A hole is cheap paper

The corner is worth four times the middle established the rule this essay is a consequence of. A flap consumes every point of paper within its own length of its tip — that is what makes a circle the right object — but only the paper that is actually there. On an edge of the sheet the flap’s disc is half on the paper, so the flap costs half. In a corner it is a quarter on the paper, so the flap costs a quarter.

The sentence that carries the result is the boundary is the cheapest paper on the sheet, and it has an obvious consequence that nobody has drawn: a sheet with a hole in it has more boundary.

A hole is not a metaphor and it is not a cut in the sense kirigami means. It is a piece of paper that is not there, of a stated size, in a stated place, and it does exactly two things. It removes paper, which is a loss and is what everybody sees. And it adds edge, which is a gain, and which one is larger is arithmetic rather than intuition.

The comparison that makes it a question #

Comparing a holed square with the square it was cut from compares two sheets with different amounts of paper, and the answer would be “the one with more paper holds more”, which is not a finding.

So the comparison here holds the area fixed. A unit square with a hole 0.28 across in the middle has an area of 0.9216, and it is compared with a solid square of side 0.9601, which has exactly that area. Same paper, arranged two ways, and the question is whether the arrangement matters.

That choice is the whole design of the measurement, and it is the reason the answer is not obvious in advance. The holed sheet’s paper is spread over a larger footprint, which is a disadvantage for anything wanting long flaps in the middle; and it is nearer a boundary on average, which is the advantage being measured.

A boundary is a boundary #

The first thing to check is that the hole’s edge behaves like the sheet’s edge, because if it does not, nothing else follows.

The claim at a point is computed by integration on an equal-area polar grid about that point: how much of the disc of radius r lands on paper. The three places the earlier rung named come back exactly.

The first three are the arithmetic anybody would do by hand, recovered by a method that knows nothing about halves and quarters. The fourth is the finding stated as flatly as it can be: a flap against the edge of a hole claims a half, exactly as one against the edge of the sheet does.

The fifth is the one that is worth a moment. The outside corner of a square hole is a three-quarter corner rather than a quarter one — the paper wraps round it the other way — so it is dearer than open paper is cheap, at 0.75 rather than 1.00. A hole has the cheap kind of boundary along its sides and the expensive kind at its four corners, and a round hole would have neither.

And the whole sheet is cheaper #

The per-place numbers are suggestive and the per-sheet number is the result. Average the claim over every point of the sheet, at several flap lengths:

Lower is better, since the number is the share of a flap’s disc that has to be paid for. The holed sheet wins at every flap length, and the margin grows: 1.9 per cent at the shortest flap and 9.8 per cent at the longest.

The direction of that trend is the mechanism showing itself. A short flap only notices a boundary if it is standing right against one, so a hole that is far away from most of the sheet buys almost nothing. A long flap notices boundaries from further off, and a hole in the middle is within reach of much more of the sheet than the sheet’s own rim is.

There is a second column worth reading, and it says the same thing more starkly. At a flap of 0.22, 99.8 per cent of the holed sheet is cheap paper — a point where a flap claims less than a whole disc — against 70.2 per cent of the solid sheet. By 0.30 the holed sheet is cheap everywhere and the solid one is cheap over five-sixths of itself.

The margin is perimeter, and it can be written down #

The table’s trend — 1.9 per cent at the shortest flap, 9.8 per cent at the longest — is not merely a direction. The absolute gap between the two columns is 0.0181, 0.0290, 0.0411, 0.0566 and 0.0737, which is very nearly a straight line through the origin, and a straight line asks to be explained.

It can be, in closed form, for flaps short enough that the boundary looks straight. A point at distance $d$ from a straight edge loses the circular segment beyond it, and integrating that loss across the strip of width $L$ along the boundary gives $\tfrac{2}{3}L^3$ per unit of edge. Dividing by the sheet’s area and by the disc’s own $\pi L^2$:

$$\text{mean claim} \;\approx\; 1 - \frac{2\,P\,L}{3\pi A}$$

with $P$ the total boundary length and $A$ the area. The only property of the sheet that appears is its perimeter.

