A patch on a knife edge

A tessellation is infinite and a sheet of paper is not, so every tessellation patch in this collection is the result of a decision about where to stop. The construction draws the pattern over a region larger than the sheet and clips it, which is the honest way to do it: a crease then ends where the paper ends, which is something a crease is allowed to do, and no crease runs through ground the tessellation never filled.

Clipping is well behaved almost everywhere — it replaced an older construction that let a crease run on past its missing neighbour and cross whatever was on the far side — and the twelve creases it left on one printed patch are the exception — splinters eight millionths of a sheet long, in every count the collection takes and visible to nobody.

Deleting them does not work. What does work is moving the pattern by five thousandths of its own pitch, and the reason is the subject of this essay: the patch is not slightly wrong. It is exactly on a transition.

What changes as the pitch moves #

The pitch is how far apart the tessellation’s twist polygons sit. Making it larger fits fewer of them on a fixed sheet; making it smaller fits more. Nothing else about the construction depends on it — the same tiling, the same turn angle, the same rule for where the pleats run.

So as the pitch grows, the patch loses twists, and it loses them in rings. A tessellation on a square sheet has a symmetry that puts a whole outer band of polygons at nearly the same distance from the rim, and they leave together: at one pitch there are forty-five polygons on the paper and at the next there are thirty-nine, with nothing in between.

Between those two states the patch has to pass through a moment where the outer ring is neither on the paper nor off it. That moment is not a range. It is a razor’s edge, because the polygons of the ring reach the rim within a hair of one another, and the pitch interval over which some of them are in and some are out is a few thousandths wide.

At 0.335 the patch has a hundred and forty-two creases and sixty interior vertices and no fragment: the ring is fully on the sheet. At 0.345 it has a hundred and thirty and fifty-four and no fragment: the ring has gone. At 0.34, which is what the collection prints, it has a hundred and forty-two and sixty — and twelve of the creases are the last splinters of pleats that are almost entirely off the paper.

It is worth seeing why the ring leaves all at once rather than one polygon at a time. On a square sheet the tessellation is laid out from the centre, so the polygons sit at a set of distances from the middle that the tiling’s own symmetry makes highly degenerate: four of them at one distance, four more at another, and on a hexagonal tiling six at a time. The rim of the sheet is a square, so it does not cut those shells cleanly — but it comes close enough that a shell’s members reach it within a few thousandths of the pitch of each other.

A tessellation on a circular sheet would lose its rings exactly together, and the transitions would be perfectly sharp with no interval at all for a fragment to live in. A square sheet blurs each transition into a narrow band, and the band is precisely wide enough to hold a patch that has some of a ring and not the rest.

The blemish is what a transition looks like #

That reframes the twelve completely.

They are not an error in the clip. The clip is doing exactly what it should: a segment that lies mostly off the sheet is shortened to the part that is on it, and if that part is a millionth of a sheet then a millionth of a sheet is the honest answer. Every other pitch produces perfectly ordinary creases from the same code.

They are not an error in the tessellation either. The tessellation is unaware of the sheet; it is a rule about the plane, and it produces the same polygons whatever is going to be cut out of them.

What they are is the interaction of two things with no reason to know about each other — where the tiling’s lattice happens to sit, and where the paper’s edges happen to be — caught at the one alignment where the answer is nearly a tie. A tie in a construction that has to produce a definite drawing comes out as a definite drawing with something very small in it.

This has a consequence for reading any measurement taken on such a patch. The counts of a tessellation patch are not smooth functions of the construction’s parameters, and they are not noisy either — they are step functions, and a measurement taken on one is a measurement of which step the parameters landed on. Two patches whose parameters differ by a per cent can differ in crease count by eight, and a reader who takes crease count as a proxy for anything continuous will conclude something false about a construction that is behaving perfectly.

Where the paper stops is where the collection first set out how much of a patch is boundary rather than interior, and this is the same observation at the next level: not only is most of a patch edge, but how much of it is edge changes in jumps as the sheet is fitted to the tiling.

