A cut that reaches the edge

A square of paper with a square hole in the middle and three creases running from the hole to the rim is this collection’s smallest object with nothing wrong with it and no folded state. It has no interior vertex at all, so developability, Kawasaki, Maekawa and the big-little-big lemma are all satisfied — vacuously, having nothing to be satisfied at. And it does not fold: its three panels form an odd cycle, so no two-colouring exists, and composing the reflections around that cycle puts the two routes to one panel 1.75 sheet widths apart.

Cut it open. One slit from the hole to the rim, along a line no crease occupies, removing no paper at all. The same three creases, the same three letters, the same three angles.

It folds.

What the cut actually does #

Not what it looks like it does. It does not remove paper — the slit has no width. It does not remove a crease — it runs where there is none. It does not change any angle at any point.

What it removes is an adjacency. Before the cut, the two panels either side of the slit’s line are neighbours: paper runs continuously from one into the other, so anything true of one constrains the other. After it, they are two pieces of paper that happen to touch, and the constraint is gone.

Every walk in this subject is a walk over adjacencies. The two-colouring flips colour across each crease and keeps it across a drawn line, walking from panel to neighbouring panel. The placement composes a reflection across each crease, walking the same graph. Both of them go round the ring because the ring’s panels form a cycle — and the cut breaks the cycle, so neither walk has two routes to compare and neither can find a contradiction.

Three kinds of line, and only one of them cuts #

A crease pattern’s edges now carry three meanings that are easy to run together and behave completely differently.

A crease — mountain or valley — joins two panels and turns the sheet over between them. Crossing one flips the colour and applies a reflection.

A raw edge is where the paper stopped when the sheet was cut out. It joins nothing, because there is no panel on the far side.

A cut is where the paper was separated afterwards. There is a panel on the far side and it is no longer a neighbour. That is the case that had no representation here until now, and adding it is two lines: a walk over the panels does not cross a cut, and a vertex on one is on the edge of the paper as far as every vertex condition is concerned.

The third is the interesting one because it is the only operation available that makes a graph smaller. Folding adds creases and adds adjacencies; drawing adds lines and changes nothing; cutting takes an adjacency away, and taking one away is what breaks a cycle.

Why the ring failed in the first place #

The failure is a parity, and the cut is a statement about where a parity can live.

On a disc, the two-colouring of panels is exactly equivalent to a condition at each vertex: each interior vertex has an even number of creases, and that is what makes the colouring consistent. Cut a hole and the equivalence breaks in one direction. The ring’s three creases run from the hole to the rim, so they meet at no interior vertex at all — every vertex condition holds vacuously — and yet the three panels form a cycle of odd length, which takes no two colours.

The parity did not go anywhere. What went away is the vertex that used to carry it, and what replaced it is a loop of panels that only a walk over the whole sheet can see.

So the cut works because it removes the loop. It is not repairing a local fault, because there is no local fault; it is changing the shape of the region, which is the only thing that could have made a difference.

The numbers, and their shape #

The rings measured here have three, five, seven, nine and eleven creases across them, and the disagreement between two routes to a panel comes out at 1.75, 1.59, 1.43, 1.32 and 1.24 sheet widths. Every one of them two-colours after the cut, and every one of them places to a part in 10¹⁶.

The trend downward is worth a sentence. More creases across a ring means a smaller angle between neighbouring ones, and the composition of the reflections around the loop is a turn by twice the sum of those angles — so the failure is largest when the creases are few and far apart, and shrinks as they crowd. It never reaches zero, because an odd number of reflections composes to a reflection and a reflection is never the identity.

That is the same distinction the ring essay makes and this measurement sharpens: an odd cycle of panels fails for a reason that is not about how far off it is. The 1.24 of an eleven-crease ring is not a near miss. It is a different kind of object from a sheet that folds.

How large the failures are, against the largest possible #

The five disagreements fall from 1.75 to 1.24 sheet widths, and the trend downward invites the reading that a ring with enough creases would nearly fold. It is worth checking that against the ceiling.