Test it. The solid comparison square has side 0.9601, so $P = 3.8404$ and $A = 0.9216$; the formula gives 0.9469 at $L = 0.06$ against a measured 0.9482, and 0.9116 against 0.9136 at $L = 0.10$. The holed sheet has the unit square’s perimeter of 4 plus the hole’s 1.12, so $P = 5.12$ at the same area; the formula gives 0.9293 against 0.9301, and 0.8821 against 0.8846.

Which says exactly what the hole is worth #

Subtracting one from the other, everything cancels but the perimeters:

$$\Delta \;\approx\; \frac{2L}{3\pi A}\big(P_{\text{holed}} - P_{\text{solid}}\big) \;=\; 0.295\,L$$

and the measured gaps at the three shortest flaps are 0.0181, 0.0290 and 0.0411 against a predicted 0.0177, 0.0295 and 0.0442. A hole 28 per cent of a side buys 1.28 units of extra edge at no cost in area — a third more perimeter than the square it replaces — and the price advantage is that number times the flap length, divided by three-π-halves of the area.

At the two longest flaps the law overshoots, 0.0648 against 0.0566 and 0.0884 against 0.0737, and the overshoot is the approximation failing rather than the measurement. A flap of 0.30 on a hole 0.28 across sees both sides of the hole at once, and edge counted twice is edge counted wrongly. That is also where corners start to matter, and the hole’s corners are the expensive kind.

What the formula makes of the design question #

Reading the result as an objective rather than a measurement changes what the question is, and the change is not comfortable.

If the paper price depends on the sheet only through $P/A$, then a designer wanting cheap paper wants maximum perimeter per unit area, which is the isoperimetric problem run backwards — and run backwards it has no answer. Two holes beat one, four beat two, and a sheet perforated finely enough has as much perimeter as anybody likes.

So the binding constraint is not the paper price at all. It is that $L$ has to be small against the spacing of the holes for any of this to hold, and that a sheet cut to lace stops being a sheet: the flaps still have to be joined to each other, and a boundary a flap cannot reach across is a boundary that has divided the model rather than cheapened it.

The right reading is therefore that the hole is worth its perimeter, up to the flap length, and no further. That is a modest claim and it is the one the arithmetic supports.

What the hole does not buy #

Three things it does not do, and being clear about them is what keeps the result from being read as a design recommendation it is not.

It does not buy area. The comparison is at equal area precisely because the hole has taken paper away, and the taking is real. A designer who cuts a hole in a square is left with less paper and has to be paid back in cheapness for it. That the payment more than covers the debt is the finding; that there is a debt is not in question.

It does not buy a flap in the hole. Every point inside the hole is not paper, and no flap stands there. The map above shows it as a blank, and the average is taken over the paper only.

And it does not respect the site’s founding rule. A sheet with a hole in it is not one square uncut. It is a legitimate object and a great many real designs use one — a box wrapper, a mask, a hinge plate — but it is a different starting object, and every number in this essay is about that object rather than about a square.

Where to put it #

The hole’s position is a free parameter and it moves the answer, so it is worth saying which way.

A hole in the middle is the best case for this effect, because the middle is the part of a square furthest from every existing boundary and therefore the part with the least cheap paper. A hole near an edge is partly wasted: the paper it makes cheap was already cheap.

A hole in a corner is not a hole at all — it is a bite out of the outline — and the measurement refuses it rather than pricing it. That is a real distinction and not a technicality: a bite changes the sheet’s shape, which is the question the first rung of this anchor asked and answered with a very spiky curve, and mixing the two would answer neither.

The hole’s size trades the two effects against each other directly. A larger hole is more boundary and less paper, and since the comparison is at equal area, a larger hole means a larger outer square as well. The measurement above uses a hole 28 per cent of the side, which is roughly the largest that leaves the sheet feeling like a sheet.

The shape of the cheap region #

The map is worth reading rather than summarising, because the two sheets fail differently.

A solid square’s expensive region is a single blob in the middle, and it shrinks from the outside in as the flap grows. Its worst point is the centre, and the centre is a single point, so a design needing one long flap can put it there and pay full price once.

