A cut is surgery

The standing account of cutting in this collection is that it buys freedom. A cut removes an adjacency, and removing an adjacency removes a constraint, so a sheet with a cut in it can do things the uncut sheet cannot.

That account is right about one kind of cut and exactly wrong about another, and the two are indistinguishable on the paper.

Two cuts #

Take a square of paper. Make a straight cut from the middle of one edge inward to the centre.

Take another square. Cut a small closed shape out of the middle — a square hole, say — and remove the piece.

Both are cuts. Both take a knife to the sheet. The first removes no paper at all and the second removes very little.

The first cut changes nothing about the sheet. It is still a disc: every closed path drawn on it can be shrunk to a point, every parity is forced by what happens at the vertices inside the path, and the sheet carries no condition it did not carry before.

The second changes the sheet. There is now a loop that cannot be shrunk — one going round the hole — and it carries a condition of its own.

The condition it adds #

Crossing a crease exchanges which face of the paper is up. A closed path on a sheet that folds flat must therefore cross an even number of creases, and on a disc that is automatic: every closed path bounds a region, the vertices inside the region force the count, and no separate condition survives.

Round a hole, nothing bounds. The count is free, its parity is not forced by anything, and the sheet folds only when it comes out even.

So cutting a hole can turn a sheet that folds into a sheet that does not, without touching a single crease.

Adding paper back #

The complementary operation makes the same point from the other side.

Take the loop of paper with three creases, which does not fold. Make one cut, from the hole out to the rim, crossing no crease.

Now it folds. Nothing about the creases changed: same spokes, same letters, same angles. What changed is that the sheet stopped being an annulus. The cut joins the hole’s boundary to the rim, the loop round the hole can now be slid off through the cut, and the condition it carried goes with it.

That is the operation the collection’s usual account of cutting describes correctly — a cut that reaches the edge removes an adjacency and buys freedom. What it does not describe is the closed cut, which adds a condition rather than removing one.

Doing both, with scissors #

The two cuts are worth making, because the difference is entirely invisible until the paper is folded and then it is obvious.

Take two squares of paper about fifteen centimetres across. On each, draw three lines from the centre outward to the edge, at roughly a hundred and twenty degrees apart, and crease all three.

On the first square, cut a small square hole out of the middle — a centimetre across, centred where the three creases meet, so that the creases now start at the hole’s edge rather than at a point.

On the second square, leave the middle alone and make a single straight cut from one edge inward, in one of the sectors between two creases, stopping short of the middle. It removes no paper.

Now press each flat.

The second goes flat easily. Three creases meeting at a point is an odd vertex, and an odd vertex cannot fold — so it does not go flat there, and the cut does nothing to help. The honest version of this experiment is to use four creases rather than three, in which case the second square folds and the cut is irrelevant to whether it does.

The first square, with the hole, has no vertex at all: the creases start on the hole’s edge and end on the rim, meeting nothing. Every condition the subject checks is satisfied. And with three creases it does not go flat, and with four it does.

That is the whole result in the hand. The paper with the hole obeys a rule that the paper without it does not have, and the rule is about a loop rather than a point.

Where the boundary is, exactly #

The distinction turns on what counts as boundary, and it is worth being precise because the word does two jobs.

The boundary of the paper is where the sheet stops. A square has one boundary circle: its four edges, joined at the corners. A square with a hole in it has two.

A crease is not boundary. It is a line the paper folds along and the paper continues across it.

A cut creates boundary. Where the knife went, the paper now stops, and the new edge is boundary in exactly the sense the old edge is.

The classification then reads: a cut whose two ends land on existing boundary extends that boundary circle and leaves the count of circles alone. A cut that closes on itself creates a new circle. And the number of boundary circles, together with the characteristic, is what names the sheet.

That is also why the collection’s construction for a holed sheet marks the hole’s edge with the same letter it marks the sheet’s outer edge with. They are the same kind of thing, and treating them differently would be a bookkeeping error rather than a modelling choice.

