A hole is an edge

A fold needs something to align is the fact the whole construction half of this subject rests on. A folder cannot fold at a coordinate. Every fold has to be specified by bringing two things together — this point onto that point, this line onto that line, this crease through those two marks — and at the very start the only things that exist are the sheet’s own edges and corners.

A square offers four of each. The sheet decides which points exist measured what follows from that and found the square to be the worst proportion in the subject: at the same area one fold reaches nine marks on a square and twenty-nine on the A-series rectangle, and the reason is the square’s own symmetry making constructions coincide that a rectangle keeps apart.

That rung varied the proportion. This one varies the topology, and the effect is larger by an order of magnitude.

What a hole gives a folder #

A square with a square hole in it has eight edges and eight corners rather than four and four. That is the whole of the input, and everything below is a consequence of it.

The eight are not interchangeable. The hole’s four edges are each parallel to one of the outer edges, at a distance the hole’s position decides, and that distance is a length the folder did not have before. Two parallel lines a stated distance apart is the thing a fold placing one onto the other bisects, so the hole supplies four new bisectors immediately — and none of them coincides with anything the plain square had.

The hole’s four corners are points, and a point is what a fold through two points needs two of. Eight points give twenty-eight lines through pairs rather than the square’s six, and each of those crosses the others.

Nine against two hundred and twelve #

The measurement uses the linear alignments only — a fold through two points, and a fold placing one point onto another — and nothing else. That restriction is deliberate: the question is what the extra edges are worth, and including the bisector or the parabola-solving axioms would answer a different question with a much bigger number.

One round of those two alignments, from the outline and nothing else:

The nine is the number this site has published: the four corners, the four edge midpoints and the centre. It falls out of intersecting the folds the axioms specify and discarding the crossings that miss the paper, and it is recovered here by a routine written for the holed case — which is the check that the two columns are comparable.

The two hundred and twelve is the finding. Twenty-three times as many references, from the same amount of paper, after the same number of folds.

Which beats a second fold #

The comparison that makes the size of the effect legible is not against one round on a plain square but against two.

A plain square reaches nine references after one round and a hundred and thirty-three after two. The holed sheet reaches two hundred and twelve after one.

So cutting a hole is worth more than a whole extra fold. That is a strange thing to be able to say, and it is worth being careful about what it does and does not mean. It does not mean the holed sheet reaches the same points more cheaply — the two sets of references are different sets, and a construction wanting a specific point may find it in neither. What it means is that the closure grows with the boundary at least as fast as it grows with depth, and the boundary is a thing a pair of scissors can change in a second.

Symmetry costs here too #

The four holes measured give 212, 273, 352 and 485 references, and the ordering is not by size or by area removed. It is by symmetry.

The centred square hole is the most symmetric of the four and gives the fewest. A square hole moved off centre gives more. A small centred hole gives more still — its edges are further from the outer edges, so fewer of the bisectors it produces land on top of bisectors the square already had. And a rectangular hole placed off centre gives 485, more than twice the centred square’s, because it shares no symmetry with the sheet at all.

That is exactly the mechanism the proportion rung found: a symmetry makes constructions coincide, and a construction that coincides with another is a reference that was already there. The square is the worst outline in the subject for locating points because it has eight symmetries; a centred square hole in a square is the worst hole for the same reason, because it keeps all eight.

The design rule that comes out of it is uncomfortable and clear. If a folder wants references, they should want an ugly sheet.

A crossing in the hole is not a reference #

One rule does the discarding and it is the same rule the plain-square measurement has always used.

A fold line extends forever and a piece of paper does not, so an axiom can specify a fold whose crossing with another fold lies off the sheet. A folder cannot put a finger there, so it is not a reference. On a plain square that rule removes the crossings outside the outline; on a holed sheet it removes those and the ones inside the hole.

The second kind is a real cost. A great many of the lines the hole’s own corners produce cross each other in the middle of the hole, which is precisely where there is no paper, and every one of those is discarded. The 212 is what is left after that.

Which means the finding survives its own biggest objection. The obvious complaint about a hole — that it removes the very region where constructions land — is true, it is applied, and the count is still an order of magnitude larger.

What kind of numbers arrive #

The references a plain square reaches in one round are all halves. That is the earlier rung’s result and it is why the second round is where thirds first appear.

A hole changes that immediately. The hole’s edges sit at whatever fractions its position gives them, and a fold placing an outer edge onto a hole edge bisects the distance between them — so a hole at 0.36 of the side puts 0.18 and 0.68 into the reference set on the first fold, and nothing about those is dyadic.

So a hole is not merely more references, it is a different kind of reference, arriving a round earlier. Whether that is useful depends entirely on what a construction is trying to reach, and this essay does not claim it is generally useful. What it establishes is that the boundary is where the arithmetic comes from, and adding boundary adds arithmetic.

How the closure is computed #

The routine is short and the two things it has to get right are both about what counts as the same.

Two lines are the same line when their direction and their signed distance from the origin agree, so a fold specified twice by different pairs of points is one fold. Without that the count would be a count of specifications rather than of creases, and a symmetric sheet would produce enormous numbers for no reason.

Two points are the same point when their coordinates agree to nine figures. That is the tolerance the rest of this repository uses to decide two marks are one mark, and it is generous relative to the spacings involved.

Then: take every pair of points and add the fold through them; take every pair and add the fold that places one onto the other; intersect every pair of the resulting lines; keep the crossings that land on paper. The plain square’s nine comes out of that with nothing special-cased, which is the anchor the holed numbers hang from.

What the routine does not do is fold the paper. A reference is a point the folder can locate on the flat sheet by making a crease, and the closure is a statement about the flat sheet — the same convention every rung of this anchor uses.

