A stub is never alone

How deep is a crossing gave a binary verdict a distance. Two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched — but how far one runs past the other is a continuum, and measured across every patch that has one it runs from 0.16 mm to 17 mm on a sheet 150 mm across. Five of the forty-seven were thinner than a pencil line.

It ended by naming a second fault it had not measured. A crease that stops in the middle of the paper has a depth too — how far from the rim it stopped — and the two on the honeycomb’s assembled patch were refused for a different reason, at a distance nobody had computed.

Computing it takes a line. What comes back is a distance, and then three facts about the distances that the crossings do not have.

What a stub is, and what its distance means #

A crease list is a set of segments. A drawing is ink. Four ways to draw a pattern is where the difference was first separated, and it has four parts: a crossing, where two segments meet at a point neither lists as a vertex; a junction, where one ends on another; a fragment, too short to see; and a stub, a segment with a free end in the middle of the paper.

A stub is refused for a reason a crossing is not, and two creases that cross is where that reason was separated from this one. A crease with a free end gives its endpoint one spoke, and a vertex of degree one satisfies no condition the subject has: the sectors round it do not alternate, Maekawa asks for a difference of two between counts that are one and zero, and the paper on either side of the crease’s end is one panel that the crease pretends to separate. It is not a near miss. It is a crease that is not there.

So its distance means something a crossing’s does not. A crossing 5 mm deep is a crease that goes 5 mm too far, and shortening it by 5 mm does not repair the pattern — it leaves the two creases meeting at a point, which is a junction and refused as well. A stub 5 mm from the rim is a crease that is 5 mm too short, and extending it by 5 mm repairs the drawing exactly. The number names its own repair.

They come in pairs, at equal depths #

The first structural fact is a count. Across the hundred and twenty patches, fifteen carry a stub, and on every one of the fifteen the number of stubs is even — two, four, eight or fourteen.

The second is sharper. Sorting the distances on each patch, every value occurs an even number of times: the honeycomb at a turn of 0.3 has two stubs and both are 0.54 mm from the rim; the rhombille at 0.2 has fourteen, and they are 1.42 four times, 1.69 twice, 2.51 four times, 22.58 twice and 35.09 twice. Sixty-six stubs are thirty-three pairs, and across all fifteen patches there are twenty-seven distinct distances.

The mechanism is the construction’s own. A pleat is two creases running the length of a tiling edge, parallel and equally long. Where the polygon at the far end of that edge falls off the paper, the construction extends both creases along their own lines to the rim — and where that extension fails, it fails for both at once, at the same distance, because the two creases are the same length and start together.

So a stub is not one fault forty-four times. It is a pleat that did not arrive, counted twice.

And they never occur alone #

The third fact is the one that decides what the measurement is worth.

Seventy-six of the hundred and twenty patches carry a crossing. Fifteen carry a stub. The fifteen are a subset of the seventy-six — there is no patch in the population with a stub and no crossing — while sixty-one patches have a crossing and no stub.

That makes the stub test redundant on this population. A checker that reads a drawing for crossings refuses every pattern a checker for stubs would refuse, and fifty more besides. The distance is real, it is measurable, it names a repair, and it adds nothing to the verdict.

This is worth stating plainly rather than buried, because a measurement that finds no new refusals is the kind that quietly does not get reported. A check that has never rejected anything proves nothing, and a test that only ever fires alongside another one is in the same position: it is not wrong, it is not independent evidence, and treating it as a second check on a drawing would be double-counting.

A distance that is not a tolerance #

It is worth separating this number from the ones the rest of this line of argument is about, because both are lengths in millimetres and they mean opposite things.

A tolerance is a direction and the allowance is spent at the end are about how far a made thing may depart from a drawing before it stops working. They are budgets: there is a right value, the world supplies a wrong one, and the question is how much wrongness the geometry absorbs.

A stub’s distance is nothing like that. The drawing is not a departure from anything — it is what the construction produced — and the distance is not an error a maker introduced but a piece of crease the rule never drew. There is no correct value it is deviating from, and no amount of care in the folding recovers it.

The two are easy to confuse because both end in a statement of the form 0.27 mm on a 150 mm sheet, and the confusion matters: a reader who takes the stub distance for a tolerance concludes that a careful folder could fold this pattern, and a careful folder cannot. Solved is not built makes the same separation from the other side — a mesh that folds because an equation holds and one that folds for a structural reason are not two examples of one thing — and the drawing faults belong on the side where no care helps.

Why the two faults travel together #

The co-occurrence has a reason and the reason limits how far the finding generalises.

Both faults are made by the same decision. The construction draws the tessellation over a region and then has to decide what happens where a polygon’s neighbour is off the paper. Extending the pleat’s creases along their own lines to the rim is that decision, and it produces a crossing whenever the extension runs through ground another polygon occupies — which is most of the time — and a stub whenever it runs out before reaching the rim. A drawing dense enough to strand a pleat is dense enough to have its extensions collide somewhere.

So the two are symptoms of one choice rather than two independent ways a drawing can be wrong. A construction that made the decision differently would have a different relation between them, and the clipped construction — which draws on a larger region and cuts the drawing to the sheet — has neither, because a crease then ends where the paper ends, which is what a crease may do.

What the fifteen patches have in common #

Fifteen of a hundred and twenty is a small enough set to look at one by one, and the list is not random.

Two tilings account for all of them. The honeycomb carries stubs at five of its twenty-four settings and the rhombille at ten of its twenty-four; the square grid, the triangular grid and the elongated tiling carry none at any period or turn tried. Those two are the tilings whose polygons sit at more than one size — the rhombille has two kinds of vertex outright, and the honeycomb’s hexagons leave the widest gaps between twists — so the pleat that fails to reach the rim is the long one on a drawing that has long and short pleats in it.

