A notch is not a hole

A hole is cheap paper: a flap standing against the rim of one claims half its disc rather than all of it, exactly as a flap against the sheet’s own edge does, so at equal area a holed sheet holds more flap than a solid one. That measurement has been on this site since it was made, and until now the machinery that made it refused a second question by name.

The refusal reads: a hole touching the edge of the sheet is a bite out of the outline rather than a hole, and it is a different object with a different boundary. It is a different object. This rung asks how different, in the only currency the design method has.

The comparison, held fair #

Two sheets, both 1 by 1 with a 0.28 by 0.28 rectangle taken out. In the first the rectangle is in the middle; in the second it opens onto the bottom edge. Both have exactly 0.9216 of paper, and both are compared against a plain square of that area — which is the fair comparison, since comparing either 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 wins”.

The price is the paper a flap claims. In the circle-packing method a flap of length r claims every point of paper within r of its tip, so a tip in the open middle of a large sheet claims a whole disc and a tip in a corner claims a quarter of one. Integrate that fraction over every point of the sheet and the result is what the paper is worth for flaps of that length — low is good, because a low number means the sheet is mostly places where a flap is cheap.

The answer, at three flap lengths #

For short flaps of 0.06 of a sheet width, the hole saves 1.91% against the plain square and the notch saves 1.11%. At 0.15, it is 4.71% against 2.58%. At 0.30 — flaps nearly a third of the sheet — 9.84% against 4.73%.

The hole is worth between 1.7 and 2.1 times the notch, at every length measured, for exactly the same paper removed.

There is a second reading in the same numbers. The share of the sheet where a flap is cheap at all — where it claims less than a full disc — goes from 47% on the plain square to 80% with the hole and 59% with the notch, at a flap length of 0.15. The hole does not merely save more; it makes most of the sheet cheap, and the notch improves a corner of it.

Why one side is missing #

The mechanism is a count of sides, and it is the whole of the difference.

A hole is a rectangle of rim with paper on all four sides of it. Every one of those four sides can have a flap standing against it, claiming half a disc, and the four sides face four different regions of the sheet, so the cheapness is distributed round the whole hole.

A notch is a rectangle of rim with paper on three sides. Its fourth side lies along the sheet’s own edge, where the paper was already ending — so that side is not new rim at all. It gives a flap nothing it did not already have, because a flap standing there was already claiming half a disc from the sheet’s edge.

Three sides against four looks like a ratio of 0.75 and the measured ratio is nearer 0.5, and the discrepancy is not a subtlety about overlapping discounts. It is that the fourth side does not merely fail to be new — it removes rim that was already there.

The count of sides, done as a signed sum #

The flap price on a sheet depends on the outline only through its total rim length: the saving against a solid sheet of the same area is, for short flaps, two-thirds of the flap length times the extra perimeter, divided by $\pi$ times the area.

So count the perimeter both ways, on a unit square with a 0.28 square removed.

The hole adds four sides of rim and takes none away: perimeter 4 plus 1.12, so +1.12.

The notch adds three sides and deletes the fourth from the sheet’s own bottom edge: perimeter 4 plus 0.84 minus 0.28, so +0.56.

Exactly half. Not three quarters — $(3-1)/4$, because the fourth side is not a side the notch fails to add but a side it subtracts.

Which is the measured ratio #

Put those into the saving formula against a plain square of the same area, whose perimeter is 3.8404.

The hole’s excess is $5.12 - 3.84 = 1.28$ and the notch’s is $4.56 - 3.84 = 0.72$, so the predicted ratio of savings is 1.78.

Measured, it is 1.72 at a flap length of 0.06 and 1.83 at 0.15. The formula is right to within a few per cent at both, and it drifts at the longest flap — 2.08 measured against 1.78 — for the reason the perimeter approximation always drifts: a flap of 0.30 sees both sides of a 0.28 feature at once, and rim counted twice is rim counted wrongly.

