The vertex the list does not have

A crease pattern arrives at a checker as three lists: the coordinates of its vertices, the pairs of vertices its creases run between, and a letter for each crease. Developability, Kawasaki’s condition, Maekawa’s and the big-little-big lemma are then evaluated at each vertex of the first list, by collecting the creases that name it and reading the angles between them. That is the right object for those four statements, because every one of them is a statement about the creases meeting at one point.

It is not the object a reader has. A reader has a sheet of paper with lines drawn on it, and lines on paper do not come with a record of which of them were meant to meet. Two segments drawn across one another look exactly like four segments meeting at a point, and the reader’s fingers cannot tell the difference either — but the paper can, and the two readings differ about whether the sheet folds.

Two readings of one drawing #

Subdividing a drawing until no two segments meet except at a shared endpoint is a routine operation, and this collection has done it since the patterns first became objects rather than pictures — the panel counts an essay quotes are read off the subdivision rather than off the sentence that designed the pattern, and the exported files carry the faces the subdivision finds.

What had never been asked is the difference the subdivision makes. It adds a vertex in two circumstances and they are not the same circumstance at all.

A junction is where one crease ends on the interior of another. Nothing is wrong with it. The fold-and-cut construction drops a perpendicular from a node of the straight skeleton onto an edge of the outline, and the foot of that perpendicular lands wherever the construction puts it — usually in the middle of the edge, because the edge was drawn between two corners and the perpendicular has no reason to arrive at either. The pattern is correct; it simply has not been cut into pieces yet. Of the eight printed patterns, five have junctions: four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle.

A crossing is where two creases pass through one another at a point neither of them has as an endpoint. That is not a subdivision waiting to happen. It is a vertex, and it is a vertex of a kind that cannot fold.

The count comes out as cleanly as a count ever does here. Across the eight patterns printed at true scale for a reader to fold, the number of crossings is zero. Across the tessellation patches, four of six carry them: twelve on a triangular grid, eighteen on a honeycomb, twelve on a rhombille, five on the elongated triangular tiling. The square grid’s patch has none.

Where the crossings actually are #

They are not in the tessellations. A twist tessellation of the plane is a perfectly good pattern, derived rather than drawn, with its polygon shapes forced by the tiling underneath it and Kawasaki satisfied identically at every corner. Nothing in that construction puts one crease across another.

The crossings are in the way a patch of it was cut out to fit on a square of paper. The rule was: keep every twist unit that fits on the sheet whole, and where such a unit has a neighbour off the paper, run that pleat’s two creases out along their own line until they reach the rim. The reason for the second half is sound and is the reason a folder would give — a fold line that stops in the middle of a sheet is a line the paper cannot obey, because there is nothing there to turn about. So the stub is extended to the edge, where a crease may legitimately end.

What the stub then does is run on through the ground its missing neighbour would have occupied. On a square grid the pleats leave in the sheet’s own two directions and the stubs on any one edge of the paper are parallel to each other, so none of them can meet another. On every other tiling they leave at two or three different angles, and near a corner of the sheet two of them cross.

The stub was itself a repair #

The part worth keeping is not that a construction had a fault in it. It is which fault, and what it was put there to fix.

An earlier version of the same cutting-out simply stopped a pleat where its missing neighbour would have been. That leaves a crease ending in the middle of the paper with nothing at its end — and a crease that ends in the middle of a sheet is unfoldable for a reason a folder feels immediately: the paper has to turn about the line, and past the end of the line there is nothing to turn about. Reading a pattern as ink counts those too, and the honeycomb’s patch still carries two of them.

So the stub exists to remove a drawing error, and it introduces a different one. Both are invisible to every condition the subject states, because both are about a crease’s ends and its neighbours in the plane, and the four conditions are about a point.

That is the shape of both faults and of the reading that finds them: an omission, where every other check here reads something that is present and asks whether it is right.

