A knife edge nine decimals wide

The Yoshimura is drawn with equilateral triangles. That is what a crushed paper bag makes, what a buckled cylinder shows, and what every diagram of the pattern uses, and it corresponds to one number: the height of a row, measured in half-columns, is $\sqrt3$. Every other proportion is a perfectly good Yoshimura and the pattern is linear in its size at all of them.

Nothing about the pattern requires it. Kawasaki’s condition holds at every row height, so the proportion is genuinely free and the family is a family. And at exactly that free parameter’s most natural value, the conditions the collection applies at a vertex change what they say — discontinuously, and in the direction that makes the pattern less determined rather than more.

The measurement #

The same Yoshimura, six columns by five rows, sixty-five panels, twenty-two interior vertices, at nine row heights.

At 1.2: thirty labellings admitted at each vertex, and fifty-seven search steps. At 1.5: thirty and fifty-seven. At 1.7: thirty and fifty-seven. At 1.73: thirty and fifty-seven. At 1.7320508: thirty and fifty-seven.

At 1.7320509: eight and nineteen. At 1.74: eight and nineteen. At 1.8, at 2.0, at 2.5: eight and nineteen.

$\sqrt3 = 1.7320508\ldots$, so the step falls between two consecutive values in the eighth decimal place, and the equilateral pattern is on the expensive side of it.

The vertex #

Six creases meet at a Yoshimura’s interior vertex: a course running horizontally in both directions, two creases going up, two going down. Write $\beta$ for the angle the zigzag makes with the course, so the six spokes leave at $0$, $\beta$, $180° - \beta$, $180°$, $180° + \beta$ and $360° - \beta$.

The sectors between them, in order round the vertex, are

$$\beta,\quad 180° - 2\beta,\quad \beta,\quad \beta,\quad 180° - 2\beta,\quad \beta.$$

Four of one size and two of another, and the arrangement is what matters: the four $\beta$ sectors sit in two adjacent pairs, and the two $180° - 2\beta$ sectors sit between them, each with a $\beta$ on either side.

Kawasaki’s alternating sum is $\beta - (180° - 2\beta) + \beta - \beta + (180° - 2\beta) - \beta = 0$ for every $\beta$, which is why the proportion is free.

Which sector is strictly smallest #

The big-little-big lemma forbids the two creases bounding a sector strictly smaller than both its neighbours from carrying the same letter. Everything turns on the word strictly and on the arrangement above.

Take a $\beta$ sector. Its neighbours are one $180° - 2\beta$ and one other $\beta$. It is never strictly smaller than both, because one of them is its own size. So no $\beta$ sector can ever satisfy the lemma’s requirement, at any row height at all.

Take a $180° - 2\beta$ sector. Both of its neighbours are $\beta$. So it is strictly smallest exactly when $180° - 2\beta < \beta$, which is $\beta > 60°$, which is a row height greater than $\tan 60° = \sqrt3$.

That is the whole derivation. Below $\sqrt3$ the small sectors are the $\beta$ ones and they are paired, so nothing qualifies. Above it the small sectors are the other two and they are isolated, so both qualify. At $\sqrt3$ exactly, all six are equal and nothing is strictly smaller than anything.

Reading it off the pattern rather than the algebra #

The derivation above is four lines and it would be a poor idea to trust four lines, so the sectors are measured off the drawn pattern instead.

At a row height of 1.2 the six sectors come out at $50.2°$, $79.6°$, $50.2°$, $50.2°$, $79.6°$, $50.2°$ and no sector is strictly smallest. At $\sqrt3$ they are six sixties and none is. At 2.2 they are $65.6°$, $48.8°$, $65.6°$, $65.6°$, $48.8°$, $65.6°$ and two of them are — the two $48.8°$ wedges, each with a larger neighbour on either side.

The measured angles agree with $\beta$ and $180° - 2\beta$ to the digits printed, which is the algebra being checked rather than assumed. And the table sizes come out at thirty, thirty and eight, which is the lemma’s answer being counted rather than predicted.

Why “unequal” is not the condition #

The natural reading of the lemma is that it applies whenever a vertex’s sectors are not all the same, and the row height of 1.2 refutes it.

At 1.2 the sectors are $50.2°$, $79.6°$, $50.2°$, $50.2°$, $79.6°$, $50.2°$. Two distinct values, differing by nearly thirty degrees; nothing equilateral about the vertex; a drawing that looks obviously irregular. And the lemma has nothing to say, because the four small sectors are in adjacent pairs.

So a vertex can be as far from equilateral as one likes and still be a vertex the lemma is silent at. What it needs is not inequality but isolation: a sector with strictly larger neighbours on both sides. That is a condition on the arrangement, not on the spread.