An odd number of reflections composes to a reflection, and a reflection moves a point by twice its distance from the mirror line. On a unit square the furthest a point can be from a line through it is the half-diagonal, so the largest disagreement geometrically available is about $2\sqrt2 = 2.83$ sheet widths.

Measured against that, the five rings sit at 62, 56, 51, 47 and 44 per cent of the maximum possible failure.

So an eleven-crease ring is not close to folding. It is failing by nearly half of the largest amount any sheet could fail by, and the falling trend is a fall from very large to large rather than toward anything.

And the floor is not zero #

Extrapolating the differences — 0.16, 0.16, 0.11, 0.08 — the sequence is flattening somewhere near one sheet width, not near nothing.

That matches the algebra. The composition is a reflection whatever the angles are, and a reflection displaces every point off its axis; crowding the creases moves the axis but cannot make the displacement vanish. A ring with a hundred creases across it would still disagree by something of order a sheet, and the only way to reach zero is to stop having an odd cycle.

Which says what the even case is #

The parity argument also settles the ring’s even cousin, and the answer is the one the collection has already found from the other direction.

An even number of reflections composes to a rotation, and a rotation can be the identity. So an even ring is not automatically refused — it folds exactly when that rotation is trivial, which is when the alternating sum of the angles between consecutive creases vanishes.

That is Kawasaki’s condition, evaluated around a loop of panels rather than around a vertex. The ring has no interior vertex to state it at, so the same equation reappears as a condition on a cycle — which is why an even ring can still fail and why the failure looks like nothing any local test would recognise.

So the two cases are one statement. Around any cycle of panels the composed motion must be the identity; an odd cycle composes to a reflection and never can be, and an even cycle composes to a rotation and is exactly when the angles alternate to zero.

What a folder sees #

The experiment is worth doing with paper, because the model’s answer and the hand’s answer agree for once and the agreement is instructive.

Cut a square of paper, make a square hole in the middle of it, and crease three lines from the hole out to the edge — one mountain, one valley, one mountain, say, at whatever angles are convenient. Try to fold it flat. The paper will not go: one of the three creases always ends up wanting to be the other letter, and pushing on any of them undoes another. It is not stiff, it is not close; the sheet simply will not lie down.

Now snip from the hole to the edge anywhere between two of the creases. Fold again. It goes immediately, and the three creases behave exactly as they did before — the same directions, the same order, the same feel.

What is instructive is that nothing about the fold changed. A folder repeating the two attempts learns that the obstruction was never at any of the creases, which is exactly what the panel cycle says and what no amount of examining the creases would have revealed. The conditions are about a point and the failure is about a loop, and the paper says so faster than the arithmetic does.

Which cuts do this and which do not #

Not every cut breaks a cycle, and the distinction is exactly where the cut’s two ends are.

Both ends inside the paper — a slit — adds a hole rather than removing an adjacency, since the panels either side of it are still joined around both of its ends. It is the case a cut that removes no paper is about, and its effect is to add boundary rather than to take a walk apart.

One end on the rim — a notch — separates the panels along the cut, and the walk has to go round the closed end. On a disc that changes nothing topologically; on a ring, if the cut runs from the hole to the rim, it turns the ring into a disc and everything above follows.

Both ends on the rim severs the sheet into two pieces, and the question stops being about one sheet at all.

So the interesting case is the middle one, and it is interesting only on a sheet that was not a disc to begin with. A cut from rim to rim across a plain square is a pair of scissors making two smaller squares; a cut from a hole to the rim is a topological operation with a folding consequence.

The cut as an operation on the graph #

There is a compact way to say what all three lines do, and it is worth having because it makes the odd one out obvious.

Think of the panels as points and the lines between them as connections. A crease is a connection with a reflection attached; a drawn line is a connection with the identity attached; a cut is not a connection. Every question this collection asks about a whole sheet is a question about that structure:

• does it two-colour? — a question about cycles;
• do the panels place? — a question about what the connections compose to around each cycle;
• can the panels be ordered? — a question about constraints between panels that share ground, which is a different structure again.

