The third fold cannot be listed

A reference point is a place on the paper a folder can put a finger on: a corner, a crossing of two creases, a point where a fold line meets an edge. Everything a construction aligns to is one, and the set of them grows with every fold made.

Counting that set is straightforward for the first two rounds. A bare square has four references. One fold reaches nine. Two folds reach five hundred and sixty-five, and the worst-covered spot on the paper is then about a twelfth of a sheet from the nearest of them.

Three folds cannot be counted at all, and the reason is worth setting out precisely, because the number where it stops is not large by any modern standard.

Where it stops #

A round of the closure does two things. It forms every fold the current points and lines specify — the fold through two points, the fold placing one point on another, the fold placing one line on another, the fold through a point perpendicular to a line — and then it intersects those folds with each other and with what was already there, keeping every crossing that lands on the paper.

From five hundred and sixty-five points and ninety-two lines, the first half of that produces three hundred and seventy-eight thousand six hundred and fourteen specified folds, of which two hundred and seventy-four thousand three hundred are distinct.

The second half is where it becomes impossible. Two hundred and seventy-four thousand lines have about thirty-seven billion pairs, and each pair needs an intersection computed and tested against the sheet. That is not a computation this collection will run, and it would produce a point set with no honest way to report it: many of those crossings sit a millionth of a sheet apart, which is not a distinction a folder can make.

So the closure is refused at the third round rather than approximated. The machinery has a cap, the cap fires, and the refusal names the number that tripped it.

Why the refusal is the right behaviour #

A cap that silently truncates would be worse than useless here, and it is worth saying why the collection insists on a refusal.

A truncated closure returns a point set. That point set looks exactly like a computed closure — same shape, same kind of numbers, same covering radius available from it — and every statement made about it would be a statement about an arbitrary subset chosen by whatever order the folds happened to be generated in. There would be no marker on the output saying this is not the answer.

The refusal is the marker. It stops the computation, names the depth, names the count, and forces whoever wanted the number to say what they will accept instead.

What can be said instead #

What replaces a closure is a sample with a direction, and the direction is what makes it worth having.

Take the two hundred and seventy-four thousand folds the third round specifies, shuffle them deterministically, and keep the first few hundred. Intersect those with each other and with the ninety-two folds already there, keep the crossings that land on the paper, and measure how far the worst-covered spot on the sheet now is from the nearest reference.

Fifty folds: three thousand four hundred and ninety-eight references, worst gap 0.076 of a sheet. A hundred: eight thousand and fifty-six references, 0.052. Two hundred: twenty-three thousand four hundred and eighty, 0.046. Four hundred: seventy-four thousand six hundred and ninety-four, 0.022. Eight hundred: two hundred and seventy thousand eight hundred and eighty-two, 0.014.

Every one of those is a lower bound on what the full round buys, and the direction is the point: leaving folds out can only leave the gap larger, never smaller. So the third fold brings the worst gap to at most 0.014 of a sheet is a statement that survives everything the sample does not contain.

How the sample is taken, and why it is honest #

A sample is only a bound if it is taken in a way that cannot be steered, so it is worth saying exactly how these are drawn.

The two hundred and seventy-four thousand distinct folds are generated in whatever order the axioms happen to produce them — which is not random at all: the folds through pairs of points come first, in the order the points were found, then the perpendicular bisectors, then the bisectors of pairs of lines, then the perpendiculars. Taking the first eight hundred of that list would sample the folds through the earliest-found points, which are the ones near the sheet’s corners, and would produce a badly skewed picture.

So the list is shuffled first, by a seeded stream, and the first k taken. The seed is fixed, so the measurement is the same measurement twice; the shuffle is uniform, so the sample is not concentrated in any region of the paper.

That is what makes each row a bound rather than an estimate. Nothing about which folds were kept is correlated with where the gaps are, so the covering radius measured is the covering radius of an arbitrary subset — and an arbitrary subset of a set can only cover the paper less well than the whole.

