than by searching for something that is not there.

And the census counts pairs of creases meeting at an interior vertex only. Two creases that share a vertex on the edge of the paper are not a move: no condition is evaluated there, so flipping both changes nothing any condition can see, and counting it would inflate the denominator with pairs that were never candidates.

The creases that reach the edge #

The rule names buried creases, and the complement is worth stating in its own right because it is what a folder can see.

A crease with at least one end on the edge of the paper is not buried, and flipping it disturbs only one interior vertex — which a move can repair, because a move has a second crease to spend at that same vertex. So the negotiable letters are exactly the ones on creases that run out to a raw edge, and the settled ones are exactly the letters on creases that do not.

On the printed shelf the split runs from everything to a little over half. The preliminary base and the fold-and-cut triangle have every crease reaching an edge. The square twist has eight of twelve; the hexagon twist twelve of eighteen; the Miura sixteen of thirty-eight, the tapered corrugation eighteen of forty-five, the waterbomb tessellation thirty-four of seventy-six, the Yoshimura thirty-eight of eighty-six.

That last column is the useful one. A tessellation’s negotiable creases are the ones round its rim, and a bigger patch of the same tessellation has proportionally fewer of them — the rim grows like the side and the interior like the square of it. Enlarging a tessellation does not give a folder more to decide; it gives them less.

What is not decided here #

Everything in this essay is about the locally admissible letterings: the ones satisfying developability, Kawasaki, Maekawa and the big-little-big lemma at every interior vertex. Whether any of them actually folds is a different question and an NP-hard one, and nothing here claims otherwise.

That matters for the interpretation rather than for the arithmetic. The pieces are pieces of the locally admissible set; a folder who cannot reach another piece by local changes could not reach it whether or not the pattern folds globally, because the route passes through letterings that fail a condition at a vertex. What the caveat costs is the right to say “four million foldings” where the figures say four million pieces of a set of candidates.

What a folder should take from it #

Three things, and the third is the one that changes what happens at the table.

A crease that reaches the edge of the paper is negotiable and a crease that does not is not. That is a rule about a drawing, and it can be applied by looking: follow each crease to both of its ends.

The bigger the pattern, the smaller the room. The number of buried creases grows roughly with the number of interior vertices, so a tessellation is not a pattern with more choices in it — it is a pattern with more decisions already taken. The Yoshimura has eighty-six creases and forty-eight of them were settled by whoever drew it.

And a re-crimp is a local move. Pushing on a pleat to make a mountain into a valley is exactly the operation the census counts, which is why a sheet that will not be argued with is not being stubborn. On the Miura, four of the ninety changes a folder might try are available, and none of them touches the creases that decide what the sheet is.

The same instrument answers the question for a designed pattern as readily as for a crumpled one, and what it counts does not depend on where a lettering came from — which is what makes the two measurements comparable at all.