Drawn by the same hand

This collection keeps four populations of crease patterns and runs its checkers over them: the eight patterns printed at true scale, twelve twist tessellations, six quadrilateral meshes and seven fold-and-cut outlines. Thirty-three patterns, and the point of keeping them is that a claim beginning over crease patterns should be a measurement rather than a gesture.

They all have one thing in common that nobody chose and nobody noticed. Not one of them was drawn with two creases across each other, and for years nothing here would have said so if one had been.

What a drawing does when nothing is choosing it #

Drop two segments on a square with all four endpoints uniform, and they cross 23.1% of the time — measured over three thousand drawings, and close enough to a quarter that the exact value is not the point.

Add more and the share climbs the way independence says it should. Three segments: 51.2%. Four: 73.5%. Six: 95.2%. Eight: 99.4%. Twelve: indistinguishable from certainty, with 15.3 crossings in the average drawing.

The mean is the more useful number, because it grows with the pairs rather than the segments: about 0.23 crossings for every pair, so a drawing of n segments has roughly 0.23 × n(n−1)/2 of them. A Miura’s fifty-eight creases, drawn at random, would cross about 390 times.

That is the baseline against which this collection’s zero has to be read. A crossing is not a rare event that the patterns happened to avoid. It is what a drawing does, and every pattern here escapes it because it was not drawn — it was constructed.

The 23.1 per cent is exactly twenty-five over a hundred and eight #

The measured crossing rate for two segments is not an empirical constant that happens to be near a quarter. It is a classical result, and identifying it is worth doing because it turns the baseline from a simulation into an equation.

Two segments with all four endpoints uniform in a square cross exactly when the four points are in convex position and the pairing is the crossing one. Given convex position, one of the three pairings crosses; given a point inside the triangle of the other three, none does.

Sylvester’s four-point problem for a square gives the probability of convex position as $25/36$, so

$$P(\text{cross}) = \frac{25}{36} \times \frac13 = \frac{25}{108} = 0.23148\ldots$$

against the three thousand drawings’ 23.1 per cent. The baseline is exact, and the mean crossing count for $n$ segments is $\tfrac{25}{108}\binom{n}{2}$ with nothing sampled at all — 15.3 for twelve segments, 390 for a Miura’s fifty-eight, 9,050 for the rhombille’s two hundred and eighty-two.

Which sharpens what the nought is measured against #

That matters because a baseline computed rather than simulated cannot be dismissed as a modelling choice. The collection’s nought is being measured against a number with a proof behind it.

It also removes the essay’s own hedge about which null to use. Whatever region the endpoints are drawn from, the answer is the region’s convex-position probability over three — a disc gives $1 - 35/(12\pi^{2})$ over three, or 22.4 per cent — so the constant moves by a per cent or two and the counts move by nothing that matters.

The crossings are clumped #

One thing the measurements say that the exact rate does not, and it is worth extracting.

If pairs crossed independently at $25/108$, the chance of a drawing with no crossing would be $(83/108)^{\binom{n}{2}}$: 45.5 per cent at three segments, 20.7 at four, 2.0 at six. The measurements give 48.8, 26.5 and 4.8.

Every one is higher than independence predicts, which means crossing-free drawings are commoner than they should be and the crossings are positively correlated.

The mechanism is segment length. A drawing whose segments happen to be short crosses nowhere; one with several long segments crosses everywhere. So the pairs are not independent — they share the same lengths — and the distribution of crossing counts is wider than a binomial, with more drawings at nought and more at many.

That makes the collection’s nought slightly less extraordinary and only slightly: at twelve segments the independent model gives one crossing-free drawing in $10^{8}$ and the measurement gives rather more, and either way the shelf’s thirty-three noughts are not a draw from it.

Why a construction does not cross itself #

Each of the four populations is produced by a rule, and each rule forbids a crossing without mentioning one.

The printed shelf is patterns whose creases come from folds. Paper turned about a line leaves a line, and two such lines meet at a point that was on both folds — which is a vertex, and which whoever recorded the pattern drew as one because they watched it happen.

The quadrilateral meshes are grids: rows and columns of quadrilaterals whose creases are the edges of the cells. Two edges of a planar mesh meet at a corner or not at all.

