What one cut buys

Fold a sheet of paper as elaborately as anybody likes and pick any point in it. Measure the paper that surrounds that point — walk right round it, adding up the angle. The answer is a full turn, and it was a full turn before the folding started.

Take a pair of scissors to the same sheet, cut out a wedge, and bring the two fresh edges together. Now the answer is not a full turn. It is short by exactly the wedge, and the sheet has no way to lie flat again.

That is the whole of this essay. One operation cannot change the angle at a point, and the other can, and everything a cut sheet does that a folded one cannot comes out of that one sentence.

Why the rule is being bent here #

This site keeps one sheet and no cuts, and that is not a matter of taste. It is a claim, and it earns its place because the theorems the subject is built out of are theorems about a sheet whose paper is all still there. Developability at a vertex asks whether the sectors sum to a full turn, which is only a question worth asking if the answer might be yes. Take the rule away and the four conditions the fold checker runs stop being conditions.

A rule of that kind can only be priced by removing it. Saying that folding preserves the angle at a point means very little on its own; saying that folding preserves it and cutting does not is a statement about the difference, and the difference is the thing worth knowing.

There is precedent, and it is worth naming rather than glossing. One straight cut is a theorem in which the cut is the output — every fold happens first, on an uncut sheet, and the scissors arrive at the end to release what the folding has already arranged. The oldest book in the subject cuts on nearly every page, because the tradition is older than the rule and never agreed to it. Both bend the rule sideways. This essay bends it head-on, exactly once, at a single point, and computes nothing that is not local to that point.

The turn of paper at a point #

The quantity is easy to state and easier to measure than it looks. At any point strictly inside a sheet, the creases through it divide the paper around it into sectors. Add the sector angles. On a flat sheet the answer is 360° because the paper is flat. On a folded sheet the answer is still 360°, because folding rearranges those sectors in space without altering any of them: a crease is a rigid motion of the paper either side of it, and a rigid motion does not change an angle.

That is worth checking rather than asserting, and it is the control the whole argument rests on.

Ninety-two interior vertices, drawn from every pattern the site has — the Miura, the Yoshimura, the waterbomb tessellation, the twist, the preliminary base — and every one returns a full turn to within about one part in ten thousand million million, which is arithmetic noise rather than a measurement. The sector angles are not read off a formula for what they ought to be; they are computed by the same routine the fold checker uses when it decides whether a pattern is legal at all.

The cut vertices never come near it. A wedge of thirty degrees taken out leaves 330°; a straight angle removed leaves 180°; a wedge of forty-five degrees let in leaves 405°. The gap in each case is the wedge, exactly, because there is nothing else it could be.

What a fold has no way to do #

The reason the folded column is so uninteresting — a perfectly straight line of ninety-two identical answers — is the reason it matters.

A flat sheet has no Gaussian curvature anywhere, an unstretched deformation cannot change it, and so nothing made by folding has any either. The theorem behind that is Gauss’s, it belongs to cartographic-projection.com in this fleet, and this site quotes it and does not re-derive it. What this site does own is the discrete form: at a vertex, where the surface is not smooth and the curvature is concentrated rather than distributed, the concentrated amount is the shortfall in the turn — and folding a whole sheet gives every vertex a shortfall of zero.

So the pattern library is not merely full of vertices that happen to hold a full turn. It is full of them because a pattern that did not hold one would have been refused, and it would have been refused because it could not have been cut from a single flat sheet. The rule and the measurement are the same rule.

How much cone a wedge buys #

Once the turn is short, the sheet has to do something about it, and what it does is entirely determined.

At slant distance r from the cut point, the paper that survives forms a circular arc of length equal to the turn that is left, multiplied by r. A cone of half-angle θ has a circle of circumference 2πr sin θ at that same distance. Setting the two equal gives the half-angle straight away: the sine of it is the fraction of the turn that is left. Nothing about the size of the sheet appears, and nothing about the shape of the cut region appears either.

