Where a rule can close a loop

Four corrugation families in this collection have now had every one of their repeating rules built and tested. Three of them produce rules whose letters send four panels round in a circle — a proof that no arrangement of the layers exists — and one produces none at all.

The counts are thirty-eight of sixty-four for the Miura, thirty-eight of sixty-four for the tapered leaf, a hundred and twenty of five hundred and twelve for the waterbomb tessellation, and nought of sixty-four for the Yoshimura.

The Yoshimura is not short of failing rules. Thirty-eight of its sixty-four break the count somewhere, exactly as in the grid families. What it is short of is a particular kind of failure, and the reason is a fact about the degree of its vertices that has nothing to do with which letters the rule chooses.

What the circle needs #

A circle of four panels goes round one interior vertex, and orienting one requires the letters round that vertex to alternate strictly: mountain, valley, mountain, valley, all the way round. The paper turns over at every crease, so the arrow a crease contributes is its letter multiplied by the near panel’s orientation; for every arrow to point the same way round the ring, the letters must flip at every step to cancel the orientation flipping at every step.

An alternation has equal counts of the two letters, and the counting theorem requires a difference of two. So no admissible lettering closes a circle at a vertex — which is why a folding pattern never does — and a rule that has already broken the count at some vertex is under no such protection there.

The question is therefore narrow and mechanical: can a repeating rule write a strict alternation round this pattern’s vertices at all?

The two counts agree family by family: nought and nought, thirty-eight and thirty-eight, a hundred and twenty and a hundred and twenty. One is read off the letters at each vertex; the other is found by folding the sheet, collecting an arrow per crease and walking the panel graph for a back edge. Neither knows about the other, and they never disagree.

A straight line carries one letter #

Here is what stops the Yoshimura.

Its interior vertices are all of degree six: two halves of a horizontal course, and four zigzags leaving upward and downward. The two course-halves are collinear — they are two pieces of one straight line running through the point — and they lie in the same row of the pattern.

A repeating rule assigns a letter from a crease’s parity class. Both course-halves are in the same row, so under any rule in the family they carry the same letter. That is not a choice the rule gets to make; it is what makes the rule a rule.

Now count positions round the vertex. Going round, the six creases are: course-left, zigzag, zigzag, course-right, zigzag, zigzag. The two course-halves are three apart — opposite one another, as two halves of a straight line must be.

A strict alternation puts one letter at positions one, three and five and the other at two, four and six. Positions three apart therefore get different letters. And the rule has already given the two course-halves the same letter.

The alternation is unavailable. Not unlikely, not rare — unavailable, to every rule in the family, at every vertex.

Why degree four is different #

The same count at a degree-four vertex comes out the other way, and the arithmetic is one line.

A Miura’s interior vertex has two collinear row-halves and two collinear column-halves — two straight lines crossing. The row-halves are opposite each other, which at degree four means two positions apart, not three. An alternation puts the same letter at positions two apart. So the alternation wants the two row-halves to agree, which the rule already forces them to.

Degree four permits it; degree six forbids it. The general statement is short: a straight line through a vertex of degree d has its two halves d/2 positions apart, and an alternation gives them the same letter exactly when d/2 is even.

Four and eight allow it. Six and ten do not. That is a fact about parity rather than about paper, and it explains the whole table without appealing to anything about the individual families.

Sixteen, thirty-eight and ten #

The grid families’ rules partition exactly, and the partition is worth writing out because the three classes add to the whole with nothing left over.

Sixteen rules fold. Thirty-eight close a circle. Ten do neither — they break the count somewhere without producing an alternation anywhere.

$$16 + 38 + 10 = 64$$

Every rule is in exactly one class, and the classes are mutually exclusive for a reason rather than by observation.

Why folding and alternating cannot coincide #

At a degree-four vertex of a grid corrugation the two row-halves carry one letter $r$, and the two column-halves carry $v$ above the row and $w$ below.

Folding requires $v \ne w$: the counting theorem wants three of one letter and one of the other, and with the two row-halves already equal, the two column-halves must differ.

Alternating requires $v = w = \bar r$: consecutive creases round the ring must differ, and the ring is $r, v, r, w$.