The period matters less than the tiling. Eight of the fifteen are at a period of 0.28, three at 0.34 and four at 0.42 — a lean toward the densest drawing and nothing like the concentration the tilings show. Whatever produces a stub is decided mostly by which tiling is being drawn and only a little by how finely.

That is a pattern rather than an explanation, and it is the shape of pattern that makes a third measurement worth taking: the same census with the extension length recorded, which would say whether a stub is a pleat that ran out of length or a pleat whose direction took it along the rim rather than into it.

Two readings of one drawing, and which one refuses #

Underneath all of this is the distinction the whole measurement depends on, and it is worth restating because it is what makes a stub findable at all.

A pattern is checked as a list: a set of segments with endpoints, from which vertices are read, sectors computed and the four conditions applied. A pattern is folded as ink: whatever a reader sees on the paper. The two readings usually agree, and every fault counted here is a place where they do not.

A stub is the case where the list has an endpoint the ink does not mark. The list says a crease ends at a point in the middle of the sheet; the ink says a line stops there, which a folder reads as a line that goes to the edge and was drawn a little short, or as a line that was meant to be there and was not finished. The list’s reading is the one the conditions are applied to, so the list’s reading is the one that refuses — and the reader’s disagreement with it is not represented anywhere.

That asymmetry is why the distance is worth having. It is the size of the disagreement, in the units a reader would notice it in, and it is the only number that says whether the two readings differ by something visible.

The shallow end, again #

The distances span a factor of a hundred and thirty: 0.27 mm at the shallowest, 35.09 mm at the deepest, on a sheet 150 mm across.

The deep ones are unmistakable. A crease stopping 35 mm from the edge of a 150 mm sheet stops a quarter of the way in, and nobody folding it would take the drawing for anything but unfinished.

The shallow ones are the same problem the crossings had. At 0.27 mm a crease stops about a quarter of a millimetre short of the rim, which on a laser printer is a pixel and a half and in pencil is nothing at all. A folder looking at that drawing sees a crease reaching the edge. The crease list says it does not, and the crease list is what decides the verdict — so the pattern is refused for a gap that cannot be seen and could not be drawn either way.

That is the same conclusion the crossings reached from the other side, and reaching it twice by different routes is the useful part: the drawing and the list disagree about things smaller than the ink, and which of them is right is a question about the intention rather than about the geometry.

What the population is #

One construction, one hundred and twenty drawings. Five tilings at three periods and eight turns, all built with the extension rule rather than the clip. Nothing here is a statement about patterns in general, or about this construction’s clipped form, which has no stubs at all.

The distance is to the nearest rim, in the sheet’s own units, converted at 150 mm because that is what these patterns print at. On a larger sheet every number scales with it and the shallow ones stop being marginal, which is the same caveat the crossings carry.

A stub is an interior vertex of degree one, found by reading the drawing rather than the list, and the reading is the shared one — the same pass that finds the crossings, so the two counts are commensurable. A crease ending exactly on the rim is not a stub and is not counted, and neither is one ending on another crease, which is a junction.

And the pairing is measured, not derived. The account above — a pleat’s two creases failing together — explains it and is not what establishes it. What establishes it is that every depth occurs an even number of times on every patch.

What the measurement cannot settle #

It does not say stubs are harmless. They refuse a pattern, and a pattern with one is not a crease pattern — nowhere to put the error is about what a real sheet does with a misplacement it is given, and a missing crease is not a misplacement it can absorb. What it says is that refusing them buys nothing a crossing test has not already bought, on this population.

It does not cover the clipped construction. That one produces no stubs by design, and whether it produces faults of some other kind at the same places is a separate reading.

It says nothing about which patch a designer should use. A drawing with fourteen stubs and a hundred and eighty-four crossings is not fourteen problems worse than one with two and ten; both are the same decision made in a place it does not work. How deep is a crossing made the same point about counting crossings, and the stubs do not change it.

And it says nothing about a pattern that folds. Every drawing in the population is refused. Whether a pattern with a stub could be repaired into one that folds — by extending the two creases the distance measured — is plausible and untested, and it is the one experiment the number is literally an instruction for.

And it is a distance, not a probability. Nothing here says how often a construction produces a shallow stub; the sixty-six measured are all of them, and they came from one rule on five tilings. Error is folded too is the standing argument that a small misplacement does not stay small, and it applies to a drawing fault the same way: the distance a stub falls short by is not the error the folded sheet ends up with.

Still open: the third distance #

Two of the four drawing faults now have a number. The junction does not, and it is the one where the number would be hardest to interpret.

A junction is a crease ending on another rather than past it or short of it, and the natural distance — how far the endpoint is from the nearest vertex of the crease it lands on — measures something different from either of the two above. A junction close to a vertex is a drawing that nearly listed a vertex there; a junction far from one is a T where no vertex was intended at all. Those are opposite situations and one distance would report them as the same, which is why the quantity has to be defined before it is measured rather than after.

The fourth, the fragment, already has a number and it is not a distance but a length — and the reason a fragment cannot simply be deleted is on the record. A census with all four quantities side by side would be the first statement of what separates a drawing from a list that was arithmetic rather than a list of kinds.

Sideways from here, the co-occurrence is worth looking for elsewhere. Two tests that always fire together are one test, and a checker accumulating conditions over years will have several such pairs in it without anybody noticing, because each was added for a reason and none was ever compared against the others. The comparison is cheap: run both over a population and look at the two sets.

The habit worth carrying is about measurements that come back empty. A quantity that turns out to add no refusals is still worth the hour, because what it establishes is that the checker is smaller than it looks — and a checker with a redundant test in it is a checker nobody can reason about the coverage of.