Six creases and the same straight line

Every interior vertex of a flat-foldable crease pattern carries an even number of creases and at least four, so nothing meets at three and the smallest vertex anybody folds is degree four. Almost everything in this collection is degree four: a grid, a Miura, a leaf, a crumple, a twist polygon’s corner. Degree six is rare enough that until now it appeared in exactly one pattern here.

That one pattern was used to make a claim. Four families cost exactly one search step per panel and the Yoshimura, at fifty-seven steps on sixty-five panels, does not — so degree six was said to break the equality, on the reasoning that a degree-six vertex has more admissible labellings and the propagation settles more of the sheet without branching.

One pattern is not a family, and the claim was recorded as needing one.

The family #

The Yoshimura is a corrugation of horizontal courses crossed by a zigzag: a straight crease running across the sheet, met from above and below by pairs of diagonal creases, repeated up the paper. It is the pattern a paper bag makes when it is crushed lengthwise, and it is the shape of an axially loaded cylinder that has buckled — the same family a corrugation’s own rules are swept over elsewhere here.

At six sizes — three columns by three rows, four by four, five by five, six by five, seven by six and eight by seven — it holds twenty-one, thirty-six, fifty-five, sixty-five, ninety and a hundred and nineteen panels, and five, eleven, eighteen, twenty-two, thirty-three and forty-five interior vertices.

Every one of those vertices is degree six, and the search costs sixteen, thirty-two, forty-seven, fifty-seven, eighty-two and a hundred and eight steps.

Linear, and not on the line #

The costs against the panel counts are 0.76, 0.89, 0.86, 0.88, 0.91 and 0.91. A straight line through the origin fits them, and the number of decisions the search takes back — across all six sizes — is zero.

So the Yoshimura is exactly as well-behaved as a grid. It propagates cleanly, it never guesses wrong, and its cost grows in proportion to its size. What it does not do is sit on the line at one; it sits a little under it, at about nine-tenths.

That is a much weaker statement than breaks the equality, and it is a much more interesting one, because the deviation is small, systematic and explicable.

Against the other families #

Set the six numbers beside the families that were said to obey the rule and the difference is small enough to need care.

A grid at nine sizes: four steps on four panels, nine on nine, sixteen on sixteen, and so on to two hundred and fifty-six on two hundred and fifty-six. Exactly equal, every time.

A tapered leaf at four widths: twelve on twelve, sixteen on sixteen, twenty on twenty, twenty-four on twenty-four. Exactly equal.

A Miura at four sizes: the same.

Six crumples: eight on eight, sixteen on sixteen, eighteen on eighteen, then thirty-four on thirty-five, thirty-eight on thirty-nine, seventy on seventy-one — one short, and the one is the last panel, which needs no decision because everything around it has already been decided.

The Yoshimura: sixteen on twenty-one, thirty-two on thirty-six, forty-seven on fifty-five, fifty-seven on sixty-five, eighty-two on ninety, a hundred and eight on a hundred and nineteen. Five, four, eight, eight, eight and eleven short.

The shortfall grows with the pattern rather than staying at one, so it is not a boundary constant. It is a slope: nine-tenths rather than one.

What degree does #

A degree-six vertex differs from a degree-four one in two ways, and they pull in opposite directions.

It covers more creases. Six rather than four, so settling one vertex settles half as much again of the sheet. Fewer vertices are needed to cover a pattern, and the Yoshimura’s forty-five vertices carry a hundred and nineteen panels where a grid’s two hundred and twenty-five carry two hundred and fifty-six. Fewer places to decide anything.

It admits more labellings. Sixty-four labellings of six creases; Maekawa’s requirement that mountains and valleys differ by two admits four-and-two or two-and-four, which is thirty. A degree-four vertex is cut from sixteen to eight. So a degree-six vertex is less decided than a degree-four one, in the sense that a larger fraction of what it could say survives.

The first effect makes a degree-six pattern cheaper and the second makes it dearer, and at the proportion the Yoshimura is normally drawn they nearly cancel. Nine-tenths of a step per panel is what the cancellation looks like.

The other direction #

The cancellation is not a fact about degree six. It is a fact about degree six at that proportion, and the same family drawn at a different row height falls to a quarter of the line.

That is the subject of the knife edge in the Yoshimura’s own proportion and only the outline belongs here. The row height decides the sector angles; the sector angles decide whether the big-little-big lemma has anything to say; and where it does, the tables fall from thirty to eight and the cost from fifty-seven steps to nineteen.