Nothing was checking #

Every one of those patches passes every condition at every vertex of its list, with zero failures reported. That is not a bug in the conditions; it is the exact statement of what they are about. Developability asks whether the sectors around a listed vertex sum to a full turn. Kawasaki asks whether the alternating sum of those sectors is zero. Maekawa counts the letters at a listed vertex. Big-little-big compares a listed vertex’s smallest sector to its neighbours. A crossing appears in none of those sentences, because a crossing is not in the list.

This is the same shape as the fault the boundary vertices had and the opposite fault to the one an odd-degree vertex has. Both of those are about a vertex the conditions can see and do not apply to. A crossing is about a vertex the conditions cannot see at all — and the reason is not subtle. Nothing looked.

The second instrument, which shares no code #

A check that a drawing has no crossing is worth very little on its own. It is a tidiness complaint: two lines meet where the file says they do not, and a reader might reasonably ask what harm that does.

The answer comes from somewhere else entirely. A flat folded state is a composition of reflections: the panel across a crease carries the near panel’s motion followed by reflection in that crease, and walking the panel graph from one starting panel places every panel in the plane. Where two routes reach the same panel they must agree, and the amount by which they fail to is a distance a reader can see — the machinery that measures it was written for a sheet with a hole in it and knows nothing about drawings, crossings or lists.

Point it at one vertex at a time and it becomes sharp. Walk the creases at a vertex in angular order, reflect in each, and what comes back is a rotation by twice the alternating sum of the sectors there. That quantity is zero exactly when Kawasaki’s condition holds, so every vertex any of these crease lists declares contributes nothing at all. The only vertices that turn are the ones the list does not have.

On the triangular patch, fifty-four vertices of the drawing compose to the identity and twelve turn — by 240° in one direction or the other. On the honeycomb, sixty-six close and eighteen turn, by 129.2°, 240° or 350.8° depending on which pair of creases crossed. On the rhombille, a hundred and two close and twelve turn by 240°. On the elongated tiling, forty-five close and five turn.

The two counts are equal on all fourteen patterns measured, which is the whole argument for reading a pattern this way. A crossing is not untidy. It is the precise amount by which the folded sheet fails to exist, expressed as a rotation.

What a crossing is, as a vertex #

Reading a crossing as a vertex settles it in two lines, and neither line is a new theorem — both are the subject’s oldest conditions, asked somewhere nobody asked them.

The four sectors at a crossing are two straight lines through a point, so they come in two equal pairs: a, π − a, a, π − a. Kawasaki asks the two alternating sums to be equal, which here means 2a = 2π − 2a, which happens only at a right angle. So every crossing that is not square fails Kawasaki, whatever its letters are.

The right-angled crossing survives that and fails the other. A drawn line is one crease and carries one letter along its length, so the four spokes at a crossing are two creases counted twice, and the mountains and valleys come out four and none, two and two, or none and four. Maekawa’s condition asks for a difference of two. Nought and four are not two.

Swept over eight angles and every lettering, that is thirty-two crossings of which none satisfies both conditions, four satisfy Kawasaki — the four right-angled ones — and not one satisfies Maekawa. The survivors appear only when the four spokes are lettered independently, which is a different object: it is four creases meeting at a point, of the kind Maekawa’s count is actually about. Eight of the hundred and twenty-eight such letterings fold, and every one of them is a right angle whose letters change across the point.

The turn is four times the crossing angle #

The rotations the second instrument reports are not incidental numbers, and deriving them says exactly what the instrument sees.

A product of reflections in lines at angles $\alpha$ then $\beta$ is a rotation by $2(\beta-\alpha)$. At a crossing of two creases meeting at angle $a$, the four spokes lie at $0$, $a$, $180°$ and $180° + a$, so composing the four reflections in angular order gives a rotation by

$$2\big[(180°+a) - 180° + a - 0\big] = 4a$$

The holonomy is four times the crossing angle, and nothing else about the pattern enters.

That reads the measurements back. A turn of 240° means $a = 60°$ — which is the angle two creases meet at on a triangular grid and on a rhombille, and those are exactly the two patches where every ringed vertex turns by 240°. The honeycomb’s three values, 129.2°, 240° and 350.8°, correspond to crossings at 32.3°, 60° and 87.7°, symmetric about sixty.