A holed square has no expensive region in the middle, because there is no middle. What it has is four expensive patches, one in each quadrant, each of them between the hole’s corner and the sheet’s corner. They are smaller than the solid sheet’s single blob and there are four of them, and the worst point in each is nearer to full price than to half.

So the two sheets do not merely differ in average — they differ in the shape of what is left. A design wanting one very long flap does better on the solid sheet, which has a single deep place to put it. A design wanting several does better on the holed one, which has four shallower places and a great deal of half-price paper between them.

How the integration is done, and what it could get wrong #

The area of a disc intersected with a square has a closed form; the area of a disc intersected with a square that has a square hole in it does not have one worth writing. So the claim at every point is integrated numerically, and the design of the integration is where a measurement like this can quietly lie.

The grid is polar about the flap’s own tip, with equal-area rings: the k-th sample radius is r times the square root of (k+½)/rings, so every sample stands for the same amount of paper and no weighting is needed. That puts the samples where the answer varies — near the boundary the disc crosses — rather than spreading them evenly over a bounding box, which is what a Cartesian grid would do.

The solid sheet is integrated the same way. That is not a detail. A hand-derived quarter for the corner and a sampled number for the hole would differ by the method as much as by the geometry, and the comparison would be between two conventions. Every number in this essay comes out of one routine.

And the routine is checked against three answers it was not given. A flap in the open middle claims 1.0000, one against an edge 0.5000, one in a corner 0.2500 — to four figures, on a grid that has never heard of a half. A hole far smaller than a flap is priced at 0.9986, which is very nearly nothing, and is what a hole two thousandths of a side across is worth.

The cheapest point on the sheet #

One more number falls out of the map and it is worth having, because it is the extreme case a designer would reach for.

The cheapest place on a solid square is a corner, where a flap claims a quarter of its disc. The cheapest place on the holed square is also a corner of the outer square, at the same quarter — a hole does not produce anything cheaper than a corner, because a corner is where two boundaries meet at a right angle and the hole’s own corners meet the wrong way round.

So the hole does not lower the floor. What it does is raise how much of the sheet is near the floor: on the solid sheet at a flap of 0.22 there are four quarter-price points and a large expensive middle, and on the holed sheet there are the same four quarter-price points and no expensive middle at all.

That is the difference between a cheaper place and cheaper paper, and only the second is what a design spends.

Why it has not been noticed #

Two reasons, and they are both about the tools rather than about the geometry.

The circle-packing method is stated for a convex sheet, and every implementation of it assumes one. A sheet with a hole is not convex, the packing search has no way to express “this region is not paper”, and the constraint that a circle lie inside the sheet becomes several constraints rather than one. None of that is hard; it is simply not what the software does.

And the design tradition starts from a square because origami paper is sold square. A designer holding a square does not ask what a holed square would do, in the same way that nobody asked what a rectangle would do until the reference measurements were repeated on one and the square turned out to be the worst sheet in the subject for locating points.

The same hole, a second time #

There is a second thing a hole does to a sheet and it belongs to a different field, so it is only named here.

A folder’s reference points come from the sheet’s own edges and corners — a fold has to be specified by aligning something with something — and a hole is four more edges and four more corners. So the same object that costs a designer paper hands a constructor references, and the count is not close: one round of alignments on a square reaches nine points and on a square with a hole reaches 212.

The two results are independent. Nothing about the paper price depends on where the folds can be specified, and nothing about the references depends on how much a flap claims. What they have in common is the object, and it is a nice demonstration that an edge is worth something in more than one currency.

What a designer could do with it #

The honest answer is: not much yet, and the reason is worth stating plainly.

What has been measured is the paper price of a hole, which is one input to a design and not a design. Turning it into a design needs a packing search that can handle a non-convex sheet, and a molecule construction that can fill a region with a hole in it — and the universal molecule fills any convex polygon and produces nothing at a reflex corner, which is exactly the corner a hole introduces.

So the result is a reason to build that machinery rather than a technique. What it establishes is that the machinery would be worth building: the price advantage is not marginal, it is between two and ten per cent of the whole sheet’s cost, and it grows with the flap length — which is to say it grows with how ambitious the design is.