The first two are cycle questions, and cutting is the only one of the three operations that changes the cycles. That is why a cut can fix a pattern that folding cannot: adding creases can only add cycles, and the ring’s problem was that it had one.

Which also says what a cut cannot fix. A pattern whose panels place perfectly and whose letters force a loop of panels above one another — the case every repaired tessellation patch here is in — has a cycle in a different structure, one about layers rather than about adjacency. A cut through the paper would break that too, and it would do it by removing the paper whose layers were in contention, which is a much larger intervention than a slit.

What this costs the site’s own rule #

This collection’s rule is one sheet and no cuts, and the rule is a claim rather than a preference: every theorem the subject has is stated about a sheet whose paper is all still there. Three essays already bend it — the one-cut theorem, where the cut is the output, the oldest book in the subject, which cuts on nearly every page, and the module that prices what a cut buys.

This is the fourth bend and it is the smallest so far: a cut of zero width, removing no paper, changing no angle, along a line no crease occupies. Everything the vertex conditions can see is unchanged, and the pattern goes from unfoldable to foldable.

That is worth stating as a caution rather than as a result. The four conditions are conditions on a disc, and every time this collection has gone outside that assumption — a hole, a ring, a patch that is not simply connected — the conditions have gone on passing while the sheet stopped folding. A checker that reads vertices is a checker that assumes the paper is a disc, and nothing in it says so.

What it means for the collection’s own patterns #

Three consequences, and the first is a piece of housekeeping that had been outstanding since the ring was drawn.

A cut is now representable. Before this, a pattern could carry creases, raw edges, drawn lines and unassigned edges, and the one thing it could not say was the paper is separated here. Every essay about cutting had to describe the operation rather than draw it. Now the drawing carries it, and everything downstream — the panel walk, the two-colouring, the placement, the vertex conditions — behaves correctly without being told.

The vertex conditions treat a cut as an edge, which is right and is worth stating because it is a choice. A vertex where a cut ends is a vertex on the boundary of the paper, so the four conditions do not apply to it, exactly as they do not apply to a vertex on the sheet’s rim. The alternative — treating a cut end as interior — would report a Kawasaki failure at a point where the paper simply stops.

And no printed pattern here has a cut in it. The shelf is eight patterns and the rule is one sheet, uncut; the fold-and-cut triangle’s heavy outline is where the scissors go and is not a crease, which is the closest the shelf comes. So the machinery is here for the arguments rather than for the patterns, which is the honest description of most of what a collection like this builds.

The same shape, four times #

This is the fourth object in this collection whose failure is invisible to every condition the subject states, and the four are worth listing together because the list is the argument for reading a pattern as a whole.

A ring has no interior vertex, satisfies everything vacuously, and does not fold. A crossing is not a vertex, satisfies nothing because nothing is asked of it, and does not fold. A lettering that forces a loop of panels satisfies every vertex condition and has no ordering. A tessellation patch cut out badly satisfies every vertex condition and cannot be placed at all.

In every case the conditions pass and the sheet does not exist, and in every case what finds it is a walk over the whole pattern rather than a reading at a point. That is not a coincidence about four objects; it is what “local” means. A condition at a point cannot see a cycle, and every one of these four failures is a cycle somewhere — in the panel graph, in the drawing, in the forced order.

The cut belongs on that list from the other side. It is the operation that removes a cycle, which is why it repairs the first of the four and none of the others.

Where the argument stops #

A cut is not modelled as two coincident edges. In the drawing a cut is one line that panels do not cross; in paper it is two edges that can move apart. Nothing here computes what happens when they do, and the folded states counted are those of the flat sheet with the adjacency removed.

The cut is straight and crosses no crease. A cut that crossed a crease would cut the crease as well, which is a different operation — a cut that turns a crease into a raw edge removes the constraint that made the folded state determined at all, and the measurement then becomes binary and uninformative.

And the ring is the smallest case rather than the general one. What has been shown is that one cut fixes one family of sheets whose panels form a single odd cycle. A sheet with several holes has several cycles, and how many cuts it takes to break all of them is a question this collection has not asked.