Those are incompatible. A rule that folds can never close a circle and a rule that closes one can never fold, and the two counts are therefore counting disjoint sets rather than overlapping ones.

Which is what makes ten the interesting number #

The remainder is where the account has something left to explain, and it is small enough to name.

The ten are the rules with $v = w = r$ — both column-halves agreeing with the row letter, giving four of one letter and none of the other. That breaks the count as decisively as anything can, and it is the one arrangement that is neither three-and-one nor an alternation.

So the three classes are $v \ne w$, $v = w \ne r$, and $v = w = r$, and the sixty-four rules fall into them sixteen, thirty-eight and ten.

The partition is a statement about three bits at a vertex, repeated across the pattern, and the counts follow from how many of the rule’s six bits are free once each condition is imposed. That is why the Miura and the tapered leaf return identical tables: the partition never mentions a length, an angle or a taper.

It also says the Yoshimura’s zero is not a smaller version of the same thing. Its vertices have six creases rather than four, so there is no three-bit partition to make, and the class that closes circles is empty rather than small — which is the parity argument arriving as a missing column rather than as a low count.

What the two counts being equal is worth #

Two numbers agreeing exactly across four families is the kind of result that can be either a strong check or a tautology, and which one it is depends on how the numbers were produced.

The alternation count reads letters. It builds the pattern under a rule, visits each interior vertex, lists the creases round it in angular order, and asks whether consecutive letters differ all the way round. It never places a panel and never computes an arrow.

The circle count folds. It reflects every panel into place across the creases on a path back to a fixed panel, collects one arrow per crease from the panel that must lie below to the one that must lie above, and runs a depth-first walk on the panel graph looking for an edge back to something already on the stack.

The only thing shared is the pattern. If the sign rule relating a letter to an arrow were wrong, the second count would move and the first would not. If the angular ordering round a vertex were wrong — the mistake that makes or breaks the parity argument — the first would move and the second would not. Six hundred and forty rules, four families, no disagreement.

That is what makes the parity argument evidence rather than a story. The story predicts that alternation and circle are the same thing; two instruments that could not both be wrong in the same direction report the same counts.

The waterbomb, which has both #

The waterbomb tessellation is the case that makes the account testable rather than merely consistent, because it has vertices of both degrees.

Its geometry is a square grid with both diagonals in every cell and no vertical grid lines. That produces exactly two kinds of interior vertex: the centre of a cell, where two diagonals cross at four right angles, and a corner of the grid, where six creases meet at forty-five, ninety, forty-five, forty-five, ninety and forty-five degrees.

A hundred and twenty of its five hundred and twelve rules close a circle. Every one of those circles is four panels long and goes round a degree-four vertex — the cell centres — and not one goes round a corner.

That is the prediction, made before the measurement was looked at and confirmed by it. A family with both degrees closes circles at one of them and not the other, and it is the one the parity argument permits.

There is a wrinkle worth noting because it strengthens the case rather than weakening it. At a waterbomb cell centre the two diagonals are also straight lines through the point — so the same forcing would apply if the rule gave each diagonal’s two halves one letter. It does not: the rule assigns the four half-diagonals of a cell from four separate bits, so the halves may differ. The alternation is therefore doubly available there, and a hundred and twenty rules take it.

The sixteen and the twenty-six #

There is one number in the family table that the parity account does not explain, and it is worth flagging rather than glossing over: the Yoshimura has twenty-six surviving rules where the grid families have sixteen.

That is a different question — how many rules satisfy the count, rather than how many break it in a particular way — and it comes from the degree rather than from the alternation. The count wants a difference of two, so a degree-four vertex needs three of one letter and one of the other, and a degree-six vertex needs four and two. Four-and-two is a looser requirement than three-and-one: there are more ways to arrange it, and more rules land on one.

Working it through at a Yoshimura vertex: the two course-halves carry one letter, and the four zigzags carry letters drawn from the row parity and the direction of lean. Several combinations of those give four-and-two, and the twenty-six are what survives when the requirement is imposed at all twenty-two interior vertices at once.

So the two families differ in both directions, and for two unrelated reasons. The Yoshimura admits more rules because its vertices are of higher degree, and it closes no circles because its vertices are of a degree whose half is odd. Neither fact implies the other, and reading the table as though a family with more survivors must have fewer contradictions would be a coincidence mistaken for a mechanism.

