A cut that removes no paper

A fold moves paper about and cannot change how much of it surrounds a point; a cut can, and that difference is what the founding rule of this subject is worth. The essay that established it measured the exchange rate: take a wedge of thirty degrees out of a sheet and the paper closes into a cone whose turn is thirty degrees short of a full one. Take sixty and it is sixty short. The relationship is not merely monotone, it is an identity — what the cut removes is what the sheet loses, to the degree.

There is a second kind of cut, and it obeys no exchange rate whatever.

The other kind #

Cut a hole in the middle of a square of paper — not a slit, a hole, with the middle piece removed and thrown away — and run three creases from the hole’s edge to the sheet’s edge.

That sheet cannot be folded flat. Not “is difficult to fold flat”: there is no flat folded state at all, because the three panels form a cycle of odd length and the two colours a folded sheet’s panels must take cannot alternate round it. Composing the reflections all the way round the loop leaves one panel 1.8654 sheet-widths from where the other route puts it, which is not a residual to be argued about.

Now shrink the hole.

The measurement #

At a hole a third of the sheet across, occupying eleven and a half per cent of its area, the gap is 1.8654. At a fifth across — four per cent of the area — it is 1.8654. At a tenth, one per cent, 1.8654. At four hundredths, sixteen parts in ten thousand, 1.8654. At one hundredth of the sheet across, one part in ten thousand of its area, 1.8654.

Not approximately the same. The same, in every decimal place the measurement reports, because the number does not depend on the hole at all: it is the composition of three reflections in three lines, and the lines are where they are whatever has been removed from the middle.

It is worth doing once with real paper, because the result is genuinely surprising in the hand. A square of copier paper, a needle hole in the middle, three creases scored from the hole out to three different edges — one to a corner and two to sides, so that they do not line up — and then an honest attempt to press the thing flat. It will not go. Something always stands proud, and moving the crease that stands proud makes another one stand proud instead.

A folder’s instinct is that the sheet is nearly right and needs persuading. It is not nearly right. The obstruction is exactly as large with a needle hole as with a hole the size of a coin, and no amount of persuading is going to move a quantity that is not there.

Two kinds of quantity #

The contrast is the essay, and it is worth stating in its sharpest form.

The wedge cut changes an amount at a point. The amount is the angle deficit, it lives at the vertex the wedge was cut from, and it is a continuous function of the cut: a wedge of nothing removes nothing, a wedge of thirty removes thirty. Every consequence of it — the cone’s half-angle, the curvature concentrated at the tip, how a growing disc has to buckle to accommodate an excess — scales with it.

The hole changes what kind of sheet it is. What is affected is not a quantity anywhere; it is whether every closed path in the paper can be shrunk away, and that is a yes-or-no question with no small version. The paper either has a way round that cannot be got rid of or it does not.

Where the discontinuity is #

There is no hole size at which the behaviour changes, which sounds like a paradox and is not.

The function “does this sheet fold” takes the value yes at hole size exactly zero and no at every positive hole size. It is discontinuous at zero and nowhere else. That is a perfectly ordinary way for a function to behave; what makes it feel strange is that the subject’s other cuts are all continuous, so the intuition that has been built up on wedges and slits does not transfer.

It also means the hole cannot be approximated away. A modeller who wants to treat a small hole as a perturbation of a sheet without one has nothing to expand in: there is no small parameter, because the size of the hole appears in no consequence.

The slit that is not a hole #

The obvious objection is that a hole is a lot of cutting and a slit is not, so perhaps the interesting boundary is somewhere in between. It is worth settling, because the answer is clean.

A straight slit — a cut with no width, ending inside the paper — does not do this. A sheet with a slit in it can still have every closed path shrunk away: a path that appears to go round the slit can be slid off its end, because the slit has an end. The sheet is still a disc, and the theorem still holds, and everything about the two-colouring goes on working.

To get the effect, a cut has to come back to itself — and a closed cut takes the middle out. So the boundary between the two kinds is not about how much is removed; it is about whether the removal disconnects a piece. That is why the smallest hole in the world does it and the longest slit does not.

There is a small piece of bookkeeping that makes the discontinuity feel less like a trick. The quantity that does change continuously is the area of paper: eleven per cent, four per cent, one per cent, and so on down. The quantity that decides whether the sheet folds is not the area, and there is no reason it should have been — the two-colouring is a fact about a graph of panels, and a graph does not know how large its faces are.

That is the same distinction a cut sheet’s ligament and its piece count turned on, from the other side. There the length fell smoothly to nothing while the count of pieces sat at one and then jumped to six; here the area falls smoothly to nothing while the obstruction does not move at all. A length is not a count, and neither of them is a parity.

The hole is a vertex with half its conditions #

There is a way of reading the discontinuity that makes it ordinary rather than startling, and it uses only vocabulary the subject already has.

Draw a small closed path just outside the hole’s rim. It crosses every crease that reaches the rim, once each, and it crosses nothing else — a crease that does not touch the hole is either wholly outside the path or crossed twice, entering and leaving. So the crossing count is simply the number of creases meeting the hole, and the colouring fails when that number is odd.

Which is exactly the condition an interior vertex is under. A vertex of odd degree has no colouring, no folded state, and no way to be argued with; a hole with an odd number of creases at it has the same three properties for the same reason. The hole behaves as a vertex whose degree is the number of creases reaching it.

It behaves as a vertex for the parity and for nothing else, and that is where the discontinuity comes from. A vertex carries four conditions. Developability and Kawasaki are statements about the sectors round it — the angles of paper between consecutive creases, summing to a full turn. A hole has no such sectors: between two consecutive creases at the rim there is paper, but it does not close round a point, and the angles at the rim sum to whatever the rim’s shape makes them sum to. Maekawa is a statement about counts at a point and there is no point.