Three in a thousand #

The striking part is how little of the round is needed.

Eight hundred folds is three tenths of one per cent of the two hundred and seventy-four thousand available, and it already brings the covering radius from 0.089 to 0.014 — a factor of six and a half. The remaining ninety-nine point seven per cent of the round can only improve on that, and cannot improve on it by much: the gap is already smaller than a crease is wide on a sheet of ordinary size.

That says something about the shape of the reachable set which the count of it hides completely. The references are not spread evenly and they do not need to be. A few hundred well-scattered folds cover the paper; the other quarter of a million are filling in regions that are already covered, adding points a folder could not tell apart from ones already there.

The count and the reach have always come apart — two folds buy sixty-three times the points for four times the reach — and this is the same divergence continued past the point where the count can be written down. The count becomes meaningless first.

What each new reference is worth #

The sample gives six pairs of numbers — references against covering radius — and they are enough to price a reference, which is a thing the count alone never does.

Points spread as well as points can be spread over a unit square cover it to a radius of about $0.62/\sqrt{N}$. Against that ideal the measured sets do badly, and increasingly so: 565 references cover to 0.089 where 0.026 was available, a factor of 3.4; the 270,882 from eight hundred sampled folds cover to 0.014 where 0.0012 was available, a factor of 11.7.

So the references are not merely uneven — they get more uneven as there are more of them, which is the arithmetic behind the essay’s own remark that the extra folds fill in regions already covered.

Which fixes the exchange rate #

Fitting the six pairs gives the rate directly. The covering radius falls as

$$r \sim N^{-0.30}$$

against the $N^{-1/2}$ a uniform set would manage. The exponent is the whole story: halving the reach costs a tenfold increase in references, where a well-spread set would need only fourfold.

That is worth carrying because it converts the uncountable round into a bounded expectation. Going from the measured 0.014 of a sheet to 0.001 — about a fifth of a millimetre on a 150 mm square, which is where a crease’s own width ends the question — is a factor of fourteen in reach and therefore a factor of $14^{3.3} \approx 6{,}600$ in references: something near two billion of them.

The third round does not contain that many. The fourth does, comfortably, and cannot be listed either. So the honest statement is that the paper is covered to a folder’s own resolution somewhere in the fourth round, and that the exponent is what says so rather than any enumeration.

The arithmetic of the explosion #

It is worth writing the growth down, because the reason the third round is impossible is not that the numbers are large but that they are large in a particular way.

Each round forms folds from the points and lines it has. The fold through two points and the fold placing one point on another are both quadratic in the point count; the bisector of two lines is quadratic in the line count; the perpendicular is the product of the two. So the folds a round specifies grow roughly as the square of the points.

Then the crossings of those folds are quadratic in the folds. So the points a round produces grow as the square of the folds, which is the fourth power of the previous round’s points.

Four points, then nine, then five hundred and sixty-five. The next term in a fourth-power growth from five hundred and sixty-five is not a number anybody needs to write down, and the fold count — two hundred and seventy-four thousand — is only the square root of it.

That is the shape that makes an empirical approach hopeless rather than merely expensive. A quantity growing as a fourth power at each step is one where the third term is comfortable, the fourth is impossible, and there is no intermediate case to work on. The same shape appears wherever this subject counts configurations, and the response is always the same: stop counting and start bounding.

What a folder is actually doing at three folds #

There is a gap between this arithmetic and the practice it is supposed to describe, and it is worth naming rather than letting it sit.

No folder works with a reachable set. A folder makes a third fold to a particular reference, chosen because it is the one the construction calls for, and the other quarter of a million folds are never considered. The closure is a description of what is available, not of what anybody does.

Where the closure matters is in questions of the form can this be reached at all — which is the question behind what numbers folding constructs and behind every impossibility result in the subject. For those, the set matters and its size does not; what is needed is an argument about which numbers appear, not a list.

So the practical consequence of the third round being uncountable is narrow. It means the empirical approach to reachability — build the set, look at it, measure it — runs out at two folds, and everything past that has to be reasoned about rather than enumerated.