The fold-and-cut outlines are skeletons plus perpendiculars, and a perpendicular is dropped from a node onto an edge and stops at its foot. The construction has an explicit end for every crease.

The twist tessellations are polygons and the pleats between them, all of them ending on a polygon — which is why the fault, when it finally arrived, arrived at the one place the construction had to invent an ending: the pleats of a unit whose neighbour fell off the paper.

So the populations are not a sample of crease patterns. They are a sample of the constructions this collection has written, and those constructions share an author, a set of habits, and a blind spot.

The moment the count moved #

The population did not go from zero crossings to one. It went from zero to thirty-two on a single patch, and to five patches carrying between 32 and 78.

That is the characteristic shape of this kind of blind spot. A fault a construction cannot produce appears in none of its output; when a construction finally can produce it, it produces it everywhere at once, because the mechanism that makes one makes them all. There is no gentle onset and no small first case to notice.

It is also why the fault survived the arrival of a test population at all. The populations were assembled to make claims about typical instances measurable, and they did that: they showed that satisfying every vertex condition does not mean folding, and that eleven of the thirty-three are refused by one proof or another. What they could not show is anything about a fault none of their members has.

The counts, pattern by pattern #

Set the two side by side and the gap is easier to feel than to argue.

The preliminary base has twelve creases. Twelve segments dropped at random cross fifteen times over; the preliminary base crosses nowhere, and its creases are four straight lines through the centre of a square, which is the most crossing-prone arrangement anybody could deliberately draw — four lines through one point would be six crossings if the point were not a vertex. What makes it a pattern rather than a scribble is precisely that the meeting is declared.

The Miura has fifty-eight creases and a random fifty-eight would cross about 390 times. The Yoshimura has ninety creases: about 920. The waterbomb tessellation has ninety-two: about 960. Every one of those patterns has nought, and every one is a grid whose creases are the edges of its cells.

The ratio is the part worth carrying, because it survives any reasonable choice of null. Whatever a random drawing is taken to be — segments with both ends uniform on a square, chords of a disc, segments of a fixed length dropped anywhere — the expected number of crossings grows with the number of pairs, and every pattern on the shelf has a crossing count that grows with nothing. Twelve creases against fifteen expected crossings is a gap no sampling error reaches, and the gap widens with every pattern on the shelf. A population’s nought is only worth remarking on when there is a baseline to remark against, and producing one was a few lines of arithmetic that nobody had a reason to write until the nought had a name.

The largest number in the collection is the rhombille patch, at 282 creases clipped from the plane — a random drawing of that many segments would carry about 9,000 crossings. Assembled the older way it carried twelve, which felt like a defect and is, against that scale, an extraordinarily small one. A construction that has gone slightly wrong is still a construction.

What a test population can and cannot do #

A population is a sample, and every sample carries the assumptions of whatever drew it. Three consequences follow, and this collection has now met all three.

A population tests a checker against the faults it contains. The four here contain patterns whose panels cannot be ordered, patterns whose panels cannot be placed, and patterns that fold — so the checkers were genuinely exercised on those, and eleven refusals is a healthy proportion for that job.

A population cannot test a checker against a fault it excludes, and the exclusion is invisible from inside. Nothing in the counts says no member of this population has a crossing, because nobody was counting a quantity that was always nought.

And a population built by the checker’s author shares the author’s model of what can go wrong. That is the deepest of the three and the least fixable. The constructions and the checkers were written from the same understanding of what a crease pattern is — a list of vertices, edges and letters — and both inherited the same gap: neither had a reason to consider the drawing.

Every quantity but one differed #

There is a sharper way to say what the populations were doing well, and it comes from the measurement that compared them.

Four ways of producing a valid pattern were asked the same four questions, and they disagreed by factors rather than by margins: how far the paper shrinks differed by twelve times across the populations, how much creasing it costs by six, how much of the pattern is boundary by two. That spread is the reason the populations exist — a sentence beginning over crease patterns means something different depending on which of the four is meant, and the difference is not a rounding.

On the quantity nobody was measuring, all four agreed exactly. Nought crossings, in every population, at every parameter, for as long as the populations have existed.

That contrast is the whole lesson in one line. Populations that disagree about everything measured can agree about anything unmeasured, and their agreement is not evidence of anything except that the same question was not asked. A spread across populations tests the sensitivity of a measurement; it says nothing at all about a measurement that was never taken.