So the whole difference between a notch and a hole is a signed count of sides, and no argument about corridors or overlapping cheapness is needed to produce it.

That also predicts the depth result before it is measured. A bite 0.12 wide and 0.8 deep adds $2(0.8) + 0.12 = 1.72$ of rim and deletes 0.12, for an excess of 1.60 — larger than the compact hole’s 1.28, which is exactly why the deep narrow bite outperforms it. And a shallow bite 0.8 by 0.12 adds $2(0.12) + 0.8 = 1.04$ and deletes 0.8, for +0.24: a fifth of the hole’s, and the measured saving of 1.38 against the hole’s 4.71 is a ratio of 3.4 where the perimeters predict 5.3.

The rule of thumb the essay arrives at — cut in, not along — is therefore the perimeter rule wearing a designer’s clothes. Cutting inward adds two long sides and deletes a short one; cutting along adds two short sides and deletes a long one.

The bridge that makes a hole a hole #

Between the two objects there is a continuum, and walking it says which of the two properties above is doing the work.

Slide the rectangle from the middle of the sheet towards the bottom edge. While it is strictly inside, it is a hole, and the strip of paper between it and the edge — the bridge — gets narrower. The flap price changes slowly at first: a flap standing on a wide bridge claims most of a disc, one standing on a narrow bridge claims very little, and the bridge is a small part of the sheet.

Then the bridge reaches zero, and two things happen at once. The paper in the bridge is gone, which is a change of nothing measurable because there was nearly nothing there. And the object stops being a hole: its fourth side stops being rim with paper on both sides and becomes part of the outline.

The flap price is continuous through that moment and the count of sides is not. That is the honest shape of the difference: the paper does not notice the bridge going, and the boundary does. It is the same distinction a cut that reaches the edge turns on from the other side of the subject, where what changes is not a price but whether the sheet is still a disc.

Shape beats size #

If a notch’s value is rim with paper on both sides of it, then a notch’s shape should matter more than its area — and it does, by a factor larger than anything the area contributes.

Four bites, all of exactly the same area, taken out of the bottom edge of a square, priced at a flap length of 0.15:

• shallow, 0.8 wide and 0.12 deep: saves 1.38%.
• wide, 0.4 by 0.24: 2.11%.
• tall, 0.24 by 0.4: 3.79%.
• deep, 0.12 wide and 0.8 deep: 6.31%.

A factor of 4.6 between the first and the last, with the same paper removed in each. The deep narrow bite is worth more than the hole was at the same flap length, and the reason is that its two long sides run most of the way into the sheet with paper on both of them, which is precisely what a hole’s four sides are.

So a deep notch is a hole that has been opened at one end, and it keeps almost all of the value. A shallow one is a piece of edge that has been moved, and it keeps almost none.

Reading the four bites #

The ordering of those four is worth one more sentence each, because they are four different design decisions and not four sizes of the same one.

Shallow, running most of the way across the sheet and barely into it, is a sheet that has had a strip taken off one edge — which is nearly the same as being a slightly smaller rectangle, and it prices like one. Its saving of 1.38% is almost entirely the two short ends it added.

Wide and tall are the same rectangle turned through a right angle, and the difference between 2.11% and 3.79% is the whole content of the claim about depth: turning it costs nothing and nearly doubles what it is worth.

Deep, a slit reaching four fifths of the way across the sheet, is worth 6.31% — more than the compact hole managed at the same flap length. Its two long sides are a corridor of rim with paper on both sides, which is a hole’s geometry with one end opened, and the opened end costs almost nothing because it was at the sheet’s edge where a flap was cheap already.

A designer reading that sequence has a rule of thumb rather than a formula: cut in, not along. The paper removed is the same and the value differs by a factor of four and a half.

What a constructor sees #

There is a second currency, and the notch does worse in it too, by an amount that can be counted exactly rather than integrated.

