How deep is a crossing

Two creases either cross or they do not. There is no third answer, and the consequence of the first is total: a crossing has no flat folded state at any angle or under any lettering, so a pattern with one in it cannot be folded, while the same pattern with the two creases a hair’s breadth apart is untouched by the whole argument.

A verdict that flips on a hair’s breadth is worth measuring rather than asserting. The question is how far inside the two creases a crossing actually sits — how much of a crease would have to be trimmed for it to stop being one.

The measurement #

For each crossing, take the four endpoints of the two creases that make it and find the nearest one. That distance is the crossing’s depth: how far past the end of a crease the meeting happens, which is exactly how much shorter one of them would have to be for the crossing to disappear.

The depths spread over two orders of magnitude. On the triangular patch the twelve crossings run from 0.00569 to 0.112 of a sheet width, with a median of 0.069. On the honeycomb, eighteen from 0.00605 to 0.102. On the rhombille, twelve from 0.00105 to 0.0835. On the elongated tiling, five from 0.00158 to 0.112.

At the 150 mm this collection prints at, that is 0.16 mm at the shallowest and about 17 mm at the deepest. The deep ones are not in doubt: a crease running a centimetre and a half past another is crossing it in any sense anybody means. The shallow ones are a different matter.

What half a millimetre means here #

Half a millimetre is not an arbitrary threshold. It is a distance this collection has already had to name three times, in three different arguments, and it came out roughly the same each time.

A drawn line has a width. A pattern printed at 150 mm with lines fine enough to be distinct is drawing at about a third of a millimetre; anything closer than that is one line rather than two.

A folder cannot place a crease better than about half a millimetre. Two folds from a bare square leave five hundred and sixty-five marks, the closest pair 0.52 mm apart, and that essay’s whole argument is that what bounds a folder is not what the axioms reach but what the paper can tell apart.

A crease is not a line at all. It has a radius — the paper turns through a region rather than at a point — and on ordinary paper that region is a few tenths of a millimetre wide.

So five of the forty-seven crossings here are shallower than the ink, the hand and the crease itself. On the paper, those five are not crossings; they are two creases meeting at a point, which is a T-junction and perfectly foldable. On the coordinates, they are crossings, and the pattern is refused.

The other margin: how close a pattern comes to an accident #

The depth says how far a crossing went past the end of a crease. The mirror question is how close a pattern that has no crossing came to having one, and it is the number that says whether a pattern is robust or lucky.

Measure it as the smallest distance from any crease endpoint to any crease it does not touch. On the eight printed patterns the answer is large: 53 mm on the preliminary base, 25 mm on the square twist, 24 mm on the fold-and-cut triangle, 20 mm on the waterbomb tessellation, 13 mm on the tapered corrugation, 12 mm on the Yoshimura, 6.2 mm on the hexagon twist — and on the Miura, 160 mm, which is more than the sheet, because no endpoint of it is anywhere near a crease it does not meet.

Those are not near misses. The closest of them is a dozen times the width of a drawn line and twelve times what a hand can place a crease to. The printed patterns are not avoiding crossings by a whisker; they are avoiding them by a construction that never puts a crease near one it does not meet.

The patches are different. On the repaired triangular patch the same margin is 1.32 mm, and on the rhombille 1.04 mm — an order of magnitude tighter, and only two or three times the distance at which a fold stops being placeable. A pattern with that margin is not wrong, and it is one bad decision away from being wrong.

Which answer is right #

Both, and the reason is that they are answers to different questions.

The coordinates are right about the construction. A crossing 0.16 mm deep is not a rounding artefact; it is where the construction put the crease, and it is there because a stub was extended until it hit the sheet’s edge and passed through something on the way. Shortening it to remove the crossing would be a change to the pattern, made for the reader’s convenience, and the pattern would then be a different one from the one the rule produced.

The paper is right about the fold. A reader folding a printed sheet cannot act on a difference smaller than the crease they are making. Whatever the coordinates say, the object in their hands has a junction there.