What a random population would and would not be worth #

The obvious repair is to add a fifth population of random drawings. It is worth doing and it is worth being clear about what it buys.

What it buys is a supply of patterns with the fault in them: a dozen random segments come with fifteen crossings, so the crossing sweep would be exercised on every draw, and so would anything downstream that must refuse them.

What it does not buy is a test of anything else. A random drawing fails the vertex conditions immediately and comprehensively — Kawasaki’s condition is an equality, and a drawing satisfies an equality with probability zero — so a random population is refused at the second rung of the ladder and never reaches the interesting ones. It is a population of one fault repeated.

That is the general difficulty with random test cases in this subject. The property being checked is a coincidence of measure zero, so the two available regimes are constructed, where everything passes because it was built to, and random, where everything fails at the first condition. The interesting instances are the ones that pass several conditions and fail a later one, and those have to be built to fail, one fault at a time, by somebody who has thought of the fault.

What this collection did about it #

Two things, and only one of them is machinery.

The crossing sweep runs everywhere, on every pattern the collection draws, as part of the same gate the vertex conditions run in. A fault that no population contains is now refused by a test rather than avoided by a habit, which means the next construction to produce one will be caught on the build that produces it.

And the twist population is pinned to the older construction on purpose. Cutting a patch out of the plane removes every crossing, so a population built the repaired way would be twelve patterns that all pass — a worse test set. The population keeps the patterns with the fault in them, and says so, because a population used to exercise a checker should contain things the checker must refuse.

The second is the one worth carrying, because it inverts the instinct. When a construction is repaired, the natural move is to rebuild everything that used it. The right move for a test set is the opposite: keep the broken output, label it, and let it go on failing.

The habit this suggests #

The general form of the fault is worth stating without the crease patterns, because it is not about folding.

A checker and its test cases written by the same person against the same model of the object will share that model’s gaps, and the sharing is silent: the test suite is green, the population is varied, the counts are real, and the gap is a quantity that is always nought and therefore never printed. Nothing distinguishes this fault does not occur here from this fault is not being looked for.

What breaks the symmetry is an instrument built on a different model of the object. The crossing sweep reads a pattern as ink; the checkers read it as a list; and the two disagree exactly where the list is missing something. That is the same shape as this collection’s other useful disagreements — the strips folded as a line and as a sheet, the vertex enumerated combinatorially and solved as a linkage — and it is the only mechanism here that has ever found a fault nobody suspected.

So the habit is not add more test cases. It is compute the same thing twice from different premises, and treat an agreement as evidence and a disagreement as a finding. A second population drawn by the same rules is more of the same evidence; a second reading of the same pattern is a second opinion.

Why so long, and not at once #

A last question worth answering plainly: the collection has had a subdivision routine almost since it began, and that routine has been counting the vertices it adds since the day it was written. Why did nobody read the number?

Because the number was never the answer to a question anybody had. The subdivision existed to produce faces — the panel counts an essay quotes, and the exported files that a folding tool can load — and the vertices it invents on the way are an intermediate step of that job. A count that is a side effect of a computation is not a measurement, however visible it is in the code.

What turned it into one was asking a different question: not how many panels does this pattern have but do the list and the drawing agree. The arithmetic was the same and already running. The reading was new, and it came from writing an essay about a patch that failed to close and having to explain why.

What is not being claimed #

Not that the populations are bad. They are the reason several claims here are measurements, and the fault they missed was found by adding an instrument rather than by replacing them.

Not that 23% is a constant of nature. It is the share for segments with both endpoints uniform on a square, which is one of several reasonable models of a random drawing; chords of a disc, or segments of fixed length, give different numbers. What survives every model is that the share rises with the pairs and reaches certainty quickly.

Not that the random model is the right null. Segments with both ends uniform on a square are one way of saying a drawing nobody chose, and it is a generous one: real drawings made carelessly are still made with intent, so the true rate for a hand-drawn pattern is somewhere between nought and a quarter per pair, and nobody has measured it.

And not that a crossing is the last such gap. The argument here is about a class of fault — one a construction cannot produce and a checker does not look for — and the only honest position on how many of those remain is that the count is unknown and is not nought.