What it costs #

Three costs, and the first is fatal to the obvious application.

The sheet is not one square, uncut. Every construction on a holed sheet is outside this site’s founding rule, and the results are about a different object. That is not a defect of the measurement — the same is true of everything in the kirigami anchor — but it does mean a folder wanting to construct a heptagon from a square cannot use any of it.

The paper in the hole is gone. The comparison holds area fixed, so the holed sheet is physically bigger and the construction lands on a larger square with a piece missing. A construction that needed the middle has nowhere to put it.

And nothing here says the extra references are good ones. They are points a folder can locate. Whether any of them is the point a construction wants is a different question, and crowding says that past a certain density the references stop being distinguishable at all — two points closer than a crease is wide are one point to a folder, and 212 references on a 150 mm sheet are considerably closer together than nine.

The other thing a hole is worth #

There is a second measurement on the same object in a different field, and the two are independent.

A hole is more boundary, and boundary is the cheapest paper on the sheet — a flap standing against the edge of a hole claims half its disc, exactly as one against the edge of the sheet does. Measured at equal area, a sheet with a hole is cheaper paper than a solid one at every flap length.

Nothing about the paper price depends on where folds can be specified, and nothing about the references depends on how much a flap claims. What the two share is the object, and it is a clean demonstration that an edge is worth something in more than one currency — and that the currencies are not exchangeable.

A hole is not a bite #

The measurement refuses a hole that touches the edge of the sheet, and the refusal is not a technicality.

A square notch cut out of the side of a sheet leaves the paper in one piece with a more complicated outline. That is a change of shape, which is what the proportion rung varies, and its effect on the closure is the effect of a different outline. A hole strictly inside the sheet is a change of topology: the paper is no longer simply connected, there is a boundary component that is not the outline, and the sheet has an inside edge.

Mixing the two would answer neither question. So the sheet builder throws on a hole that touches or leaves the edge, with the reason in the message, and every number in this essay is about a hole strictly inside.

The distinction also says where the effect comes from. It is not that the outline got longer — a sufficiently wiggly outline would be longer still and is a shape question. It is that there are now two boundary components at a stated separation, and a separation is a length the folder can bisect.

What the extra references are made of #

The 212 points are not a uniform scattering and it is worth saying what they are, because “more references” would otherwise be an unhelpfully bald statement.

They come in three groups. There are the originals — the nine a plain square reaches, which are still there. There are the points where a fold specified by the hole’s own features crosses a fold specified by the sheet’s, which is the largest group and is where the non-dyadic fractions come from. And there are the points where two hole-specified folds cross, which is the smallest group because so many of those crossings land inside the hole and are discarded.

The middle group is the one that matters, and its size is roughly the product of the two boundaries’ contributions rather than their sum. Four extra edges and four extra corners produce twenty-eight folds through pairs of points instead of six, and those cross each other and the originals — so the growth in references is quadratic in the number of lines, which is quadratic in the number of boundary features.

That is the arithmetic behind the factor of twenty-three, and it says the effect should keep growing with a second hole rather than saturating immediately.

One number does both sheets #

The account of where the extra references come from is a description of three groups, and there is a single relation underneath it that covers both sheets and every hole measured.

Take the distinct folds a round specifies and count the pairs of them. The plain square specifies eight folds, so twenty-eight pairs; the holed sheet thirty-six folds, so six hundred and thirty pairs. Divide each by three: 9.3 and 210, against the measured 9 and 212.

A third of the pairs of folds cross on paper, on both sheets. The same third on a square with four edges and on a square with eight — which is a considerably stronger regularity than anything the essay claims, and it means the entire twenty-three-fold gain is carried by the fold count.

Run it backwards on the other holes. Two hundred and seventy-three references implies about forty distinct folds; three hundred and fifty-two implies forty-six; four hundred and eighty-five implies fifty-four. So the four holes differ by specifying thirty-six, forty, forty-six and fifty-four folds respectively — and the ordering by symmetry the essay reports is an ordering by how many of the folds a hole’s features specify fail to coincide with one another.

Which turns the next measurement into a prediction #

That relation makes the two-hole question answerable in advance rather than only by running it.

Twelve boundary features give twelve points, and every pair of points specifies two folds — the line through them and their perpendicular bisector — so a hundred and thirty-two before coincidences. On a sheet with two holes placed asymmetrically, few of those coincide; call it a hundred distinct. Pairs of a hundred folds is four thousand nine hundred and fifty, and a third of that is about sixteen hundred references from a single round.

That is a definite figure with a stated derivation, and it is the kind of prediction the measurement it anticipates can refuse. If two holes come back at four hundred rather than sixteen hundred, the one-third rule has broken and the reason will be crowding — two thousand points on a hundred-and-fifty-millimetre sheet are eight millimetres apart on average, which is beginning to approach the region where two marks stop being two marks.

So the next measurement is not merely cheap. It is the one that decides whether the growth is the arithmetic of pairs or the paper running out of room, and the two answers differ by a factor of four at a size somebody could fold.

What would be worth measuring next #

Two things, and both are cheap.

The bisector. This measurement uses two alignments out of seven, so the numbers are a floor. Adding the fold that bisects an angle would put irrational references in on the first round, as it does on a plain square, and the interaction between a hole’s new angles and that axiom is not obvious in either direction.

More than one hole. Two holes give twelve edges and twelve corners, and the growth in the reference count against the number of holes is a curve nobody has drawn. Whether it saturates — and it must, since crowding puts a ceiling on how many distinguishable points a sheet of a given size can carry — is the question worth asking.