What this does not say #

The claim is about repeating rules, and every part of that qualification is load-bearing.

An irregular lettering of a Yoshimura is under no obligation to give the two course-halves the same letter, so it can alternate round a degree-six vertex — and if it does, it fails the count there and closes a circle. Nothing in this essay says a Yoshimura’s letters cannot contradict themselves. It says no rule can make them. In fact the Yoshimura is among the most consistent patterns this collection has measured — a hundred and ninety of two hundred redrawn letterings agree with themselves — and that is a fact about its having only twenty-two independent chains of panels, which is a different fact again.

The claim is also about the shortest circle. A longer circuit — six panels round two adjacent vertices, eight round a twist polygon — is a different structure and this argument is silent about it. Across every population measured here circles of six, eight, ten and up to twenty-two occur, and none of them is reachable by the reasoning above.

And it is not a claim that the Yoshimura is safer to letter. Its rules fail the count thirty-eight times in sixty-four, which is a worse rate than the grid families’ forty-eight in sixty-four is better. What it is, is a family where a failing rule fails in the milder of the two ways: a vertex that will not close, rather than a set of layer demands that contradict one another.

A pattern that was found rather than designed #

There is a pleasing coincidence in which family turns out to be the exception, and it is worth a paragraph because it is a coincidence rather than a cause.

The Yoshimura is not a designed pattern. It is what a thin cylinder does when it is crushed — the buckling mode of a shell under axial load, named for the analysis that described it in 1955 rather than for anyone who invented it. Its vertices are degree six because the diamond tiling it forms is what the shell’s geometry produces, not because six was chosen.

So the family that cannot close a circle under any rule is the one nobody arranged. There is no mechanism connecting those two facts, and it would be easy to write a sentence implying one. What can be said is narrower and still worth saying: the structural feature that protects it — a straight course running through every vertex — is also the feature that makes it a corrugation of a cylinder rather than a flat tessellation, and both come from the same shell.

The tapered leaf sits at the opposite end of the same comparison and shows that the protection is not about being found in nature. A leaf’s corrugation is as found as the Yoshimura — it is what a plicate leaf packs into, and nobody designed it either — and its rules close thirty-eight circles, because its vertices are of degree four. Being a natural pattern buys nothing here; being a pattern with a straight line through a degree-six vertex buys everything.

The same question asked of the patterns that are not corrugations #

It is worth checking the account against the rest of the collection, because a parity argument that only ever describes four families is a description rather than a rule.

Every pattern this collection prints has its vertices at degree four, six or eight, and none of their own letterings closes any circle — which is expected, since all of them are admissible and an admissible lettering never closes the shortest circle at any degree. So the printed shelf says nothing either way.

The place the account is testable outside the rule families is the tessellation patches, whose irregular letterings close circles constantly. There the prediction is that no circle of four should ever appear, because those letterings are admissible and the count closes the four-panel case at every degree. And across every population measured — over a thousand circles — not one has four panels in it, and not one has an odd number. The shortest is six.

So the rule families and the patches are the two halves of one statement. Where the letters are admissible, four-panel circles are impossible and everything observed is six or more. Where a rule has broken the count, four-panel circles are the only thing observed — and whether they are available at all is decided by the degree, which is what this rung measures.

Where the ladder goes next #

The parity statement predicts a case nothing here contains: a repeating pattern with vertices of degree eight. Half of eight is four, which is even, so the alternation should be available and rules should close circles there — and no family in this collection has a degree-eight vertex to check it on.

The preliminary base has one, but it is a single vertex on a single sheet rather than a repeating pattern, so it has no rule space to sweep. Building a degree-eight tessellation and sweeping its rules would be the direct test, and it is the obvious next measurement on this ladder.

The other open direction is the one the wrinkle above points at. Whether a rule can write an alternation depends on which creases the rule forces to agree, and that is a property of how the rule is parameterised rather than of the pattern. A waterbomb rule with one bit per diagonal instead of four would close no circles at its cell centres, on the same pattern. So the count of circle-closing rules is a joint fact about a pattern and a description of it — which is an uncomfortable thing for a measurement to be, and worth saying out loud rather than leaving in the machinery.