Spending the cheap paper

A hole is cheap paper priced a hole and stopped there. A flap claims every point of paper within its own length of its tip, and only the paper that is actually there — so a flap against the sheet’s edge claims half a disc and one at a corner a quarter. A hole is more edge, and at equal area a square with a hole in it came out cheaper than a solid square at every flap length measured, by 1.9 per cent at the shortest and 9.8 at the longest — the same reasoning that makes a corner worth four times the middle.

A price nobody can spend is a curiosity, and the essay said so: turning that into a design needs a packing search that can express “this region is not paper”, and this collection had none. It has one now, and building it produced a different answer from the one the price predicted.

The condition a hole changes #

In the circle-packing method, two flaps may not claim the same paper. On a solid sheet that reduces to a statement about distance: the discs must not overlap, so the tips are at least twice the flap length apart. That reduction is why a flap costs a circle and why the whole method is a disc-packing problem — and why the packing is the hard part of designing a base at all.

On a holed sheet the reduction fails. Two discs that overlap are still legal if everything they share is hole, because what they share is not paper and nobody is claiming it. The condition has to be written as it actually is:

for every pair of flaps, either the tips are at least 2r apart, or the paper their two discs share is empty.

The second clause is the whole difference, and it is not a discount. It does not make paper cheaper; it makes two flaps able to stand where they could not stand before.

The size of it, isolated #

The cleanest measurement puts two flaps at fixed points and varies only the paper between them.

Two tips six tenths of a sheet width apart. On solid paper each carries a flap of exactly 0.300 — half the gap, which is what two discs meeting halfway means. Put a slot between them and the answer climbs:

At a slot four tenths wide the flaps reach two thirds further than on solid paper. The tips have not moved and no paper has been added anywhere; the only change is that some of what used to be between them is not there.

That is a large effect and it is invisible to the price. The rim discount at these flap lengths is a few per cent. This is sixty-seven.

The gain is half the slot, exactly #

The table climbs and the climb has a formula, which is worth extracting because it turns six measurements into a rule that covers every slot and every separation.

Two tips a distance $d$ apart, discs of radius $r$ about each. On solid paper the discs may not overlap, so $r = d/2$ — the 0.300 in the first row. Put a slot of width $w$ between them and the discs may overlap, provided everything they share is hole. Along the line joining the tips they share the interval from $d - r$ to $r$, and that interval has to lie inside the slot. The slot is $w$ wide, so the shared interval may be $w$ long, which gives

$$r = \frac{d + w}{2}.$$

Each flap gains exactly half the slot’s width. Put the table’s numbers in with $d$ = 0.6: a tenth of a slot predicts 0.350 against a measured 0.357, two tenths predicts 0.400 against 0.404, four tenths predicts 0.500 against 0.500 exactly. Take the differences down the measured column and they average 0.050 per tenth of slot — a slope of one half, to two figures.

The small excesses are the sheet’s own edges doing what the hole does. A disc reaching the rim of the square is clipped there too, so the pair can afford slightly more overlap than the one-dimensional argument allows, and the effect is a per cent or two on a sheet this size.

Which gives the relative gain a formula too #

Divide through and the fractional improvement is $w/d$: the slot’s width as a share of the distance between the tips.

That reproduces the headline. A slot four tenths wide between tips six tenths apart is $0.4/0.6$, which is sixty-seven per cent — the number the essay quotes, arrived at without measuring anything.

And it explains why the whole-sheet search finds so little. A search spreads its tips across the paper, so for most pairs $d$ is a substantial fraction of the sheet while $w$ is the width of one hole; $w/d$ is small for nearly every pair, and zero for every pair whose discs do not cross the hole at all. Two per cent is what a handful of pairs with a favourable ratio contribute to an average over all of them.

It also says what a designer should do about it, and the advice is sharper than the essay’s own. Put the hole between the two tips that are closest together, not between the longest flaps — because the gain is the slot divided by the separation, and a small separation is what makes the ratio large. A pair of tips a tenth apart with a slot half that width between them gains fifty per cent; a pair half the sheet apart gains a few.

Why the overlap is not cheating #

The clause looks like a loophole and it is worth being clear that it is not one, because the whole rung rests on it.

A flap of length r is a point of the folded object that is r of paper away from the tip in every direction the paper goes. What it needs is a disc of paper, and the tree method’s circles are exactly that requirement drawn. Where the sheet stops, the requirement stops with it: a flap at a corner claims a quarter disc and is not thereby a worse flap. That is not an approximation of the method — it is the method, and it is why a corner is the best place on a square to put a long flap.

A hole is the same statement about a different piece of missing paper. The rim of a hole is a raw edge like any other, the disc is clipped by it like any other, and two discs clipped by the same hole may overlap in the region that is gone. Nothing about the flaps has been relaxed; the sheet is a different sheet.

The test that keeps it honest is that the two discs must share no paper at all, not that they must share little. Where the hole is narrower than the overlap, the pair is refused exactly as it would be on solid paper — which is what the first two rows of the table above are, and why the numbers climb rather than jumping.

What a search actually gets #

The isolated case is a construction. The design question is what a search can do with a whole sheet, and the answer is much smaller.

Across two to eight flaps the holed sheet is ahead six times out of seven, by between 0.7 and 4.2 per cent, and behind once by 0.27. The mean advantage is a little under two per cent.

Two per cent is not nothing and it is nothing like sixty-seven. The reason is that the overlap trick needs the hole to be between two flaps, and a search placing eight flaps on a square with one small hole in the middle mostly places them where the hole is irrelevant. The effect is available and it is available in a few places.