The operation that undoes a hole #

There is an asymmetry between the two directions that is worth noticing.

Cutting a hole adds a condition. Cutting from the hole to the rim removes it again. So the two operations are inverse in their effect on the sheet, and neither is inverse in its effect on the paper: the first removes material and the second does not, and no amount of the second puts the removed piece back.

What that means is that the sheet’s shape is a coarser thing than the paper. Two sheets can be the same shape and different objects — a square, and a square with a slit in it, are the same sheet and not the same piece of paper — and every condition in this essay is about the first.

That is a good deal of what makes the topological account useful. It throws away everything about the paper except the one feature that decides which conditions apply, and the feature is not one anybody would have picked by looking.

Slits, and why kirigami is unaffected #

Almost all of this collection’s cutting work is about slits, and none of it moves.

A kirigami pattern is a field of straight cuts, each running between two points that are already boundary — either the sheet’s edge, or the end of another slit, or a previous cut. The result is a sheet that is still a disc, no matter how many slits are in it, and it carries no condition beyond the ones its vertices carry.

That is why a cut buys freedom is the right account of kirigami, why one cut short of falling apart is about connectivity rather than about topology, and why a cut that reaches the edge is the case the collection has been analysing.

The exception is a slit whose two ends are both in the interior — a cut that starts and stops in the middle of the paper. That is not a closed cut and it does not create a new boundary circle either: the new edge doubles back on itself, and the sheet stays a disc. Such cuts are common in kirigami and they are topologically inert, which is a mildly surprising thing to be able to say about a cut that goes right through the material.

What actually distinguishes them #

Not the length, not the amount of paper removed, and not whether the knife left the sheet connected. All four of those are the same for the two cuts.

What distinguishes them is whether the cut’s two ends are on the existing boundary.

A cut running from boundary to boundary leaves the sheet a disc, because the two pieces of new edge join up with the old edge into a single boundary circle. A cut that closes on itself in the interior creates a second boundary circle, and a sheet with two boundary circles is not a disc.

That is a statement about the sheet’s shape and it can be computed. Euler’s number is one for a disc, nought for a sheet with one hole, and it falls by one for every additional hole.

Why the usual account got it right anyway #

Almost every cut anybody makes in practice runs from edge to edge, and the reason is that a closed cut removes a piece and most cutting is not meant to.

A slit, a fringe, a taper, a notch: each starts on the boundary. Kirigami patterns are made of slits, and a slit is topologically nothing. So the whole of the collection’s kirigami work is about cuts of the first kind, and for those cuts a cut buys freedom is the correct and complete account.

The closed cut turns up in exactly two places. One is a hole cut deliberately, to make a sheet cheaper to design on. The other is a hole that is part of the object — a window, a mounting point, a hole for a stem.

The arithmetic #

The bookkeeping is simple enough to state in full and it is the useful part of the essay.

Start with a disc: Euler’s number one, no loops that cannot be shrunk, no conditions.

Every closed cut in the interior takes the number down by one and adds a loop, and every loop carries a parity condition on the creases crossing it.

Every cut between boundaries leaves the number where it is and adds nothing.

Every glued pair of edges takes the number down by one as well, and adds a loop in the same way — which is why a cylinder is the same sheet as an annulus and behaves identically.

So the count of conditions a sheet carries is a property of its shape, and it can be computed from three integers before any pattern is drawn on it.

Gluing, which is the same operation backwards #

A gluing and a cut are opposite operations and they have opposite effects on the count, which makes the whole family easy to hold.

Cutting a hole takes Euler’s number down by one and adds a loop. Gluing a pair of a rectangle’s edges does exactly the same: down by one, one more loop, one more condition. So a cylinder and an annulus are the same sheet, reached from opposite directions, and every condition either of them carries the other carries too.

