Closer than a crease is wide

Every axiom names things that must already be on the paper — a point to fold onto a point, a line to bring to a line — so what a folder can build is bounded by what they can refer to, and that set is finite at every depth. Nine references after one fold, several hundred after two, and every one of them computable in advance.

That is a real bound and it is not the binding one.

The three rounds #

One fold. Nine marks: the four corners, the four edge midpoints and the centre. The closest pair is seventy-five millimetres apart on a 150 mm sheet, which is half the sheet. Every mark is unmistakable, and a folder working at this depth has no precision problem at all.

Two folds. Five hundred and sixty-five marks. The closest pair is now 0.52 mm apart. The median mark’s nearest neighbour is 2.7 mm away, and 5.7 per cent of the marks have a neighbour within a millimetre.

Three folds. Here the enumeration has to be restricted: the full operation set specifies over two hundred and seventy thousand distinct folds from the second round’s marks, and their closure will not fit in memory. So the third round is computed with the point-onto-point axiom alone — one of the seven — and it reaches 553,823 marks.

The closest pair is at the limit of double arithmetic. The tenth percentile mark has a neighbour 0.016 mm away. The median has one 0.058 mm away.

What a crease is wide #

A crease in ordinary paper is not a line. It goes round a small arc, the arc uses more paper than the stack advances by, and the width of the mark it leaves is a fraction of a millimetre — a quarter of a millimetre is a reasonable figure for copier paper folded with a bone folder, and a great deal more for anything thicker or folded by hand.

Two marks a fifth of a millimetre apart are, to a folder, one mark. Not hard to distinguish: the same mark. The crease made at one of them is the crease made at the other, and any construction that depends on the difference has failed before it has begun.

At three folds, 94.4 per cent of the reachable marks are in that condition. At three tenths of a millimetre it is 98.5 per cent, and at half a millimetre 99.8.

There is an exact way of saying it. Five hundred and sixty-five points in a unit square have a mean spacing of about one over the square root of the count — around four per cent of a side, or six millimetres — and the observed median of 2.7 mm is smaller because the points are not uniformly spread. Half a million points have a mean spacing of about a tenth of a millimetre, and the observed median of 0.058 is again smaller for the same reason. So the crowding is not an artefact of any particular axiom or any particular construction; it is what a large set of points in a small square is.

The two bounds part company #

So there are two bounds on what a folder can construct and they are of completely different kinds.

The reachability bound is combinatorial: a mark exists or it does not, the set is finite, and it grows very fast — nine, then five hundred and sixty-five, then half a million with one axiom of seven. That is the bound the axioms are usually discussed under, and it is the bound that makes the subject algebraic.

The resolution bound is metric: two marks are usable only if they are further apart than the tool that makes them. It grows the other way — as the reachable set fills the square, the typical gap falls as one over the square root of the count, which is what any set of points in a square does.

Those two cross somewhere between the second fold and the third, and after they have crossed the reachable set is a mathematical object rather than a set of instructions.

Which round a folder actually works in #

Real constructions do not push into the crowded region, and this says why they cannot.

Haga’s construction uses two folds and lands on marks that are well separated. Fujimoto’s method for thirds is an iteration rather than a construction precisely because a third is not reachable in a small number of folds at all, and its error halves at every step until it is under the crease width — which is a convergence argument and not a reachability one. The systematic route to an nth takes n folds and is worse than the short one everywhere, and the reason it is tolerable is that n is small.

Which theorem was checked, and how #

The reachable sets come from the closure routine this site keeps for the purpose, which builds them round by round: every fold the given axioms specify from the current points and lines, every new point where those lines cross, deduplicated exactly. The counts at one and two folds — nine and 565 — are the ones this site has quoted before and are reproduced here rather than recalled.

The spacings are computed by a grid rather than by comparing every pair, because half a million points is a hundred and fifty billion pairs. Every point is bucketed by position and compared only against the nine buckets around it, with a cell size chosen so that a nearer neighbour cannot be outside them.

The third round’s restriction is stated rather than hidden, and it matters in the right direction: using one axiom instead of seven produces fewer marks than a folder can actually reach, so the crowding measured is an understatement. With the full set the marks would be denser and the share within a crease width higher.

Where the model stops #

The measurement is of nearest-neighbour distance, which is the right statistic for “can these two marks be told apart” and the wrong one for “can this particular construction be executed”. A construction uses a handful of specific marks, and whether those are separated is a question about them rather than about the distribution. A folder who knows which marks they need may be working in a sparse corner of a crowded set.

The crease width used is a single number for a single paper. Thin tissue creases much finer than a quarter of a millimetre and thick card much coarser, so the round at which the two bounds cross is a property of the material as much as of the axioms — which is the idealisation this essay is really about.

And nothing here says anything about accumulated error. Each fold is assumed exact and the marks are placed where the arithmetic says; a real folder’s second fold is placed against a first that was already a little wrong, and error compounds in a way this account does not model at all. That makes the practical situation worse than the one described.

There is a fourth limit worth naming because a reader will suspect it. Half a million points includes a great many that arise twice — the same point reached by two different fold sequences — and the deduplication uses a tolerance. Two marks a millionth of a sheet apart are counted as one; two a thousandth apart are counted as two. Where that line is drawn changes the count, and it changes the tail of the spacing distribution, which is why the closest pair is reported as being at the limit of the arithmetic rather than as a number. The median and the deciles are far from the tolerance and are unaffected.

How many marks a sheet holds #

The two bounds cross somewhere between the second fold and the third, and where is computable rather than merely observable.

