A cut is a licence

A fold moves paper about and cannot change how much of it surrounds a point; a cut can, and that difference is the whole of what the one-sheet rule is worth. Take a wedge out and the sheet closes into a cone. Let a wedge in and it has more paper at that point than a plane does.

That is the metric account and it is right. It is also not what most cuts do. A slit — a cut that removes nothing, whose two sides can be brought back together — leaves the angle at every point exactly as it was, and it buys a great deal.

What it buys can be counted, and the unit is not area. It is vertices.

What a slit does to a vertex #

Every condition this subject has is stated at a vertex with paper all the way round it — Kawasaki and Maekawa at a point, and the lemma about the smallest sector beside them. At a vertex where the paper stops — where the sheet’s edge passes through — none of the four applies at all: the sectors do not close, there is no cycle to alternate round, and the folded cross-section never comes back for Maekawa to count.

A slit puts an edge of paper where there was none. The vertices at its ends are now vertices the sheet’s boundary reaches, so they stop being interior vertices and stop being subject to anything.

The end of the slit is the interesting case and it has its own shape. The paper goes all the way round the tip of a cut, so the fan there spans a full turn rather than a straight angle — and it is still a fan rather than a ring, because walking round it starts on one side of the cut and finishes on the other. So a slit tip has two π of paper at it and satisfies no vertex theorem, which is the sharpest statement of what a cut is for.

The arithmetic, on one pattern #

A square twist has twelve creases and four interior vertices. The conditions admit 256 of its 4,096 letterings — one in sixteen.

Cut one crease and count again. There are two kinds of crease to cut and they give two answers.

Cutting a pleat crease — one that ran from a vertex out to the sheet’s edge — releases one vertex and doubles the share. Cutting one of the four sides of the central square releases two and quadruples it.

A factor of two per vertex released, exactly, on all twelve creases of the pattern.

Why a factor of two #

The factor is not a coincidence of this pattern and it is not deep either.

An interior vertex of degree four admits, of the sixteen letterings of its four creases, the ones that satisfy Maekawa and the lemma — and at a generic degree-four vertex that is four of sixteen, a quarter. Release the vertex and its creases become unconstrained, so the four become sixteen.

But the creases are shared. Cutting one crease also removes that crease from the count of things the pattern has to letter — the cut crease is no longer a crease — so the space being counted halves at the same time as the admitted set grows. What comes out of those two movements is the doubling above, and the reason it is exact rather than approximate is that the released vertex’s conditions were exactly a factor on the count.

The useful way to state it is as a ledger. Every interior vertex costs the pattern a factor in the number of letterings it admits, and a cut refunds it.

The three rows are one formula #

The table’s three rows are usually read as a before-and-after, and they are tighter than that: all three are the same expression evaluated at three points.

Write $c$ for the pattern’s crease count and $v$ for its interior vertices. Then the admitted letterings number $2^{c-v}$, on every row.

Uncut, the square twist has twelve creases and four interior vertices: $2^{8}$, which is 256. Cut a pleat and it has eleven creases and three vertices: $2^{8}$ again, which is why that row’s admitted count is unchanged at 256 while its share doubles. Cut a buried crease and it has eleven creases and two vertices: $2^{9}$, which is 512.

So the two movements the essay describes — the space halving and the admitted set growing — are one subtraction, and which of them dominates is decided by how many vertices the cut releases. A cut that releases one vertex leaves the admitted count exactly where it was and only improves the share; a cut that releases two doubles the count as well.

That is worth having because the two quantities answer different questions. A folder wants the count: how many letterings are there to choose among. A checker wants the share: how much of the space it has to sift. Cutting a pleat helps the second and not the first, and nothing in the table says so until the formula is written down.

The share is the vertex count and nothing else #

Divide by the total and the crease count cancels. The share a pattern admits is $2^{-v}$: one in sixteen at four interior vertices, one in eight at three, one in four at two.

The share does not depend on how many creases the pattern has. A twist with twelve creases and a tessellation with a hundred admit the same share if they have the same number of interior vertices, and a cut changes the share by exactly the factor of two per vertex it releases — no more, and not less either.

Which turns the ledger’s instruction into an inventory a designer can take off a drawing before cutting anything. The pattern’s constraint budget is its interior-vertex count; each cut spends against it by however many vertices its ends release; and the arithmetic never consults the creases at all.

It also makes the two-cut prediction exact rather than a plausible extension. Two cuts through four distinct vertices take $v$ from four to nought, so the share goes from one in sixteen to one in one — every lettering of the remaining ten creases is admitted, because there is no interior vertex left to admit anything. Two cuts meeting at a shared vertex release three, take $v$ to one, and leave a share of a half. And a second cut through a vertex the first already released takes $v$ nowhere, so it multiplies the share by one: the ledger’s warning that such a cut buys nothing is not an intuition about waste but the same formula returning a factor of $2^{0}$.

What this says about kirigami #

Cut sheets are usually explained by what the cuts let the material do: the sheet stretches, twists, opens into a lattice with a Poisson’s ratio of exactly minus one, takes a curvature it could not take before. Those are mechanical descriptions and they are true.

The count above is a different explanation of the same thing, and it is prior to the mechanics. A cut sheet has fewer constraints than an uncut one, and how many fewer is a property of the cut pattern rather than of the material. A slit array that turns half the interior vertices into slit tips has released half the pattern’s conditions, and everything the sheet can then do is happening inside a space that got larger for a reason that can be counted before any material is chosen.

That also explains why a cut that removes no paper is not a small version of one that does. Removing a wedge changes the angle at a point, which is a metric change and is graded — twice the wedge is twice the effect. Releasing a vertex from four conditions is not graded at all: the conditions are there or they are not.