Gluing the second pair of edges takes the number down again — from nought to minus one, by the arithmetic — except that it does not, because of the corner. The four pieces of paper at the rectangle’s corners are all one panel, and that single extra merge is what keeps the number at nought and makes a torus a torus rather than something with a pinch in it.

So the arithmetic is: cut a hole, minus one; glue a pair of edges, minus one, with a correction of plus one for the second pair. Every operation this collection can perform on a sheet is one of those three, and the number they produce says how many conditions the result carries.

Why this belongs under cutting rather than under folding #

A reader might reasonably file the whole essay under flat-foldability, since every consequence above is about which sheets fold.

It belongs under cutting because the cut is the operation being classified. The conditions are consequences; what is new is that two operations spelled the same way in ordinary language are different, and one of them is the operation that produced the collection’s oldest example of a global obstruction.

There is also a practical reason. Anybody designing a folded object with a hole in it — a window, a port, a mounting point — is performing the second operation, usually without noticing, and the pattern they had that folded may stop folding. The condition is cheap to check and it is not a condition anybody would think to check, because holes are usually treated as decoration on a pattern rather than as a change to the sheet.

What a cut does not do #

Three things worth ruling out, because a result about cutting invites all three.

It does not change any vertex. A cut running between existing boundaries meets no crease; a closed cut in the interior meets none either, in the constructions here. The vertex conditions hold exactly as they did.

It does not change the crease pattern. Nothing is drawn or erased. The claim is entirely about the sheet the pattern is drawn on.

It does not make the paper weaker in the sense that matters here. A physical sheet with a hole in it tears more easily and that is a real fact and not this one. The condition above is exact and holds for ideal paper of no thickness, and it is about which folded states exist rather than about which the material survives.

A short catalogue #

For reference, since the classification is more useful as a list than as a rule.

A slit from the edge inward, ending in the interior. No change. The sheet is still a disc.

A slit right across, edge to edge. No change to the sheet’s shape, and it may fall into two pieces, which is a different matter.

A slit entirely in the interior, both ends free. No change. The new edge doubles back on itself and no boundary circle is created.

A closed cut, piece removed. One more boundary circle, one more loop, one more condition.

A closed cut, piece left in place. The same, since the sheet’s shape is decided by where the paper is joined rather than by whether a piece happens to be sitting in the gap.

A pair of edges glued. One more loop, one more condition — the same as a hole.

Every operation this collection performs on a sheet is in that list, and the only two entries that change anything are the last two.

What the model is ignoring #

The account above treats paper as an ideal surface with no thickness and no strength, and three physical facts are being set aside.

A hole weakens the sheet. Real paper tears from a hole, and the tearing starts at the corners of a square one. Everything above is about which folded states exist rather than about which the material reaches, and a designer cutting a hole has both problems.

A cut has width. A knife removes a strip, however thin, so the two sides of a slit are not quite the same points of the paper. The collection has an essay about exactly that gap, and the condition here is unaffected: the parity is a count of creases and does not care whether the cut is a line or a narrow strip.

A cut can be undone with tape. Physically, joining the two sides of a slit back together restores the sheet, which is the same operation as gluing and has the same effect on the count: minus one, one more loop, one more condition. So a taped repair is not a return to the original sheet unless the tape rejoins exactly what the knife separated.

That last one is more than a curiosity. A slit taped shut in the wrong place — joining the left edge of the cut to a different part of the right edge — is a gluing rather than a repair, and produces a sheet that is not the sheet anybody meant to have.

The sentence that was missing #

The collection’s standing sentence about cutting has been a cut removes an adjacency and buys freedom, and it needs a clause.

A cut between two points of the boundary removes an adjacency and buys freedom.

A cut that closes on itself in the interior removes an adjacency too, and adds a loop, and the loop brings a condition that no vertex can see.

The second is what turns a square of paper into a sheet where every theorem in the subject can hold vacuously and the paper still refuses, which is the collection’s oldest example of a global obstruction and has been sitting under the word cut the whole time.