Which gives the instrument a blind spot #

The formula also says where the holonomy test sees nothing, and the answer is worth having because it is not a hypothetical.

$4a$ is a multiple of a full turn when $a = 90°$. A crossing at a right angle composes back to the identity, so the reflections close, and an instrument that rings the vertices which fail to close would ring nothing there.

That is exactly the crossing the essay’s other argument singles out. The sectors at a square crossing are 90° four times, so the alternating sums are both 180° and Kawasaki passes too. Two independent tests, one blind spot, and it is the same one.

So the square crossing is the case where only Maekawa fires, and it fires for a reason that has nothing to do with angles: a drawn line is one crease carrying one letter, so the four spokes are two letters counted twice and the counts come out four-and-none or two-and-two, neither of which differs by two.

The three instruments are therefore not redundant. The holonomy and Kawasaki agree on everything, including on being silent at ninety degrees, and the letter count is what covers the gap. A crossing test built on holonomy alone would pass every square crossing in existence, and a square grid is where a designer is most likely to draw one.

Where the reading stops #

Three limits, and the first is the one that matters most for what a reader can conclude.

A drawing with no crossing is not thereby a pattern that folds. The absence of a crossing removes one obstruction; local conditions at every vertex remove another; and the sheet can still fail for reasons neither of them touches, because deciding a general pattern is NP-hard and the layers have to be ordered as well as placed. The reading is a refutation, not a certificate. It says no, and when it says nothing it has said nothing.

The reading is of straight segments. A crease pattern here is a set of line segments, and two of them either cross or do not. A curved crease is sampled into segments before anything is asked of it, so a crossing found between two samples is a fact about the sampling as much as about the drawing — the same qualification the ruling checks carry, and for the same reason.

Two creases that nearly cross are decided by a tolerance. The subdivision treats a meeting as a crossing when both parameters are strictly inside their segments by more than a fixed margin. Two creases drawn to pass a hundredth of a sheet width apart get one answer, and the same pair drawn a thousandth apart get whichever answer the margin says. That is a real question about drawn patterns rather than an implementation detail, and it is the reason the tolerance is stated rather than tuned.

What the reading costs #

Almost nothing, and the comparison is worth making because it is the reason this belongs in front of everything else rather than beside it.

Asking whether two creases cross is a comparison of two segments: four subtractions, two products and a division, with two range tests. Asked of every pair, the largest patch here — the rhombille’s, at two hundred and six creases — is twenty-one thousand of those, which is a few milliseconds and no state at all. Nothing has to be folded, no panels have to be found, and the answer does not depend on the letters.

Set that against what it refuses. The rhombille patch has a hundred and twenty-five panels, and deciding whether a set of panels can be ordered is a search over their permutations that is refused outright past eighteen. The patch was therefore never going to be decided by the expensive instrument, at any price, and the cheap one settles it in a sweep — not probably unfoldable, not undecided, but no, with a rotation of 240° at a named point as the reason.

What it changes about the collection #

The patches were figures. Nothing in this collection ever printed one at true scale for a reader to fold, and that is the only reason a reader was not handed a sheet that could not be folded — the printed shelf is chosen by judgement and the judgement happened to keep the tessellation patches off it. Judgement is not a check.

It matters more than it sounds. A figure is not a lesser artefact here — the whole proposition of the collection is that a crease pattern is the argument written in the only notation the subject has, and a patch drawn to illustrate a tessellation is making a claim about paper exactly as a printed sheet is. A reader who copied one of those four patches onto a square and folded it would have got a sheet that jams, and no gate would have said why.

What the reading gives is a check that runs everywhere: over the printed shelf, over every patch, over anything drawn later. It costs one sweep over the crease list, it needs no folded state, and it is the cheapest refutation this collection has — the whole ladder of them, ranked by cost and by what each refuses first, begins here.

And it names something to repair rather than merely something to report. The crossings belong to the boundary of a patch and not to the tessellation, so the repair is to cut the patch out of the plane instead of assembling it on the sheet — a change to how the paper is chosen, not to the pattern. The four patches then carry none.