So the same family, at the same size, with the same degree at every vertex, costs 0.88 steps per panel at one proportion and 0.29 at another. Degree is held still across that comparison and the cost moves by a factor of three.

Degree is therefore not the variable. It is a variable, in that it changes how many creases one vertex covers, and it is not the one that decides the constant.

Which of the two effects wins, and where #

The cancellation being near-exact at $\sqrt3$ is a coincidence of that proportion, and it is worth seeing what each effect is worth on its own.

Vertices per panel is a fact about the lattice and does not move with the row height. The Yoshimura at eight by seven has forty-five vertices on a hundred and nineteen panels — 0.38 — where a grid at sixteen divisions has two hundred and twenty-five on two hundred and fifty-six — 0.88. Fewer than half as many places to decide anything, per panel of paper.

Labellings per vertex is thirty at $\sqrt3$ and eight past it, a factor of nearly four.

Multiply the first by something increasing in the second and the answer at $\sqrt3$ lands near one and past it lands near a quarter. That is the arithmetic, and it is arithmetic rather than a derivation: nothing here says what function of the table size the cost is, and the association across families is offered with its magnitude unexplained.

What can be said exactly is that the two effects are separable, because one of them can be moved with the other held still, and moving it moves the answer by a factor of three.

The Yoshimura’s vertex, in detail #

The vertex is worth drawing because its structure is what makes all of this possible.

A course runs horizontally across the sheet. Two creases leave the vertex upward, one to each side; two leave downward. So the six spokes sit at zero and a hundred and eighty degrees — the course — and at plus and minus $\beta$ above, and their mirror images below, where $\beta$ is the angle the zigzag makes with the course.

The six sectors are then $\beta$, $180° - 2\beta$, $\beta$, $\beta$, $180° - 2\beta$, $\beta$. Four of one size and two of another, in that order round the vertex.

Kawasaki’s alternating sum is $\beta - (180° - 2\beta) + \beta - \beta + (180° - 2\beta) - \beta$, which is zero for every $\beta$. So every row height folds flat, and the proportion is genuinely free — which is what makes the family a family rather than a single object with a fixed shape.

Why the drawing has to be careful #

There is a modelling trap in this pattern that has nothing to do with any of the above, and it is worth repeating because it makes every count in this essay possible.

A course is a single straight line across the sheet, and it is tempting to draw it as one crease from edge to edge. Do that and the routine that finds the creases at a vertex looks for creases incident on the point, finds a course passing through rather than ending, and reports four creases where there are six. Every vertex of the pattern is then analysed as though it were degree four, every table comes out at eight, and the whole essay evaporates.

So a course is drawn as a chain of segments, one between each pair of crossings. That is not cosmetic: it is the difference between measuring a degree-six family and measuring a degree-four one that does not exist.

Where the deviation actually is #

There is one more reading of the six numbers that is worth taking, because it explains why the ratio is not quite constant.

The costs per panel run 0.76, 0.89, 0.86, 0.88, 0.91, 0.91 — rising slightly with size and settling near nine-tenths. That is the signature of a boundary effect: the smallest pattern has five interior vertices and a great deal of rim, and the largest has forty-five and proportionally less.

The Yoshimura’s rim is not a cut. Its creases end at the paper’s edge because the construction put them there, so the rim vertices are ones the pattern intends to have rather than ones a clip produced — but they are still vertices with fewer creases and fewer conditions, and the letters near them are freer. So the smallest pattern is cheaper per panel than the largest, and the ratio rises toward its asymptote from below — the same effect that makes a cut sheet cheaper than the pattern it was cut from, at a much smaller scale and for the same reason.

A grid does the same thing and it does not show, because on a grid the numbers are exactly equal at every size. That equality is a coincidence of the grid’s arithmetic rather than a law, and the Yoshimura is what the same effect looks like when the arithmetic does not conspire.

The degree-six vertex on its own #

Everything above is about patterns, and it is worth checking the single vertex, because this collection has measured one before and the numbers have to agree.

A lone degree-six vertex with six equal sectors admits thirty labellings of its creases. That is the same thirty the Yoshimura’s tables hold, which is the arithmetic saying the pattern’s vertices really are the ordinary object and not something the corrugation does to them.

