The crease the drawing cannot show

A crease pattern can be read two ways and the readings disagree. Every theorem here reads a list — coordinates, segments, letters — and asks its questions at the points the list calls vertices. A reader reads ink, which comes with no list attached and no statement about which lines were meant to meet.

Three ways those readings can come apart were already named, and all three are the drawing having more than the list admits. A crossing is two creases passing through one another where the list has no vertex. A junction is one crease ending on the interior of another. A stub is a crease that stops in the middle of the paper touching nothing at all.

There is a fourth, and it runs the other way.

Ink nobody can see #

A fragment is a crease the list has and the drawing cannot show. On the hexagonal patch there are twelve of them, six at 7.9 × 10⁻⁶ of the sheet’s side and six at 5.9 × 10⁻⁵.

Those numbers are worth converting. On a sheet of paper fifteen centimetres square — the size the collection prints its patterns at — the first six are a tenth of a micrometre long and the second six are nine tenths. A laser printer’s finest dot is around twenty micrometres. A pencil line is a hundred and fifty. Human vision at reading distance resolves perhaps forty.

So these are not short creases. They are not creases at all in any sense a folder would recognise: nothing marks them, nothing could be folded along them, and the toner that would represent them does not exist as a physical possibility. And they are in the crease count, the interior vertex count, the chain count and the arc count of every measurement this collection has published about that patch.

It is worth dwelling on how they survived. Every gate this collection has asks whether something is wrong: whether a label fits inside its box, whether a vertex satisfies a condition, whether two creases cross where they should not, whether a folded state closes. A fragment passes all of them, because it is not wrong. It is a perfectly legal crease with a perfectly legal length, satisfying every theorem in the subject, and the only thing peculiar about it is a comparison with the physical world that no check performs.

That is the same shape as the tick labels that were absent rather than misplaced: a defect whose symptom is nothing at all. An instrument that asks “is this right?” of everything present will never notice something whose problem is that it should not have been present, and there is no general repair for that beyond looking.

Why a threshold here is not a judgement #

Calling something a fragment requires a length below which a crease stops counting, and picking such a number is normally an argument rather than a measurement — somebody’s opinion about how small is too small, dressed as a definition.

Not here, and the reason is the shape of the distribution.

The twelve fragments run to 5.9 × 10⁻⁵ of a sheet. The next shortest crease on the same patch is 2.9 × 10⁻², five hundred times longer. Every threshold anywhere in that band catches exactly the same twelve creases and nothing else, so the number chosen carries no information and cannot be wrong.

That is what a categorical distinction looks like when it happens to be measurable: not a line drawn through a continuum, but a gap with nothing in it. A quantity whose values cluster at 10⁻⁵ and at 10⁻² with a vacancy between them is telling anybody who plots it that two different things are being measured, and the plot is the whole argument.

There is a second reason the gap is trustworthy, which is that it is not an artefact of this patch. Across the whole grid of tilings, turn angles and pleat widths, the shortest crease of a patch is either under a thousandth of a sheet or over a fiftieth — never between. Ten patches fall on one side of the vacancy and a hundred and ten on the other, and the vacancy is in the same place every time.

That is what would be expected from the mechanism below, and it is worth noticing that the prediction runs the right way round: the mechanism was found by asking where the twelve came from, and the census was run afterwards to see whether it left the signature it should. It did.

Where they come from #

The patch is not drawn unit by unit. The construction lays the tessellation out over a region larger than the sheet and then clips the drawing to it, so a crease that would run off the paper is shortened to the rim and ends there — which is what a crease is allowed to do, and which was itself a repair for an earlier construction that let creases run on past the sheet and cross things.

A clip is a well-behaved operation almost everywhere. It shortens a crease, or removes it entirely, and either is fine. The one case it handles badly is the one where a pleat happens to reach the rim of the sheet within a hair of a corner of itself: the clip then leaves not a shortened crease but a splinter, the last fraction of a percent of a segment that was almost entirely off the paper.