The resolution is not to pick one. It is to notice that a pattern whose verdict depends on a tenth of a millimetre is a pattern in trouble either way — because a folder’s error is folded too, and a crease placed half a millimetre off carries that half-millimetre into every layer above it. A shallow crossing is a place where the construction has left no margin at all, and the honest response is to repair the construction rather than to tune the threshold.

Where the threshold actually sits #

The subdivision that finds crossings uses a tolerance, and it is worth stating rather than leaving implicit, because a reader is entitled to know how the verdict was reached.

A meeting counts as a crossing when the intersection point lies strictly inside both segments — by more than a part in 10⁹ of each one’s length. On a sheet 150 mm across that is 0.15 micrometres, which is far below anything physical: it is a numerical guard against two creases that share an endpoint being read as crossing there, and nothing else.

So the threshold is not doing any of the work. All forty-seven crossings measured here are between a thousand and a hundred thousand times deeper than the guard. Moving the guard by a factor of a thousand either way would change no verdict on any pattern in this collection. The tolerance is stated so that it can be seen not to matter, which is the only useful thing to do with a tolerance in a decision that has no middle.

What a tolerance would look like if there were one #

There is a version of this check that reports a distance rather than a bit, and it is worth describing because it is the one an engineer would ask for.

Instead of do these two creases cross, ask how far apart are they at their nearest approach — a number that is negative when they cross, by the depth measured above, and positive when they do not. A pattern’s score is then the smallest such number over all pairs, and it says how much the drawing could be perturbed before the pattern changed character.

That number would be useful for exactly the reason the bit is not: it is continuous, it degrades gracefully, and it can be compared between patterns. What stops it being the answer here is the same thing that makes the bit correct — the consequence is not continuous. A pattern with a crossing 0.16 mm deep does not nearly fold; Maekawa’s count refuses it outright, by the same margin it refuses one with a crossing a centimetre deep.

So the distance is a good diagnostic and a bad criterion, which is the ordinary relationship between a tolerance and a theorem in this subject. A near miss on Kawasaki’s condition is nearly as rare as an exact hit, and for the same underlying reason: the conditions are equalities, and equalities do not have neighbourhoods.

What a printer does to it #

The numbers above are geometry. What a reader gets is a printed sheet, and printing has its own resolutions, all of them near the same half millimetre.

An ordinary office printer lays down dots at 600 to the inch, which is 0.042 mm — finer than any depth measured here, so the geometry survives the raster. What does not survive is the ink: a laser toner line spreads by a few hundredths of a millimetre and an inkjet line by more, so two lines drawn 0.16 mm apart arrive as one mark about half the time and as a very thin gap the rest.

Then there is the printer nobody controls. A sheet printed with fit-to-page silently rescales, which is why every printable pattern here carries its intended size on the sheet — and a rescale changes every depth in this essay in proportion. A pattern printed at 70% has its shallowest crossing at 0.11 mm, which is under the toner’s own spread.

None of that changes a verdict, because the verdict is computed from the coordinates and not from the print. It changes what a reader can see of the verdict, which matters for a collection whose whole proposition is that the reader can execute the figure. A fault a reader cannot see on the sheet is a fault they will meet in their hands instead, three folds later.

Why the shallow ones are shallow #

The five shallowest crossings are not scattered at random through their patterns. They are all in the same place, and knowing where is what turns the measurement into a repair.

Every one of them is where a stub extended to the rim passes just beyond the far end of another stub near a corner of the sheet. Two creases arriving at the same corner region from slightly different directions will either miss each other or overlap by a little, and which of those happens is decided by where the lattice fell against the paper’s edge — a quantity nobody chose and nothing checks.

That is why the depths have the distribution they do: a few very shallow ones, from stubs that nearly missed, and a body of deep ones, from stubs that ran well past each other. The shallow tail is not evidence that the crossings are marginal. It is evidence that the construction was producing them by accident, at whatever depth the arithmetic happened to give.

Cutting the patch out of the plane instead removes all forty-seven, deep and shallow alike, because it removes the stubs. No threshold is involved, which is the mark of a repair rather than a workaround.

What this rung is for #

Three things, and the third is the general one.

