The cut that changes nothing

A cut is surgery: a closed cut in the middle of a sheet adds a boundary circle, a loop that cannot be shrunk, and a parity condition that no vertex can see.

The complement of that result is the more useful half for anybody who cuts paper, and it is that almost every cut anybody makes does none of it.

Which cuts are inert #

A cut leaves the sheet’s shape unchanged when its two ends land on existing boundary — the sheet’s edge, or the end of another cut.

The reason is that such a cut extends the existing boundary circle rather than creating a new one. The knife’s two faces become new edge, they join the old edge at both ends, and what was one closed curve of boundary is still one closed curve of boundary — longer, more convoluted, and topologically the same.

So the sheet is still a disc. Every closed path on it can be shrunk, the two-colouring is implied by the vertex conditions, and there is no condition to satisfy that was not there before.

Even a cut in the middle #

The case that surprises people is a slit whose two ends are both in the interior of the sheet — a straight cut that starts and stops in the middle of the paper, removing nothing.

That is still inert. The new edge runs out along one side of the cut, round the far end, and back along the other side, returning to where it began without meeting the sheet’s edge at all. It is a new piece of boundary and it does not enclose anything, so it does not create a loop that cannot be shrunk.

Draw a loop round such a slit and it can be slid off the end and shrunk to a point. Draw a loop round a hole and it cannot, because the hole is in the way and there is no way past it.

That is the whole distinction: whether the removed region separates the loop from the rest of the paper.

Which is why kirigami is untouched #

A kirigami pattern is a field of straight cuts, each running between two points that are already boundary or free in the interior. So a kirigami sheet is still a disc, however many cuts it has, and every result about it stands unchanged.

A cut buys freedom is therefore the correct and complete account of kirigami, and the connectivity results are about whether the sheet stays in one piece rather than about its shape.

Doing it with scissors #

The claim that a slit through the material changes nothing is worth testing, because it sounds wrong.

Take two squares of paper. On the first, cut four straight slits, each a few centimetres long, somewhere in the middle, not touching each other or the edge. On the second, punch a hole.

Now crease both with four lines from the centre to the corners, and press.

Both fold. Now do it with three lines rather than four. The first square still folds — the slits do nothing, and a four-crease vertex is what it always was. The second does not: three creases round a hole is an odd count, and the sheet refuses.

The slitted square has had far more material cut than the holed one. It behaves exactly as an uncut square does. The holed one has lost a centimetre of paper and behaves differently.

That is the distinction in the hand, and doing it once makes it very hard to forget.

Why enclosure is the criterion #

The reason a closed cut matters and a slit does not is worth stating carefully, because encloses a region is doing all the work.

A loop drawn on the sheet either can or cannot be shrunk to a point while staying on the paper. Round a slit, it can: slide it off the slit’s end and it contracts freely, because there is paper all round the end.

Round a hole, it cannot. There is no paper in the hole, so the loop cannot pass through, and there is no route round it either, since the hole’s boundary is a closed curve.

The parity condition attaches to loops that cannot be shrunk, because those are the loops whose crease count is not forced by anything inside them. A shrinkable loop bounds a region, the vertices inside the region force its count, and there is no independent condition.

So the criterion is not about material removed. It is about whether the removal creates an obstacle to shrinking, and only an enclosed removal does.

Counting a sheet’s conditions #

For a sheet with any number of cuts, the number of independent parity conditions is easy to compute.

Count the boundary circles. A plain square has one. Each closed cut adds one. Slits, notches and edge-to-edge cuts add none.

The number of conditions is one fewer than the number of boundary circles. A square: one circle, nought conditions. A square with a hole: two circles, one condition. Two holes: three circles, two conditions.

Equivalently, Euler’s number falls by one for each closed cut, from one downward, and the number of conditions is one minus the number.

That is a complete accounting for sheets made by cutting, and it can be done by looking at the pattern rather than by computing anything.

Where a slit is not quite nothing #

Two qualifications, both physical rather than topological.