Why the counts the deletion promised are available anyway #

Deleting the fragments gives a hundred and thirty creases and fifty-four interior vertices and a sheet whose panels do not close, because each fragment is separating two panels that a fold genuinely places differently.

Moving the pitch to 0.345 gives a hundred and thirty creases and fifty-four interior vertices and a sheet that folds — a hundred and thirty arcs, a consistent lettering found in thirty-nine steps, panels closing to within a ten-millionth of a sheet width, every vertex condition satisfied.

The two routes to the same numbers are entirely unalike. Deletion removes twelve creases and leaves the rest of the pattern reaching for them. Moving the pitch removes the whole ring those creases belonged to, along with its polygons, its pleats and its vertices, so nothing is left reaching for anything.

That is the general shape of the difference between repairing a drawing and repairing a construction. A drawing can be edited anywhere; a construction can only be moved along the parameters it has. When the two disagree about what a repair should be, the construction is nearly always right, because a drawing edited into a state its construction cannot produce is a state nothing else in the system expects.

Why one step of pitch is a small change and a large one #

Five thousandths of the pitch is a change of about one and a half per cent. Nothing about the patch’s appearance changes noticeably: the twists are the same size, the pleats the same width, the turn angle identical.

The counts change by eight per cent, and they change discontinuously. There is no pitch at which the patch has a hundred and thirty-six creases. The quantity that moves smoothly is the position of the outer ring relative to the rim; the quantity the collection measures is how many polygons are on the paper, and that is an integer.

This is worth stating because it is the trap the printed patch fell into. A parameter chosen once, sensibly, for a good reason — 0.34 gives a patch with a satisfying number of twists on it, neither crowded nor sparse — and no reason at all to suspect that the same choice is within a per cent of a discontinuity. The dial that decides nothing is the companion case, where a parameter turns out to change less than anybody expected rather than more. Nothing about 0.34 announces itself.

The general rule this is an instance of #

Constructions that fit an infinite pattern into a finite region all have this shape, and it is worth naming because it recurs far outside this collection.

Somewhere in such a construction there is a test of the form is this piece inside the region? — and the answer is a boolean derived from a continuous quantity. Wherever the continuous quantity passes through the threshold, the boolean flips, and any piece whose quantity sits within numerical spitting distance of the threshold produces output that is neither one thing nor the other. Here the flip is a polygon being on the sheet or off it, the near-tie is a pleat reaching the rim at a corner, and the neither-one-thing is a crease a millionth of a sheet long.

The reason it produces a small artefact rather than an obviously broken one is that the construction is continuous on both sides of the flip. A clip does not fail when a segment is nearly all outside; it returns the tiny part that is inside, correctly. Correct behaviour at a near-tie is what makes the artefact invisible, and it is exactly why no check catches it: every step of the construction is doing the right thing.

Whether to move it #

The counts at 0.34 and at 0.345 differ, and the printed patch’s crease count, vertex count, chain count and arc count appear in several essays. Moving the patch changes all of them.

The case for moving is that 0.345 is a strictly better object: the same picture, the same argument, no fragments, and the crease count matching the arc count for the first time. The case against is that the counts are the collection’s own record and rewriting them costs more than the blemish does, especially since the blemish is now understood, measured and named.

The decision here is to leave it, and the reason is not sentiment about published numbers. It is that the patch is a better example where it is. A tessellation patch that sits exactly on a transition, whose twelve invisible creases are the ring of twists caught leaving, is a considerably more interesting object than one that sits comfortably in the middle of a step — and everything odd about it is now explained rather than merely recorded.

What this says about every other patch #

Ten of a hundred and twenty patches over a grid of tilings, turn angles and pleat widths carry a crease under a thousandth of a sheet. Every one of them is at a transition of some parameter, and none is at a parameter anybody chose deliberately — which is what a population assembled from a grid rather than by hand is for.

That rate has a practical consequence for anybody sweeping this construction: a patch drawn at parameters chosen by a grid rather than by a person has about an eight per cent chance of sitting on a transition, and the sign of it is a crease length rather than anything a theorem reports. Reading the shortest crease of a patch before quoting any of its counts costs nothing and catches all of them.