A verdict can have a size even when it has no middle. The crossing test returns a bit, and behind the bit is a distance that says how firmly the bit was earned. Reporting the distance costs nothing and it is what distinguishes a construction that failed narrowly from one that failed comprehensively.

A pattern with no margin is a pattern that will not survive being folded. Every allowance in a folded structure is spent at the end, and a feature that sits a tenth of a millimetre from changing the pattern’s verdict has spent all of its allowance before the paper is touched.

And a tolerance should be stated in order to be shown irrelevant. The most useful thing to say about the threshold in this check is that no verdict here is within three orders of magnitude of it. A check whose answers cluster near its own tolerance is measuring the tolerance.

The depths in order #

Read as a distribution rather than as a minimum, the forty-seven depths say something about the construction that produced them.

On the triangular patch, twelve crossings spanning 0.00569 to 0.112 of a sheet width — a factor of twenty, with the median at 0.069, so most of them are deep and two are shallow. The honeycomb’s eighteen span 0.00605 to 0.102 with a median of 0.0876, the same shape. The rhombille’s twelve are the shallowest set: 0.00105 to 0.0835, median 0.0345, with four of the twelve under half a millimetre at printed size. The elongated tiling has five, one of them at 0.00158 and the rest deep.

The pattern in that is the rhombille having both the most crossings for its size and the shallowest, and the reason is that it has two sizes of twist in a fixed ratio and therefore twice as many creases arriving at the rim from twice as many directions. More directions means more chances for two stubs to meet, and more chances for two to nearly meet.

That is the honest summary of the whole measurement: the depths are the output of an accident, they are distributed the way an accident distributes, and their minimum is small because nothing was choosing it.

The shallowest one is a sample size #

The closing reading — that the rhombille has both the most crossings and the shallowest — is two statements about one thing, and the arithmetic says the second follows from the first with no mechanism in between.

Take $n$ depths spread over a range of length $L$. The expected minimum of $n$ such values is about $L/(n+1)$. So doubling the number of crossings halves the shallowest one, automatically, whatever produced them.

Run it against the four patches. The honeycomb has eighteen crossings over a range of 0.102, predicting a minimum near 0.0054; the measurement is 0.00605. The triangular has twelve over 0.112, predicting 0.0086 against a measured 0.00569.

Two of the four land within a factor of two of a prediction that assumes nothing but the depths were scattered without anybody choosing them — which is the essay’s own account of where they come from.

Which makes the minimum the wrong statistic #

That has a consequence for how the measurement should be reported, and it undercuts the number the essay leads with.

Five of forty-seven under half a millimetre is a count whose expected value rises with the number of crossings and not with anything about how bad the construction is. A construction producing a hundred crossings, all of them from the same accident at the same depths, would produce a smaller minimum and more sub-millimetre cases, and would not be worse in any sense that matters.

So comparing patches by their shallowest crossing is comparing sample sizes. The rhombille’s 0.00105 is smaller than the triangular’s 0.00569 partly because it is the same accident sampled from a different range, and the two patches have the same crossing count.

The statistic that survives is the range rather than the minimum — how deep the deep ones go, which is a property of how far a stub can run past another before hitting the rim, and which is a property of the construction. On that measure the four patches are alike: 0.112, 0.102, 0.0835 and 0.112, a spread of a third.

That is a better summary and a duller one. The construction produces crossings of every depth up to about a tenth of a sheet, in whatever number the tiling gives it, and the shallowest one in any given patch is a draw from that. Nothing about the five sub-millimetre cases is a fact about the construction’s margins; they are the low end of a sample.

Where the measurement stops #

It is a depth, not a probability. Nothing here says how likely a construction is to produce a shallow crossing; the forty-seven measured are all the crossings this collection has, and they came from one construction on four tilings.

It assumes the printed size. The millimetres are 150 mm across the sheet, which is what these patterns print at. On A4 the same depths are half again as large and two of the five shallow ones clear half a millimetre; on a business card everything is marginal.

And it says nothing about the other drawing fault. A crease that stops in the middle of the paper has a depth too — how far from the rim it stopped — and the honeycomb’s assembled patch has two of those. They are refused for a different reason and the distance is a different distance, and the collection has not yet measured it.