A sheet of side $s$ whose marks must be at least $w$ apart holds about $(s/w)^2$ of them. That is the whole of it: fill a square with points no closer than $w$ and the count is the area divided by the area each point claims. On a hundred and fifty millimetre sheet with a quarter-millimetre crease it is six hundred squared, which is three hundred and sixty thousand distinguishable marks, and it is the sheet’s capacity in the plainest sense — the number of different things a folder can point at on it.

Set the rounds against that. Nine marks after one fold is six hundred times under capacity. Five hundred and sixty-five after two is still six hundred times under. Five hundred and fifty-three thousand after three, computed with one axiom of seven, is over it — not by much, half as much again, but over. The crossing does not fall neatly between two rounds; it falls inside the third, which is why that round’s marks are usable at their sparse edge and unusable in the middle.

Why a bigger sheet does not rescue it #

Capacity grows with the square of the sheet, and that invites the obvious remedy. It does not work, and the reason has two halves that are worth separating because only one of them is arithmetic.

The arithmetic half is discouraging on its own. Each round multiplies the mark count by something like a thousand — nine to five hundred and sixty-five is sixty-three times, and five hundred and sixty-five to half a million is nine hundred and eighty. Absorbing a factor of a thousand needs the side multiplied by its square root, which is about thirty-two. So buying one more fold’s worth of room on a hundred-and-fifty-millimetre sheet means folding a sheet four and three quarter metres across, and the fold after that would want a hundred and fifty.

The measured half is worse. A sheet two hundred and ten millimetres across has a capacity of seven hundred thousand marks, comfortably more than the third round’s half a million — so by the uniform reckoning above it should have room to spare. The census says 87.4 per cent of its marks still have a neighbour within a fifth of a millimetre.

The gap between those two statements is the whole of what the capacity bound leaves out. The reachable set is not spread evenly over the square; it clusters, heavily, around the marks the early rounds produced, because every new mark is built from old ones and the operations that build them keep landing near where they started. So the effective capacity is more than an order of magnitude below the geometric one, and no sheet anybody could fold closes the gap.

That is the sharpest version of the essay’s finding. It is not that the third round is slightly beyond the paper. It is that the reachable set outgrows any sheet faster than any sheet can be enlarged, and it does so by the third fold on the paper people use.

What the picture cannot show #

The third round cannot be drawn. Half a million points in a square is a grey square, and any figure of it is a figure of the ink rather than of the set. The picture that would be wanted — a magnified corner showing the marks resolving into a crowd — is a picture of a region chosen for being interesting, which is a different kind of evidence from a census.

Nor can a figure show a crease’s width honestly at the same scale as the sheet. A quarter of a millimetre on a hundred and fifty is one part in six hundred; drawn at that ratio it is invisible, and drawn visibly it is a lie about the proportion.

The generalisation #

The result generalises past folding, and stating it in the general form makes it obvious what has been measured.

A constructive theory says which objects exist: which numbers a compass reaches, which points a fold reaches, which programs a machine computes. Constructive theories are combinatorial, their sets are closed under operations, and their interesting questions are about membership. An executable theory says which of those objects can be produced by an agent with a real tool: a pencil with a width, a fold with a radius, a machine with a clock.

The two coincide for small objects and part company for large ones, and where they part is a property of the tool rather than of the theory. In folding the tool is a crease, its width is a fraction of a millimetre, and the parting happens at the third fold — which is an extraordinarily shallow depth for a theory that reaches cubics and trisects angles.

That is worth holding beside what a fold reaches algebraically. The field of origami numbers is closed under six operations and is infinitely deep; the marks a folder can distinguish on one sheet of paper number a few hundred. The algebra is a statement about the closure, and the closure is not what anybody can hold in their hands.

There is a question this raises about the site’s own habits, and it is worth answering rather than leaving implied. Every construction figure here is drawn in exact arithmetic, at coordinates computed to fifteen decimal places, and printed at a stated millimetre size. The marks in those figures are separated by amounts a folder can work with — they were chosen from the sparse part of the reachable set, like every construction in the tradition — but nothing in the drawing tells a reader which marks would survive the printer, the paper and the hand.

The honest position is that a printable construction is a claim about geometry with an unstated claim about execution attached, and the second claim gets easier to break the deeper the construction goes.

Who found it, and when #

The reference-point closure is Robert Lang’s, from ReferenceFinder, which computes short fold sequences to a target point and has been the practical answer to this problem since the late 1990s. That program is built around the same tension: it searches the reachable set and it ranks candidates by error, because a mark that is reachable in six folds and a mark that is reachable in three and is within a tenth of a millimetre of it are, for a folder, the same mark and the second is better.

So the observation is not new to the practice. What is new here is the census — the whole distribution rather than one target’s neighbourhood — and the finding that at the third round the typical mark is unusable rather than the exceptional one.

One consequence is worth stating for anybody reading a table of constructions. A published sequence of folds to some proportion is a claim about arithmetic, and a claim about arithmetic in this subject is checkable exactly: the marks are rational or algebraic, they can be computed in exact arithmetic, and a construction either lands on the number or does not. What it is not is a claim about paper. Two published constructions to the same number, one in three folds and one in five, may be equally exact and wildly unequal in how well they can be executed, and nothing in the exactness distinguishes them.

Where the ladder goes next #

The obvious continuation is the third round with the full operation set, which needs the closure to be computed without being stored — a streaming enumeration that reports the spacing distribution without ever holding the points. That is a piece of engineering rather than a piece of mathematics, and it would sharpen the numbers rather than change them.

The other direction is the one that matters for practice. Given a target point and a paper, the useful question is not which folds reach it but which short fold sequences land within a crease width of it — and how the number of such sequences grows is a question about the density of the reachable set near a point, which is exactly what the distribution above is a coarse summary of.