A slit changes the mechanics. A field of slits is what makes kirigami expand: the material can rotate and stretch in ways an uncut sheet cannot, and that is the whole of what the technique is for. Topological inertness is not mechanical inertness.

A slit can disconnect. Enough cuts in the right places and the sheet falls into pieces, which is a connectivity question with its own answers and is a different question from the sheet’s shape.

So the claim is narrow and exact: a slit changes nothing about which folded states the sheet has, by way of the global conditions. It changes a great deal about how the material behaves and it can change whether the sheet is one object.

Keeping those three apart — shape, mechanics, connectivity — is most of what it takes to reason about cutting clearly.

A rule of thumb, and when it fails #

For anybody who wants one sentence: if the scissors come back out through the edge, nothing changed; if they go round and meet themselves, something did.

That is right for straight cuts and it has an exception worth knowing. A cut that starts on the edge, wanders into the middle, and comes back out to the edge at a different point does not change the sheet’s shape — but it may cut it into two pieces, and two pieces is a different situation again.

So the full rule is: a cut whose ends are on existing boundary leaves the sheet’s shape alone and may disconnect it; a cut with free ends leaves both alone; and a cut that closes on itself changes the shape and adds a condition.

Three cases, and the last is the rare one.

Where a closed cut turns up #

Rarely, and deliberately.

A hole for a fixing. A mounting point, a screw, a stem. Any hole a designer puts in for a reason external to the folding.

A window. A hole to see through, in a folded shade or a packaging insert.

A hole to save paper. The centre of a sheet is what a uniaxial base uses least, so removing it costs little in what the design can produce.

Each of those is one closed cut or two, deliberately made, and each adds its condition. That is a small number of cases and they are exactly the cases a designer should check.

The check, for anybody cutting #

Two questions, and both are about the cut rather than about the pattern.

Do the cut’s ends land on existing boundary, or are they free in the interior? If yes, nothing changes.

Does the cut close on itself, enclosing a region? If yes, count the creases a loop round the removed region crosses and require the count to be even.

That is the whole procedure. It costs a count per closed cut, most patterns have none, and a pattern with one has one condition more than it had.

Why the distinction is invisible #

The two kinds of cut look identical at arm’s length, and that is the practical difficulty.

A very narrow hole and a slit are the same picture. So are a slit whose ends nearly meet and a closed cut. The difference is whether the removed region is enclosed, which is a fact about the ends of the cut rather than about its length or its width or how much material it takes out.

And the amount of paper removed is no guide at all. A closed cut can remove a square millimetre and change the sheet; a slit can run right across and remove nothing and change nothing.

What the collection has measured #

Since the essay is largely a classification, the measurements behind it deserve naming.

Sheets with one hole, at crease counts from three to eight, checked by two computations that share no code: a panel colouring and a composition of reflections. They agree at every count, refusing the odd ones.

Sheets with two holes, at six arrangements of the three crease families, with the two loops’ parities set independently. The colouring refuses exactly when either loop is odd, and Euler’s number is minus one on all six.

Rings with a cut from the hole to the rim, at three, five and seven creases. Every one of them refuses before the cut and folds after it, on identical creases, letters and angles.

That last is the direct evidence for the classification: the cut changes nothing about the drawing and changes the verdict, because it changes the sheet.

Nothing has been measured on a slitted sheet, because a slitted sheet is a disc and there is nothing new to measure. That is the claim rather than an omission.

The gluing, for symmetry #

The operation opposite to cutting behaves the same way and it is worth completing the picture.

Joining two boundary edges — taping a sheet into a tube — removes a boundary circle and adds a loop that cannot be shrunk. Same effect on the count as cutting a hole, opposite effect on the amount of paper.

Taping a slit shut restores the sheet, provided the tape rejoins exactly what the knife separated. Nothing changes.

Taping a slit shut wrongly — joining the left side of the cut to a different part of the right side — is a gluing rather than a repair, and produces a sheet that is not the sheet anybody meant to have.