This collection has met the same distinction once before, at a vertex whose two smallest sectors happen to be equal, and the situation there is a tie between two sectors. Here it is not a tie: the four small sectors are all the same size by construction at every row height, and the arrangement makes that permanent rather than accidental.

The other threshold, which is not one #

There is a second angle in the problem and it does nothing, which is worth recording because it is the natural candidate for a threshold and it is not one.

At $\beta = 45°$ — a row height of exactly one — the sectors are $45°$, $90°$, $45°$, $45°$, $90°$, $45°$ and the vertex looks as though something ought to happen: right angles, a pattern on a square grid, everything tidy. Nothing happens. The four small sectors are still adjacent in pairs and the lemma is still silent, exactly as at 1.2 and at 1.5 and at 1.7.

At $\beta = 90°$ the pattern degenerates: $180° - 2\beta$ is zero and the zigzag creases lie along the course. That is not a threshold either; it is the end of the family.

So the only place anything changes in the whole range is $\beta = 60°$, and it changes there because that is where the two sector sizes swap places. A parameter with one interesting value in its whole range, and the pattern everybody draws is at it.

What the discontinuity costs #

Thirty labellings against eight, so the tables are nearly four times as large on the equilateral side. The search costs fifty-seven steps against nineteen, so the pattern is three times as dear.

Both numbers are stable. Every row height at or below $\sqrt3$ that was tried gives exactly thirty and exactly fifty-seven; every one above gives exactly eight and exactly nineteen. There is no transition region and no gradient: the quantity being decided is which of two arrangements the sectors are in, and an arrangement does not change continuously.

That is unusual in this collection, where most parameters move things smoothly. It is the same shape as a tessellation patch going from having no lettering to having one across a hundredth of a radian — and it is the same cause, a sector crossing a threshold and changing which sector is smallest, arriving in a different family.

The cost is a fact about six sizes, not one #

The step happens at the same place at every size the family is drawn at.

At three columns by three rows the two costs are sixteen and nine. At four by four, thirty-two and fourteen. At five by five, forty-seven and seventeen. At six by five, fifty-seven and nineteen. At seven by six, eighty-two and twenty-four. At eight by seven, a hundred and eight and twenty-seven.

The ratio grows with the pattern — 1.8 at the smallest size and 4.0 at the largest — because the equilateral version’s cost is nearly linear in the panels and the tilted version’s is nearly linear in something smaller. So the penalty for drawing at the natural proportion is not a constant factor; it widens as the pattern gets bigger.

Why nobody had looked #

The proportion was never a parameter here until this measurement wanted it to be.

The pattern was drawn with its row height fixed at $\sqrt3$ from the day it was first built, written into the construction as a constant, because that is the shape a Yoshimura is. Nothing was wrong with that — the constant produced the pattern everybody means by the name — and a constant is invisible in exactly the way this essay is about: a number that never varies is a number whose effect is never measured.

Making it an argument took one line and produced two facts that had been sitting in the pattern for as long as it existed. That is the ordinary way a parameterisation pays here: a generator run at one point of its own argument space is a generator whose behaviour has only ever been observed at that point.

What this is not #

It is not a claim that the equilateral Yoshimura is harder to fold, or worse, or wrong. It folds at every row height and the equilateral one folds exactly as well as the others; a folder would notice nothing.

Nor is it a claim about a physical threshold. A row height of $1.7320508$ and one of $1.7320509$ produce drawings that differ by less than the width of a pencil line at any scale anybody would print, and no measurement of paper could distinguish them. What differs is what the conditions say, and the conditions are exact statements about exact angles.

The discontinuity is therefore in the mathematics rather than in the material, and the honest description is that the collection’s own vertex conditions are a step function of a parameter that the subject treats as continuous.

Why a step function is not a defect #

It is worth defending the lemma here, because a rule that changes its mind on the eighth decimal invites suspicion.

The lemma is a statement about which letterings can fold, and it is true. At a vertex with a strictly smallest sector, the two creases bounding it cannot carry the same letter, because the paper on either side would have to pass through itself. At a vertex without one, they can — and the eight labellings the lemma removes at $1.7320509$ are genuinely impossible, while at $1.7320508$ they are genuinely possible.

So the step is in the world. What changes across $\sqrt3$ is not the strength of the rule but the set of foldings, and thirty is the right answer on one side and eight on the other. A vertex with six equal sectors really does have more ways to fold than a vertex with four equal ones and two smaller ones.

What a designer should do about it #

Nothing, if the pattern is being folded. Something, if a lettering is being checked by hand.

The equilateral Yoshimura is the proportion at which the most mountain-and-valley arrangements pass every condition a person can check at a vertex. Thirty per vertex rather than eight means that a lettering drawn by eye is nearly four times as likely, per vertex, to satisfy every local test — and a lettering satisfying every local test is still a long way from a lettering that folds.