It is also the count that makes a single vertex’s own arcs unable to close a loop: at one vertex there is nothing for a loop to run through, and every one of the thirty is fine. What a lettering of a pattern has to satisfy on top of that is a condition among vertices, and the thirty is the number of ways each vertex can contribute to it.

So the Yoshimura’s difficulty, such as it is, is entirely in how the thirties interact. Each vertex on its own is settled; the sheet is what has to be decided.

What is still missing #

A second degree-six family. Everything above is one construction measured at six sizes and two proportions, and while that is a good deal more than one pattern, it is not two independent constructions agreeing.

The waterbomb has degree-six vertices among its degree-four ones and behaves like a mixture — 0.93, 0.92 and 0.89 at three sizes, between the grid and the tilted Yoshimura and much nearer the grid, which is where the proportions of the two kinds of vertex put it. That is consistent and it is not the same as a second pure family.

What would settle it is a corrugation on the triangular lattice: six creases at every vertex, all sectors equal, built by a construction with nothing in common with the Yoshimura’s. It is not built here, and the honest position is that degree six has been measured once, thoroughly.

What the claim was, and what it is now #

The sentence being corrected is worth quoting in substance because the correction is a narrowing rather than a reversal.

It said: the equality between step count and panel count is a fact about degree-four vertices laid out regularly, and the Yoshimura’s degree-six vertices break it, with the count coming out below the panel count because a degree-six vertex has more admissible labellings and the propagation settles more of the sheet without branching.

Two of those three clauses survive. The count does come out below the panel count. A degree-six vertex does settle more of the sheet, in the sense that it covers more creases. What does not survive is the reason offered for the direction: having more admissible labellings makes a vertex settle less, not more, and it is the covering that makes it settle more. The two were run together, and they pull opposite ways.

The correction matters because it predicts something the original does not. On the original account, degree six should always sit below the line. On this one it sits below the line by an amount that depends on the sector angles, and can be moved from nine-tenths to a quarter by a change too small to see — which is what happens.

The arithmetic that would settle it #

There is a version of this claim that would be a derivation rather than a measurement, and it is worth stating what it would take, because the gap is small and specific.

The step count is the number of creases the propagation never settles on its own. On a grid that number is exactly the panel count, and the reason is short: the propagation stops at one crease per panel, because a panel’s own commitment forces its neighbours and nothing forces the panel. On the Yoshimura the equivalent sentence is not known. The count comes out at nine-tenths of the panels at $\sqrt3$ and at a quarter past it, and both are stable across six sizes, so both are almost certainly countable from the pattern’s structure rather than merely measured on it.

What is missing is the argument. A degree-six vertex with thirty labellings, arranged in courses, leaves some countable number of free creases per unit cell; the same vertex with eight leaves another. Working those out would turn two empirical constants into two derived ones and would predict the constant for any degree-six corrugation before it is built — including the one this essay says has not been built.

Until then the honest description is that the linearity is established, the two constants are measured, and the reason for their particular values is not.

What a folder takes from it #

The Yoshimura is a pattern people fold, and one thing here is practical.

Its proportion is normally chosen so that the triangles are equilateral, because that is what a crushed cylinder makes and what looks right. That choice puts the pattern at the one row height where the big-little-big lemma is silent at every vertex, which means it is the proportion at which the most mountain-and-valley arrangements pass every local check.

For a folder that is neither good nor bad; the pattern folds either way. For anybody checking a lettering by eye it matters, because more arrangements passing locally means more of them failing globally, and the local check is the one a person can do.

What the picture cannot show #

A ladder of six points shows that the cost is linear and does not show that it is exact. The claim that no decision is ever withdrawn is asserted directly rather than inferred from the counts: a search that visits exactly n nodes while taking a decision and giving it back would produce the same line, and the requirement here is on the backtracks rather than on the total.

Nor does a picture of the pattern show its degree. Six creases at a point and four creases at a point look nearly the same at the scale a whole pattern is drawn at, and the difference between a course drawn as one line and a course drawn as a chain of segments is invisible in ink and decisive in every number above.

And nothing in a ladder distinguishes a pattern that is cheap because it is well-behaved from one that is cheap because it is over-constrained. Both look like a straight line through the origin, and the difference between them is whether a lettering exists at all — a question these six patterns all answer yes to and nine patches over a grid of parameters answer no to, at the same cost per panel.