That is an accident of alignment between two things that have no reason to know about each other — where the tiling’s own lattice sits, and where the sheet’s edges are. It happens when they nearly coincide, which is to say rarely and unpredictably.

Ten of a hundred and twenty. About one patch in twelve, on a construction that has been drawing patches for the whole life of this collection, with no check anywhere that reports it.

The word rarely is doing real work there and deserves a number rather than an adverb. Ten of a hundred and twenty is eight per cent, which is far too common to be dismissed as a freak and far too rare for anybody to have met it twice. A construction that misbehaves once in twelve invocations, silently, producing output that passes every check, is close to the worst possible frequency: often enough to be in the published record and seldom enough that nobody forms a habit of looking for it.

And the collection did meet it once. The hexagonal patch’s crease count exceeding its arc count by exactly twelve was noticed, reported and explained — with the wrong mechanism, as six vertices whose paper the clip had removed rather than six vertices carrying fragment pairs. The number was right and the story was not, which is the ordinary way an unnoticed cause gets an explanation attached to it.

The repair that suggests itself #

The obvious response is to delete them. A crease nobody can see is doing no work, the argument runs, so removing it leaves the pattern the reader sees exactly as it was while tidying the list to match.

The arithmetic even comes out right. Deleting the hexagonal patch’s twelve fragments gives a hundred and thirty creases and fifty-four interior vertices, down from a hundred and forty-two and sixty — which are precisely the counts that had been predicted for the repair, worked out from the fact that each of six vertices carries a pair of fragments.

The repaired pattern’s panels do not close. Following two different routes round the sheet, composing the reflection at each crease, and asking whether both arrive at the same place for the same panel gives a disagreement of more than two sheet widths. That is not a small numerical error. It is the signature of a pattern with a freedom in it — a drawing rather than a folded object — and it means the repaired list describes no flat folded state at all.

The failure is worth being precise about because “the pattern breaks” is vague and the check is not. Two routes through the panel graph to the same panel, each composing the reflection at every crease it crosses, must arrive at the same rigid motion — that is what it means for a crease pattern to have a flat folded state at all, and it is computed by machinery that knows nothing about letters. On the patch as drawn the worst disagreement over every loop is under a ten-millionth of a sheet width. On the repaired one it is 2.3 sheet widths.

A disagreement of that size is not a numerical wobble that a tolerance could absorb. It says the two routes place one panel a couple of sheets apart, which is the signature of a pattern that has stopped being a folded object and become a mechanism with a freedom in it.

Why an invisible crease is load-bearing #

The reason is that a crease’s length and a crease’s job are unrelated quantities.

A crease separates two panels and states how they are joined. In the folded state each panel is placed by reflecting its neighbour across the crease between them, so a crease with a genuine fold in it puts its two panels in genuinely different places, however short it is. Delete it and the two panels merge into one — but their two placements do not merge, because they were never the same, and the resulting object has one panel that has to be in two places.

The twelve fragments are not degenerate leftovers with no fold in them. They carry real letters, they separate real panels, and each one is the last surviving trace of a pleat that the clip has otherwise removed. What they hold together is the combinatorics of the sheet, and combinatorics has no size.

So a fragment is a genuinely awkward object: too small to draw, too small to fold, too small to print — and structurally identical to any other crease, with a job that a full-length crease could not do better.

Why the failure is two sheet widths #

The closure gap of 2.3 sheet widths is not an arbitrarily large number, and deriving it says something the essay’s argument needs.

Deleting a crease merges its two panels. One placement survives — whichever the spanning tree hands the merged panel — and the other route to it still composes the reflection that is no longer there. So the two routes differ by exactly that reflection, and the distance between the two placements of a point $P$ is

$$2 \times \text{(distance from } P \text{ to the crease's line)}$$

The fragments sit at the rim of the sheet, and the panels they separate are up to about a sheet width from the line they were reflected across. Twice that is about two, and the measurement is 2.3.

Which makes length and consequence independent #

That is the essay’s central claim arriving as arithmetic rather than as an argument.