Two cuts, and where the ledger stops predicting #

One cut is a clean experiment because it releases at most two vertices and the count is small. Two cuts are where the arithmetic could break, and it is worth being explicit about why it might.

Cutting two creases that share no vertex releases four vertices, and the ledger predicts a factor of sixteen. Cutting two creases that meet releases three — the shared vertex is released once, not twice — and predicts eight. So the prediction is a statement about the set of released vertices rather than about the number of cuts, and two cuts can buy anything from four to sixteen depending on where they are.

That is a design statement rather than a caveat. A designer cutting to free a pattern should cut at distinct vertices, and cutting twice through the same vertex is a second cut that buys nothing at all — the vertex was already released by the first, and every condition it carried was already refunded.

The measurement here stops at one cut because the enumeration does: the square twist has twelve creases, so a single cut leaves eleven and two thousand letterings to sift, and two cuts leave ten and a thousand. Both are exhaustible. What is not exhaustible is a pattern large enough for the question to be interesting, which is the same ceiling every counting argument on this site runs into.

Which theorem was checked, and how #

Every crease is cut in turn and the whole pattern re-counted. The comparison is over all twelve of the square twist’s creases rather than over a chosen one, and the two kinds fall out of the measurement rather than being defined in advance.

The counts are exhaustive. Each is an enumeration of every lettering of the pattern’s remaining creases put past every condition at every remaining interior vertex; nothing is sampled.

A slit tip is handled rather than refused. The fan at the end of a cut spans a full turn, and the machinery that decides a boundary fan takes the span from the sheet’s own outline rather than assuming a straight angle. A vertex whose creases lie on both sides of the paper’s edge is refused instead, because that is a drawing that has run off the sheet rather than a fan.

The uncut pattern is the control, counted the same way by the same code.

Where the model stops #

One pattern, one cut. The arithmetic is measured on a square twist with a single crease cut. Two cuts release up to four vertices and the ledger predicts a factor of sixteen; the enumeration at that size is still exhaustive but the pattern has few enough creases that the sample of cases is small.

A lettering is not a folding. What is counted is the letterings the conditions admit, which is necessary and not sufficient for the sheet to fold. A cut that quadruples the admitted set has not been shown to quadruple anything a folder can make, and the global question remains the intractable one.

Cuts here are along creases. A slit that runs somewhere else — across a panel, at an angle to everything — creates new vertices as well as releasing old ones, and the ledger says nothing about the new ones.

And a released vertex is not unconstrained. It is subject to the strip condition, which is exact and which is weaker than the four; the arithmetic above counts a vertex as free because on this pattern its fans admit every lettering, and a fan with a small sector between two large ones would not.

What the picture cannot show #

The bars in the first figure are shares of a count, and a share of a count has no appearance. What a reader would like to see is the new letterings — the ones a cut makes available — and there are 256 of them on the pattern above, which is a list rather than a picture.

The slit itself is almost invisible in a crease pattern. A cut removes no paper, so it is drawn as a line like every other line, and the difference between a cut and a crease in a printed pattern is a convention rather than a shape. That is a real hazard for a reader with a printer, and it is why every printable pattern here is a fold pattern with no cuts in it at all.

The idealisation, named #

The slit has no width and its two sides can be brought back together exactly, so the sheet’s area and every sector angle are unchanged. Real scissors take a fraction of a millimetre and a real slit has two edges that fray, which is a small effect at the sizes anybody folds and a decisive one in the applications kirigami is used for — a cut lattice in a thin metal foil has kerf, and the kerf is a design parameter.

The other idealisation is that a cut is instantaneous and free. In a manufactured sheet a cut is a process step, and the number of cuts is a cost; the ledger above prices cuts in released conditions and says nothing about what they cost to make.

The generalisation #

Freedom in a constrained system can be counted in constraints removed, and the count is often exact. One cut short of falling apart measures the material end of the same trade, where what is being spent is the ligament between one cut and the next. The mechanical story about cut sheets — they stretch, they open, they buckle in new ways — is a description of consequences. The prior fact is that a sheet with a cut in it is subject to fewer conditions than a sheet without, and how many fewer is arithmetic on the pattern.

The surprising part is that the arithmetic is per vertex rather than per cut. A cut is a line and its effect is felt at its two ends, so a long cut through the middle of a panel releases nothing at all while a short cut through a vertex releases everything that vertex was carrying. That is the opposite of the intuition that a bigger cut does more, and it is the same shape as the finding that a crease’s two ends decide whether it can be re-lettered: in a pattern, what matters about a line is where it stops.

The rule a designer can use #

Stated as an instruction, the ledger is three lines.

Count the interior vertices, not the cuts. A pattern’s constraint budget is one factor per interior vertex; cutting is how a designer spends against it, and a cut that releases no vertex has spent nothing.

Cut through vertices rather than across panels. A slit whose two ends land in the middle of panels creates two new boundary vertices where there were none and releases nothing, so it costs a manufacturing step and buys the arithmetic nothing at all.

Prefer the buried creases. A crease with an interior vertex at each end is worth two vertices and a crease reaching the paper’s edge is worth one, so the same single cut buys twice as much when it is placed in the middle of the pattern — which is also where a folder is least likely to think of putting it.

Where the ladder goes next #

The immediate continuation is the slit array, where the ledger becomes a design tool rather than an observation: a lattice of cuts releases a countable number of conditions, and the count is available before the sheet is made.

The harder one is the direction this rung does not go. A released vertex is subject to the strip condition rather than to nothing, and a pattern whose released vertices are constrained by it is a pattern whose cut bought less than the ledger says. Measuring that gap means asking, of every fan a cut produces, whether it admits every lettering — which is a question this collection can now answer and has not asked at scale.