Read the ends of that curve rather than the middle. A wedge of nothing leaves a half-angle of ninety degrees, which is the flat sheet the sheet already was — a cone whose surface is the plane. A wedge of very nearly a full turn leaves a needle: at 350° removed the half-angle is under 1.6°. And the wedge that is easiest to check by hand sits in the middle.

Half a turn removed leaves half a turn, and the half-angle whose sine is one half is thirty degrees exactly. That is a case anybody can verify with a protractor and a paper plate, and it is worth having in an essay whose other numbers all arrive from a computation.

The first degree of wedge is worth four #

The relation $\sin\theta = 1 - w/2\pi$ is stated by its ends, and its slope is where the useful content is.

Differentiating gives $d\theta/dw = -1/(2\pi\cos\theta)$, which blows up as $\theta$ approaches ninety degrees. Near the flat sheet the cone’s departure from flat goes as the square root of the wedge:

$$90° - \theta \approx \sqrt{w/\pi}.$$

Put a small number in. A wedge of one degree gives a cone 4.3° off flat — four times the amplification. A wedge of a hundredth of a degree still gives 0.43°, which is forty-three times.

At the other end the exchange rate collapses. At 350° removed the slope is $1/2\pi$: a further degree of wedge buys a sixth of a degree of cone. Across its range the relation runs from about 43× to 0.16×, and the steep end is the end everybody works at.

Which is why the gate cannot be lenient #

That settles something about the rule this essay exists to price.

There is no regime in which a sheet is nearly flat. The response at zero is infinitely steep, so any shortfall in the turn at a vertex — however small, whatever its cause — produces a cone disproportionately larger than itself, and the sheet has left the plane before the error is visible in the drawing.

Which is exactly why the developability check on every pattern here is an equality tested to arithmetic noise rather than a tolerance. A tolerance of a hundredth of a degree would be a tolerance of nearly half a degree of cone, and the ninety-two vertices sitting at a full turn to a part in $10^{13}$ are not evidence of a fussy implementation. They are what the square root demands: on this curve, being close to right is a different thing from being right, and the difference is a factor of forty.

The other sign, and the shape it does not choose #

Everything above assumed paper was taken away. Adding it is the other half of the story and behaves quite differently.

Insert a wedge — slit the sheet from the edge to a point and let a triangle of extra paper in — and the turn at that point is more than a full one. There is now too much paper for the plane to hold, and the sheet has to buckle out of it. The sheet ruffles.

What it does not do is close into anything. A cone is a surface of revolution with a shortfall at its apex; there is no surface of revolution with a surplus. The machinery says so by refusing: asked for a cone at a negative wedge it declines to produce one rather than returning an angle that would be meaningless. That refusal is a check as much as any assertion is, because the tempting mistake here is to take the same formula, feed it a fraction greater than one, and get a number back.

Which ruffle appears — two large waves or eight small ones, and where the crests sit — is a question this essay does not answer and cannot. The geometry fixes what the excess is and is completely indifferent about how it is arranged; that indifference has its own essay here, and choosing among the members of the family requires something this site does not do, which is a statement about how the material resists being bent. The word for that lives on another site in the fleet. Everything here is lengths and angles.

The connection to a leaf #

The surprising part of all this is not on the cutting side at all.

A leaf that grows more at its rim than at its middle ends up with more length round its edge than a flat disc of that width can hold, and it ruffles. A leaf that grows more at its centre domes. Neither leaf has been cut, and no leaf has a special point anywhere; the extra length is smeared continuously over the whole surface. And yet the sign of what happens is the same sign, arrived at by the same accounting — length that the plane cannot accommodate, and a sheet that has to leave the plane to accommodate it.

So a cut and a growth field are two ways of arranging the same quantity. One puts it all at a point and leaves every other point of the sheet exactly as flat as it was; the other spreads it evenly and leaves no point unremarkable at all.

That is a genuinely odd pairing to arrive at from a pair of scissors, and it is the reason the cut belongs on this site rather than being handed over whole. The scissors and the leaf are doing the same arithmetic.

Which theorem was checked, and how #

Three things are asserted here, and the third is the one that took the most care.