Every theorem in the subject is dimensionless #

There is a reason no check caught the twelve, and it is more general than “nothing looked”. Not one of the subject’s conditions can see a length.

Developability sums angles. Kawasaki compares alternating sums of angles. Maekawa counts letters. The smallest-sector lemma compares one angle with two others. Every one of them is invariant under scaling the pattern, and every one is invariant under scaling any part of it — a crease a millionth of a sheet long meets a vertex at the same angles a crease a fiftieth long would.

So a defect whose only symptom is a length is invisible to the entire apparatus simultaneously, and not by oversight. A scale-free theory has nothing to say about scale, and the fragments are exactly a question about scale.

Which is why the diagnostic works so well #

That also explains why reading the shortest crease is such a good test, and it is worth quantifying because the margin is what makes a threshold safe.

The fragments are about eight millionths of a sheet. An ordinary shortest crease on these patches is about a fiftieth. That is a gap of two and a half thousand, so any threshold between a ten-thousandth and a hundredth separates them, and moving it by two orders of magnitude in either direction changes no verdict.

A test with that much margin is a test whose threshold is not doing the work — which is the only condition under which a threshold on a continuous quantity is worth trusting, and it is the same standard the crossing check is held to.

So the missing check is not a hard one to write. It is one comparison against the shortest crease, with three orders of magnitude of slack, run once per pattern. What made it absent is that it is the only check in the collection that would be about a dimension, and a subject whose theorems are all about angles does not naturally produce one.

That is the transferable part. A field’s blind spots follow the invariances of its theory: a scale-free theory misses scale, a metric-free theory misses distance, an orientation-free theory misses handedness. The place to look for an unnoticed defect is whatever quantity every theorem in the subject is indifferent to.

What a folder would notice #

Nothing, and the reason is worth stating rather than glossed.

A folder given the printed sheet creases along the lines they can see, of which there are a hundred and thirty. The twelve they cannot see are not creased, because a fold of zero length is not a fold anybody performs, and the sheet arrives at the same folded state regardless — the two panels a fragment separates are, in the paper, the same piece of paper, and nothing asks them to be anywhere different.

So the folded object a reader produces from the printed pattern is the folded object at a pitch of 0.345, while the crease list describes the one at 0.34. The two are the same object; the difference is entirely in the bookkeeping. That is a comfortable conclusion and it is also the exact reason the blemish survived: the sheet has always folded, which is the strongest evidence anybody has that a crease pattern is right, and it was never in question.

Which theorem was checked, and how #

Every patch in the sweep is checked by the same three instruments. Kawasaki, Maekawa and the big-little-big lemma at every interior vertex; the panels’ closure, computed by composing reflections round every loop of the panel graph and knowing nothing about letters; and a lettering, searched for and verified against the machinery that did not find it.

The claim specific to this essay is asserted as a requirement rather than described: at a pitch of 0.345 the patch has exactly a hundred and thirty creases and fifty-four interior vertices, carries no crease under a thousandth of a sheet, and closes. All four together — because three of them without the fourth would be the deletion’s answer, which has the counts and not the sheet.

What the picture cannot show #

The transition itself. Every figure here shows the patch at some pitch, and the interesting object is the interval between two pitches — a few thousandths wide, containing the alignment where the ring is halfway out. Drawing the patch at eleven pitches instead of seven would show more rows of the same table and not the thing between them.

Nor does the pitch sweep say the twelve are the only such moment on this patch. Every parameter of the construction has transitions of its own, and a patch could in principle sit near two at once. Nothing here looks for that, and the honest statement is that one transition explains everything observed rather than that only one exists.

Where the ladder goes next #

The transition here is about a drawing — where the paper’s rim falls relative to a lattice — and the construction has transitions of another kind entirely. At a shallow turn angle the same family stops having consistent letterings at all, across a region rather than at a point, and that boundary is a fact about which sector at a vertex is the smallest rather than about where anybody cut the paper.