The fragment’s own length appears nowhere in the expression. A crease a tenth of a micrometre long and a crease half the sheet long, sitting on the same line, produce identical closure failures when deleted, because the failure is a property of where the mirror line is and not of how much of it was drawn.

So there is no length below which a crease may be dropped. A crease a millionth of a sheet long is worth exactly as much to the folded state as one a full sheet long on the same line, and the only quantity that decides the damage is how far the affected panels sit from that line.

That also predicts the shape of any future attempt at the same repair. Deleting a fragment near the middle of the sheet would produce a smaller failure — the panels are closer to the line — and deleting one at a corner a larger one, up to about $2\sqrt2$ for a panel at the opposite corner. The failure is bounded above by twice the sheet’s diagonal and below by nothing, and it is never zero for a crease that carries a letter.

The one case that would be safe is a crease with no fold in it — a line marked but not creased — and none of the twelve is that. Each carries a letter, each separates panels the folding genuinely places differently, and each was the last surviving trace of a pleat rather than a degenerate artefact of the clip.

A crease list that cannot be simplified #

This is the reading that makes fragments the odd one of the four.

A crossing is a fault, and repairing it means adding a vertex — the list gains something the drawing already had. A junction is not a fault at all; the list simply has to be told about a vertex it can compute. A stub is a fault, and repairing it means removing a crease or extending it to somewhere it can end.

All three are repaired by changing the list to agree with the drawing. A fragment cannot be, because the drawing is the thing that is wrong: the ink is missing something the list correctly contains. Repairing it means changing the construction, so that the pleat is either drawn whole or not drawn at all, and that is a change to the geometry rather than to the bookkeeping.

The option that exists to measure the mistake #

The deletion is available as a setting of the construction, and it is not there so that anybody can take it.

It is there so that the failure can be measured rather than described — so that the counts on both sides, and the closure gap on the repaired side, are numbers this collection computes rather than sentences it asserts. A repair that is known to be wrong is worth keeping executable, because the alternative is a note in a document saying that somebody once tried it, and a note in a document cannot be re-run when the construction changes underneath it.

The default draws them. That is a deliberate choice and the reason is the paragraph above rather than deference to any published number: a construction whose default silently produced a sheet that does not close would be a considerably worse construction than one with a known, measured and explained blemish.

Which theorem was checked, and how #

The patch as drawn is a flat-foldable crease pattern and remains one. Every interior vertex satisfies Kawasaki, Maekawa and the big-little-big lemma; the panels close to within a ten-millionth of a sheet width, checked by composing reflections round every loop of the panel graph; and the pattern has a consistent lettering, found by search and verified against a folded sheet rebuilt from scratch. The fragments do not break anything.

The repaired patch is checked by exactly the same instruments and fails the third of them. That is the assertion the account above stands on, and it is stated as a requirement: if deleting the fragments ever produced a pattern whose panels closed, this essay would be wrong and the check would say so.

The five-hundred-fold gap is asserted too, rather than described, because it is what makes the threshold defensible and because a construction change could close it without anybody noticing.

What the picture cannot show #

It cannot show a fragment. That is the entire difficulty and no figure escapes it: the circles in the first figure mark where they are, at a magnification the page cannot provide, and the length plot shows them as points on an axis rather than as ink. A reader who wants to see one has to enlarge the sheet by about twelve thousand times, and at that magnification nothing else in the figure is on the page.

Nor does any of this say the printed patch is wrong to print. It folds, it has been folded, and the twelve creases nobody can see are twelve creases nobody needs to make — the sheet arrives at the same folded state without them being creased, because a fold of zero length is not a fold anybody performs. What the fragments break is the count, and counts are what this collection makes its arguments out of — most of a patch is edge is a sentence about counted vertices, and the counting is where a fragment does its damage.

Where the ladder goes next #

There is a repair, and it is not this one. The patch sits at one particular setting of the construction’s pitch, and one step of that setting to either side removes the fragments entirely — one step down keeping the ring of twists whole at a hundred and forty-two creases, one step up dropping it at a hundred and thirty. The counts the deletion promised are available after all, from a pattern that folds.