The cone relation is computed twice. The cut machinery reaches the half-angle through a deficit in radians; the check reaches it from the wedge in degrees, by an expression that shares no line of arithmetic with the first. The two must agree at every wedge sampled, and they agree to machine zero. Had the drawn curve been the source of its own check, a sign error would have produced a perfectly self-consistent picture of the wrong relation.

The cone is verified to be a cone. Every point of the drawn profile must lie on one straight line from the apex, to within a part in a million million. That is what makes the surface flat everywhere except at the one point — and it is the geometric content of the claim that no paper was stretched.

The control is the strong half. The ninety-two folded vertices are not a decoration on the figure. If any pattern in the library had returned anything other than a full turn, the essay’s opening sentence would be false, and the failure would have been in the sentence rather than in the arithmetic. The check demands that every folded vertex be within a part in a thousand million of a full turn and that no cut vertex be within a millionth of a degree of one, so that the two populations are being told apart by the quantity the figure is about rather than by the labels on the rows.

The refusals matter as much. A wedge past a full turn is not a wedge and is rejected; a request for a cone on a sheet with paper added is rejected; and an unsigned formula that would happily return an imaginary angle instead returns nothing at all.

What the picture cannot show, and what it assumes #

The hero figure shows a cone and it cannot show the thing the cone is evidence for. It cannot show that no paper was stretched, because the drawing has no access to the paper’s own distances — the assertion behind it does, and that is the division of labour between the picture and the check.

It also cannot show the other sign. There is no second panel with a saddle in it, because there is no surface of revolution with more than a full turn at a point and drawing an approximate one would be drawing something that does not exist. The absence is the finding.

The idealisations are the usual ones and one that is specific to cutting. The sheet has no thickness, so the two cut edges are lines rather than faces. The cut itself has no width: a real blade removes a kerf of its own, which is a small extra wedge nobody accounted for, and on a small enough disc it is not small. And the two edges are assumed to be joined perfectly, edge to edge with no overlap and no gap, which in practice means tape — and tape is a second material with a stiffness of its own, which is exactly the sort of thing this essay’s geometry cannot see.

Who noticed it, and when #

The angle at a point is old. It is the discrete form of curvature that shows up wherever polyhedra are studied, and cutting a wedge out of a disc to make a cone is the demonstration of it that every geometry teacher reaches for. What is more recent is the deliberate use of it as a design tool.

Cut paper as a craft is very old and largely unconcerned with any of this — the Chinese tradition of jianzhi and the Japanese kirie are pictorial, and the cut is a way of making an image rather than a shape. The 1797 book that opens the recreational tradition slits its sheet into a grid to make connected cranes, and that too is a cut for connectivity rather than for curvature.

The word kirigami as a name for cutting used as a geometric operation is a twentieth-century coinage, and the systematic treatment — cut patterns designed so that the shortfalls at the cut points add up to a target surface — belongs to the last twenty years. The mathematics it uses was available the whole time. What arrived late was the intention.

Where the ladder goes next #

The immediate rung above is the other thing a cut buys, and it buys it without removing any paper at all. A sheet slit into squares that stay joined at their corners opens in both directions at once, holds its ratio of the two exactly, and pays for the property in holes — the same behaviour a Miura-folded sheet reaches by folding, bought with different geometry.

Two directions lead away rather than up. One runs into biology, where growth changes the metric continuously and nothing is ever concentrated at a point; the cut is that subject’s limiting case, with all of the curvature at one place and none anywhere else. The other runs back into the sheet, where a fold genuinely carries no curvature even though it looks like the sharpest curvature paper could have, and where the third way out is to break the assumption that the sheet cannot stretch and buy about twenty degrees of sphere with it.

Two questions belong elsewhere and are worth naming so they are not looked for here. Adding the shortfalls over a whole region and relating the total to the shape of that region is cartographic-projection.com’s result and is used there for a purpose this site does not have. Working out how much angle a cap of a given size needs removing in total — the question a dressmaker asks when cutting a dart — is textile-structure.com’s, where it is a statement about cloth and about a pattern that has to be laid out flat before it is sewn. Neither is derived here, and the local statement at one point is the whole of what this site claims.