And the one negative row is the search, not the sheet. A search returns a lower bound: it found 0.3164 on the holed sheet and 0.3173 on the solid one, which says the holed sheet is worth at least 0.3164 and says nothing about what it is worth. Reporting that row as “a hole hurts at five flaps” would be reporting the optimiser.

Why the comparison is at equal area #

Comparing a holed square with the square it was cut from compares two different amounts of paper, and the answer would be “the one with more paper holds more”, which is not a finding. So the solid sheet here is the square of the same area — side √(1 − wh) rather than 1 — and the question is: given this much paper, is it better arranged with a hole in it or without?

That convention comes from the pricing rung and it is the only one that makes the question a design question. It also makes the holed sheet larger across than its rival, which is part of why it wins: a bigger outline gives the tips further to spread, and spreading is what a packing wants.

So the two per cent is a mixture — some of it the overlap, some of it the extra span — and this measurement does not separate them. The two-flap table separates them completely, which is why it is the one the claim rests on.

What the search is and is not #

It is simulated annealing from a grid start and from random starts, seeded, so a rebuild gets the same packing every time. The flap length for a given set of tips is found by bisection, because the pair test is monotone in the length and is not differentiable in anything.

It is a search and not a solver. For a solid square the best packing is known for a handful of flap counts and unknown for the rest; for a holed square nothing is known at all, and there is no published value anywhere to check against. Every number here is a lower bound, and the comparison drawn is between two searches with the same budget on the same day.

That is a weaker kind of statement than most of this site makes and it is stated as one. The two-flap table is not of that kind — it is a bisection on a configuration nobody chose, and its numbers are exact to the tolerance printed.

The three checks the module gets #

Two flaps on a plain square reach half the diagonal. They go to opposite corners, each claims a quarter disc, and the two quarters stay clear until the discs meet along the diagonal — so the answer is 0.7071 and not the disc-packing answer of a quarter of the square. A module that returned the second would have quietly adopted the wrong model of what a flap costs.

Two flaps either side of a wide hole reach further than the same two on solid paper. That is the claim the module exists for, and it is asserted rather than merely measured: if the overlap clause were dropped, the two answers would be equal and the assertion would fail.

And a flap whose tip is inside the hole is refused rather than priced. A tip that is not on paper is not a flap, and giving it a length would be pricing a flap that cannot exist.

Where the gain actually sits #

The two per cent is an average and it is worth asking which flaps earn it, because a design decision needs to know where to put the hole rather than whether to have one.

At two flaps the gain is 4.2 per cent and the reason is visible: with only two tips to place, the search puts them at opposite corners of the larger outline, and the whole advantage is the extra span. No overlap is involved at all — the two discs meet along the diagonal and the hole is nowhere near them.

At six, seven and eight flaps the gain falls to about one per cent, and the tips are spread over the sheet with the hole in the middle of them. Here the overlap clause does come into play, on the pairs whose discs cross the hole, and the extra span is worth less because the flaps are shorter.

So the two mechanisms trade places as the flap count rises: span at low counts, overlap at high ones, and neither dominates. That is a less tidy answer than either alone and it is the answer the search gives.

What it suggests for a design — and it is a suggestion rather than a result — is that a hole is worth putting between the flaps that are longest, since a long flap’s disc is the one most likely to reach across it. Nothing here tests that, because the search chooses the tips and is not told where the hole should go.

What is still owed #

The rung’s original description asked for two things and this delivers one.

The packing search is built. It expresses a region that is not paper, it finds the flap lengths, and it says what a hole is worth.

The molecule construction is not. Turning a packing into a crease pattern needs the region between the flaps filled, and the universal molecule fills any convex polygon and produces nothing at a reflex corner. A hole puts reflex corners into the region to be filled — four of them, for a square hole — so the step from a packing to a crease pattern is exactly the step that has no construction.

That is not a gap in this essay’s argument; it is the reason the argument stops where it does. What is measured is how much flap a holed sheet can carry, not how to fold one.

The rim price, still true #

Nothing here retires the earlier measurement, and it is worth saying which parts of it survive intact.

A flap against a hole’s rim still claims half a disc. That was computed by integrating over a polar grid about the tip and counting what is paper, and it comes back as ½ exactly on the rim, ½ on the sheet’s own edge, and ¼ at a corner. Those numbers are not affected by anything above.

The holed sheet is still cheaper by the square inch. At equal area, the average price of a flap over the whole sheet is lower with a hole than without, by 1.9 to 9.8 per cent depending on flap length, and that is a fact about where a flap can be put rather than about how many of them fit.

What has changed is what the discount predicts. A cheaper flap does not automatically mean a longer one, because a packing is a constraint problem and price is not one of its constraints — the constraint is that claims are disjoint. The discount says how much paper a flap consumes; it says nothing about whether another flap can stand nearby, and it is the second question a search answers.

Two measurements of the same object, agreeing about the sheet and disagreeing about the reason. That is a better position than either alone, and it is why the rung was worth writing rather than assuming.

What a folder should take from it #

A hole is worth more as a gap than as a discount. The rim price is a few per cent and the overlap across a slot is worth tens of them, and the two are different mechanisms with different geometry. The discount is about how much paper a flap consumes; the gap is about where another flap may stand.

Two flaps can share a hole and not share paper. That is the sentence the whole module exists to make true, and it is the one thing about a holed sheet that a solid sheet of any outline cannot do.

And a search that reports a lower bound should say so every time. Six of seven rows favour the hole; the seventh does not, and the seventh is the optimiser rather than the paper. That is the same discipline the square-versus-rectangle sweep needed, and for the same reason: a spiky curve made of search results is mostly a picture of the search.