The practical form of the advice is unglamorous. On a pattern drawn at a proportion where the lemma is silent, checking vertex by vertex is worth proportionally less, and the arrangement has to be tested as a whole. That is true of every pattern here to some degree; it is worst exactly where the drawing is most regular.

Tilting the rows is not the answer, because the equilateral proportion is chosen for reasons that have nothing to do with checking — it is what the physical object does. What the measurement offers is a reason to be more careful rather than a change to make.

Where else the arrangement matters #

The mechanism is about arrangement rather than about the Yoshimura, so it should show up wherever a vertex has repeated sector sizes, and it does.

A twist polygon’s corner has sectors of four different sizes on the square tessellation — $46.15°$, $90°$, $133.85°$, $90°$ — and the smallest is isolated between two right angles. The lemma applies, the table falls from eight to four, and that is why the patches are the most decided vertices in the collection.

A degree-four vertex with two equal smallest sectors is the case where the lemma is silent at degree four, and it is what the preliminary and waterbomb bases have everywhere — which is the reason those two bases admit eight labellings where a twist’s corner admits four.

The Yoshimura is the degree-six version of the same phenomenon, with one extra feature: because it has a free parameter, both sides of the arrangement are available in one family, and the comparison can be made with nothing else moving.

What the free parameter is worth #

It is worth noticing how unusual it is to have this comparison at all, because most of the collection’s vertices have no free parameter to move.

A grid’s vertices are right angles because a grid is a grid; nothing can be varied without making it something else. A twist polygon’s corner has angles set by the tiling and the turn, and moving the turn moves several things at once. A crumple’s angles are wherever the folds landed and cannot be set at all.

The Yoshimura has exactly one number in it, that number changes exactly one thing, and Kawasaki does not care what it is. So the pattern is a controlled experiment that happened to already exist — the same size, the same degree, the same construction, the same panel count, with a single condition switched on and off.

Experiments like that are rare enough here that when one turns up it is worth running at more than one size, which is why the step is measured at six.

Which theorem was checked, and how #

The sector angles are measured off the pattern rather than computed from the row height. Each vertex’s creases are found by incidence, sorted by the angle at which they leave, and the sectors are the differences — so a pattern drawn wrongly would give wrong angles rather than the intended ones, and the derivation above would be checked against the drawing rather than assumed by it.

The tables are enumerated: all sixty-four labellings of six creases, filtered by Maekawa’s difference-of-two and by the lemma’s requirement applied at each sector in turn. Thirty and eight are counts of what survives, not formulas.

And the search cost is measured under a fixed letter order rather than a random one, so that the same pattern gives the same number every time and the comparison between two row heights is a comparison of two patterns rather than of two runs.

A pattern with vertices of both kinds #

The Yoshimura moves a whole family across the edge at once, and there is a pattern here that sits on both sides of it simultaneously.

A waterbomb tessellation has degree-four vertices where its diagonals meet the grid and degree-six vertices where three creases cross, and its tables run from eight to thirty inside one pattern. Its cost per panel is 0.93, 0.92 and 0.89 at three sizes — between the grid’s 1.00 and the tilted Yoshimura’s 0.29, and much nearer the grid, because most of its vertices are the four-crease kind.

That is what a mixture should do, and it is worth having because the alternative account — that the constant belongs to a construction rather than to its vertices — predicts nothing in particular for a pattern built of two kinds.

What the picture cannot show #

A drawing of the vertex at $1.7320508$ and one at $1.7320509$ are the same drawing. The figures here use $1.2$, $\sqrt3$ and $2.2$ instead, which are far enough apart to see, and the reader has to take on trust that the transition is at $\sqrt3$ rather than somewhere in the visible gap — which the nine row heights measured establish and no picture could.

Nor does the shading in the sector figure show why a shaded wedge matters. That a sector strictly smaller than both its neighbours forbids its two creases from agreeing is a theorem about paper passing through itself, and it is argued elsewhere; here it is applied.

And no figure here shows the thing a reader most wants to see, which is what the twenty-two letterings that exist on one side and not the other actually look like. They are letterings of a whole sheet rather than of a vertex, and the difference between thirty and eight is a difference in what each of twenty-two vertices may say, multiplied together — a number far too large to draw and, as a rarity that falls while a cost stays flat shows, not a number that predicts anything about the search on its own.

There is also nothing in these pictures about the other threshold in the family, which is where the pattern stops being drawable rather than where the lemma changes its mind. As the row height grows the triangles get taller and thinner and the pattern approaches a set of parallel courses with nothing between them; as it shrinks toward zero they flatten out. Neither end is a threshold in the sense this essay is about, and neither is anywhere near $\sqrt3$.