So the hole inherits the one condition that is about a walk and none of the three that are about a place. Parity survives having the middle removed because a loop can still be drawn; the angle conditions do not, because there is nothing left for them to be conditions on.

Which is why zero is the only special size #

Now shrink the hole and watch which statements change.

The parity does not change at any size, including sizes far too small to see, because the loop can always be drawn and the creases reaching the rim are the same creases throughout. That is the measurement: 1.8654 at a third of a sheet and 1.8654 at a hundredth.

The angle conditions appear only in the limit, and they appear all at once. At hole size exactly zero the rim has collapsed to a point, the paper between consecutive creases closes round it, and developability and Kawasaki become meaningful statements — which a three-crease vertex then fails, on grounds of odd degree, which is the parity again wearing a different hat.

So the whole discontinuity is the arrival of three conditions at a single value of a parameter, and the one condition that was there all along is the one that decides the case. Nothing jumps. Three statements that were undefined become defined, and the statement that was defined throughout never moved.

That also explains why the repair is what it is. Adding a fourth crease to the hole fixes the sheet, and adding a crease anywhere else does not, because only creases at the rim are in the count. On a pattern with a thousand creases and one hole, the whole question is settled by however many of them reach the rim — which is a number a reader can take off the drawing without checking anything else at all.

What this does to the founding rule #

One sheet, no cuts is the rule this whole collection is built on, and it has always been justified by the exchange rate a wedge buys: cutting gives access to angle deficits, folding does not, and admitting cuts admits a whole different subject.

The measurement above adds a second, sharper reason. Cuts do not merely buy an extra quantity; some of them change what the theorems are about. Every vertex theorem in the subject is proved for a sheet on which any closed path can be shrunk away — which is every uncut sheet, and every sheet with slits in it, and not every sheet with a hole in it.

That is a stronger statement than “cuts are outside the rule”. It says the rule is not merely a matter of taste about materials: it is the hypothesis several of the results depend on, and dropping it costs theorems rather than costing purity.

What a maker does with it #

Kirigami is a real practice and a real branch of engineering, and none of this is a warning against it. A cut sheet gets its behaviour out of the ligaments between its cuts, and an array of square holes buys a Poisson’s ratio of exactly minus one — both of them things an uncut sheet cannot do, both of them measured, both of them useful.

What a maker gets from the measurement is a rule of thumb with an unusual shape. Ask of any cut: does it leave the sheet in one piece and leave every loop shrinkable? A slit does. A hole does not, however small. And the flat-folding conditions — which are how anybody checks a pattern before committing paper to it — are only trustworthy on the first kind.

Why nobody had reason to notice #

Cut-and-fold work is old and there is a great deal of it, so it is worth asking why this has not come up.

The answer is that the cuts people make are not the cuts that do it. A traditional cut-and-fold decoration — the paper-cut stars that were made for centuries before anybody proved anything about them — cuts through the folded stack, which unfolds to a pattern of cuts reaching the sheet’s edge or meeting one another, and those leave the sheet as a disc or in several pieces. Engineered kirigami uses slits, which have ends. Neither practice has any reason to put an isolated hole in the middle of a sheet and then try to fold it flat.

And the mathematical literature has no reason either, because its subject is the uncut square. The gap is between two practices that each have a good reason to stay out of it, which is the usual shape of a thing that stays unnoticed for a long time.

Two cuts, one table #

The whole of the finding fits in a comparison, and it is worth making it explicit.

A wedge removes an angle at a point. The sheet loses exactly that angle; the effect is proportional to the cut; halving the cut halves the effect; a cut of nothing does nothing. Everything downstream — the cone the paper closes into, the curvature concentrated at the tip — scales with it.

A hole removes a disc of paper. The sheet loses nothing at any point: every angle everywhere is what it was. What it loses is the property that every closed path can be shrunk away, and that loss is the same for every hole of every size. A cut of nothing does nothing; a cut of anything does all of it.

There is no third behaviour in between, and there is no version of the hole that behaves like the wedge. That is the practical content of calling one of them metric and the other combinatorial: quantities can be traded and parities cannot.

Where the model stops #

One hole, three creases, one sheet. The measurement is on a family of patterns designed to make the point cleanly, not a survey. What a hole does to an arbitrary pattern depends on how many creases reach it, and the answer is the parity of that number.

The shape of the hole appears nowhere. Square holes were used because they are easy to draw and check. Nothing in the argument uses the shape, and a round hole or a ragged one would behave identically.

The topological reading is named and not developed. That a loop round a hole cannot be shrunk is a statement about loops up to deformation, which is a piece of mathematics with a large literature and no part of this subject. Everything argued above is settled by counting creases and colouring panels.

Nothing here says an even hole folds. Four creases to a hole gives an even cycle and a colouring, and a colouring is necessary and not sufficient as always.

One further thing is worth recording, because it is the kind of consequence that only shows up when a claim is turned around. If a hole of any size does this, then so does a hole that a maker did not intend — a punched filing hole, a tear that has been repaired badly, a rivet. None of those is a folding decision and all of them change what the sheet can be asked to do, by an amount that does not scale with how careless the damage was.

Where the ladder goes next #

The natural next question is quantitative in a different direction: given a sheet with a hole and a pattern that does not fold, how much has to change to make it fold? Adding one crease to the hole changes the parity and fixes it, which is a cheap answer; moving an existing crease does not. What the cheapest repair is, in some measure a maker would care about, is a question with an answer and no figure here.

The other direction is two holes, and it is the one that turns the parity into arithmetic: with two independent ways round, there are two parities, they can be set independently, and a sheet can be made to fail in one direction and succeed in the other.