Before any fold is made, a sheet offers its own outline: lines to align to and corners to bring together. A plain square offers four lines and four points. A square with a hole offers eight and eight, and the hole’s four lines are each parallel to one of the outer edges at a distance the hole’s position decides, which is what makes them produce new reference points rather than repeat old ones.

A square with a notch offers eight points and seven lines. The fourth side of the notch lies along the sheet’s own edge, so it is a line the folder already had — the same missing side, arriving in a completely different measurement.

That is worth noticing as a pattern rather than as a coincidence. The flap price and the reference count are computed by different machinery, one by integration and one by counting alignments, and both of them find the notch short by exactly the side it spends on the outline.

Where a notch is the right choice anyway #

The measurements say a hole is worth more per unit of paper removed. They do not say a hole is what to cut, and three considerations point the other way.

A hole has to be cut from inside. Making one means starting a cut in the middle of a sheet, which needs a blade rather than scissors and is awkward at any scale. A notch is cut from the edge inward, which is the cut a pair of scissors makes.

A hole weakens what surrounds it. The bridge between a hole and the sheet’s edge is a strip carrying whatever the sheet carries, and in a manufactured folded structure that strip is where the tearing starts. A notch has no bridge.

And a notch can be deep. The measurement above says a deep narrow bite is worth more than a compact hole, and depth is available to a notch in a way it is not to a hole — a hole 0.8 deep in a sheet of side 1 leaves bridges of a tenth at each end, and those bridges are the weakest paper on the sheet.

So the honest summary for a designer is that a hole is the better rate and a notch is often the better object, and the rate is what this measurement supplies.

The same currency, three features #

This is the third feature of a sheet’s outline this collection has priced with the same integral, and putting the three together says what the currency is actually measuring.

A corner is worth four times the middle, because a flap in a corner claims a quarter of its disc against a whole one in the open. A hole is cheap paper, by 1.9% at short flaps and 9.8% at long ones, because its rim is edge with paper on both sides. A notch is worth about half of that, because one of its sides is not new.

In every case the quantity is how much rim the sheet has, and how much paper each piece of rim has beside it. A corner is two edges meeting, so it is rim twice over. A hole is rim with paper on both sides, so its rim counts double. A notch is rim with paper on both sides for three of its four sides.

That is a single sentence covering three measurements made a long way apart, and it also says what would be worth measuring next: a sheet whose outline is a comb, or a spiral, is nothing but rim, and the flap price should approach some floor that this argument does not predict.

What a packing search does with it #

The price above is what a flap claims at a point. What a designer wants is what a whole set of flaps can do, and that is a search rather than an integral — but the search’s answer moves the same way.

Packing several flaps onto a holed sheet is not the disc-packing problem it is on a solid square, because two discs may overlap where everything they share is hole: what they share is not paper, so nobody is claiming it. That is the extra freedom a hole buys, and it is worth more than the price above suggests, because it changes the rule rather than the rate.

A notch buys the same freedom. Two flap tips either side of a deep bite may stand closer than twice their length, for exactly the same reason — the lens between their discs is off the paper. So the search’s structure is identical and only the geometry differs, which is why the notch is worth a fraction of the hole rather than nothing at all.

What is being measured, and what is not #

The price is a claim, not a packing. What is integrated is what a flap of a given length claims at each point, which is the quantity the disc model is built on. It is not a packing search: finding the best arrangement of n flaps on a holed sheet is a separate and much harder problem, and its answers are stochastic where these are deterministic.

The bites are rectangles. A rectangle is what the machinery represents and a real bite might be any shape. The two quantities the measurement is sensitive to are how much rim the bite adds and how far into the sheet it reaches, so a shape with more perimeter for its area — a slit, a V — would do better on both, and by how much is not measured here.

And the sheet is square. Everything above compares against plain squares of equal area, so the numbers are savings against a square. On a long rectangle the sheet’s own edges are already doing most of what a notch does, and the notch would be worth correspondingly less — which is the same overlapping-discount effect as within the notch itself, at the scale of the whole sheet.