That last one is more than a curiosity. A repair to a torn sheet is a gluing, and a careless one changes the object’s shape. Whether it does depends on exactly which points were joined to which, which is not something anybody inspects.

So the operations that change a sheet’s shape are: a closed cut, and a join of two boundary edges. Everything else — slits, notches, edge-to-edge cuts, careful repairs — leaves it exactly as it was.

Slits that meet #

One more case, since kirigami patterns are full of it: a slit whose end touches another slit.

Two slits meeting at a point form a single connected piece of new boundary, shaped like a cross or a T, with free ends. It still encloses nothing, so the sheet is still a disc and no condition arrives.

Three or more slits arranged to form a closed circuit are a different matter. A triangle of three slits, meeting end to end, encloses a region — and the region is not removed, since nothing was taken out, but the paper inside it is joined to the rest only if the slits fail to meet exactly.

That is the case where a kirigami pattern would acquire a condition, and it is the case kirigami avoids by construction: a closed circuit of cuts detaches a piece, which is a failure rather than a feature, and every pattern is designed so that it does not happen.

So kirigami stays on the harmless side not by luck but because the alternative is the material falling apart, which is a constraint the technique was always working under.

A note on notches #

There is a fourth case that sits between the others and it has come up before in this collection.

A notch is a bite out of the sheet’s edge: a piece removed, with the removal touching the boundary. It changes the outline, it removes real material, and it leaves the sheet a disc — one boundary circle, deformed.

So a notch is inert in the sense of this essay, which is worth stating because it does remove paper and it does change what a designer can do. A notch is not a hole is a result about design that this essay makes topological: a notch adds no condition, a hole adds one, and the reason is that a notch’s removal touches the existing boundary and a hole’s does not.

That is the same criterion again — enclosure — applied to a case where the removed region is large and the answer is still nothing.

Why this is worth an essay #

A classification with three cases and one rare exception could be a footnote, and there are two reasons it is not.

The first is that the exception is the case people meet when they are not thinking about folding. A hole for a fixing, a window, a mounting point: each is put in for a reason external to the pattern, usually late, often by somebody other than whoever designed the folding — and each adds a condition that can turn a working pattern into one with no flat state.

The second is that the classification runs the opposite way from intuition. More cutting means more change is the natural belief, and it is wrong in the strongest possible sense: a hundred slits change nothing and one small hole changes the object.

An essay is warranted when a rule is both easy to state and reliably guessed wrong, and this one is.

What the paper does not know #

A closing observation, since the whole essay is about a distinction the material cannot feel.

Paper has no idea whether a cut in it encloses a region. The fibres are severed the same way, the sheet is weakened the same way, and the two sides of the cut fall apart the same way, whether the knife came back to its start or ran out to the edge.

What differs is a fact about the set of points the paper occupies, and the folding conditions read that set rather than the material. That is a good description of why the topological conditions are so unlike the material ones: they are about where the paper is, and the paper has no opinion about that at all.

The three vocabularies #

It is worth ending on the separation, because the essay’s practical value is in not confusing three things that all get called cutting.

Shape. Which cuts change the sheet: only the closed ones. Measured by boundary circles and Euler’s number. Decides how many parity conditions the sheet carries.

Connectivity. Which cuts separate the sheet into pieces. A different question, with its own results, and orthogonal to the first — a closed cut never disconnects, and an edge-to-edge cut may.

Mechanics. What the cuts do to how the material stretches and rotates, which is what kirigami is for and what a cut buys.

A given cut has an answer in each of the three, and the answers are independent. A slit: no shape change, possible disconnection, large mechanical effect. A hole: shape change, no disconnection, small mechanical effect.

Nothing about a cut’s appearance predicts its answers, which is why the three have to be asked separately.

The sentence that covers both #

A cut removes an adjacency, and buys freedom. True of every cut.

A cut that closes on itself also adds a boundary circle, a loop that cannot be shrunk, and a parity condition. True of the small minority of cuts that do.

Which of the two a cut is depends on where its ends are, and nothing else — not its length, not its width, not what it removes.