What the sheet’s shape does to all of this #

Everything above is measured on a square, and the square is the worst possible case — which makes the numbers a floor rather than a typical value.

The same axioms to the same depth on five proportions of the same area reach wildly different numbers of marks, and the square reaches the fewest by a wide margin, because its own symmetry makes folds that would have been distinct coincide. A rectangle of no particular proportion reaches a couple of orders of magnitude more references at two folds than a square does.

So a non-square sheet hits the enumeration wall earlier. Its second round already produces enough lines that the third would be worse than a quarter of a million folds; on some proportions the second round is itself uncomfortably large.

That is worth knowing for the practical reason that almost nobody folds a square. A4 paper is the common case, and its reachable set at two folds is far larger than a square’s and correspondingly further past the point where it could be listed.

The two limits, side by side #

This collection now has two independent reasons why the reachable set stops being a useful object, and they arrive at different depths.

The first is resolution. A reference a folder cannot distinguish from its neighbour is not a second reference, and the crossings at two folds are already dense enough that thinning them to the marks a folder could tell apart on a sheet of a hundred and fifty millimetres changes the counts substantially on some proportions.

The second is enumeration, which is this rung: the folds themselves become too numerous to list at the third round, before their crossings are even considered.

Both limits say the same thing about what a count of references is worth, from opposite directions. One says many of the points are not really distinct; the other says there are too many to write down. A quantity with both problems is a quantity to stop quoting, and the covering radius — which has neither — is what this collection reports instead.

What is actually being counted #

One clarification, because the word fold is doing two jobs in this essay and they differ by a factor of nearly a third.

The third round specifies three hundred and seventy-eight thousand six hundred and fourteen folds and contains two hundred and seventy-four thousand three hundred distinct ones. The difference is a hundred and four thousand constructions that name a crease some other construction already named.

That duplication rate — about twenty-seven per cent — is much lower than it is early on. At the first round from a bare square, the four axioms specify thirty-eight folds and reach twelve: sixty-eight per cent duplication. At the second, three hundred specified against ninety-two distinct.

So coincidence is a small-configuration phenomenon here, and it thins out as the paper fills. On a bare square the corners and midlines are so symmetric that most constructions land on each other; by the third round the points are irregular enough that two constructions agreeing is a genuine accident. That is the same effect which makes the square the poorest sheet to fold on, seen at a different depth, and it means the count of distinct folds is close to the count of constructions once the configuration is large — which is bad news for anyone hoping the explosion would be tamed by duplication.

What the axioms contribute, at this depth #

There is one more number the third round yields cheaply, and it settles a question the earlier rounds left ambiguous.

At the first round from a bare square, one of the four elementary axioms supplies every fold the others cannot and the other three supply nothing — and two rounds later that ranking has inverted. The natural next question is whether it keeps moving.

It does not, in the only sense the third round can report. The fold through two points and the perpendicular bisector between them are both quadratic in a point count that is now five hundred and sixty-five, so between them they specify the overwhelming majority of the round; the bisector of two lines is quadratic in ninety-two, and the perpendicular is the product of the two. The round is, by count, almost entirely the two point-based axioms.

That is a statement about arithmetic rather than about geometry, and it is worth separating from the earlier finding. Which axiom contributes folds nobody else could is a question about marginal value and it inverted between rounds. Which axiom contributes the most folds is a question about how many pairs of things there are, and once the points outnumber the lines by six to one the answer stops being interesting.

Where the ladder goes next #

Two directions, and the first is nearly free.

The sample can be run to convergence. Eight hundred folds gives 0.014; sixteen hundred would give something smaller, and the curve is clearly flattening. Finding where it stops improving would establish, empirically, what the third round is worth without ever enumerating it — and the answer would be a number this collection could quote with a bound attached.

The second direction is the one a mathematician would take first, and this collection has not: the covering radius after k folds ought to have an asymptotic form, and the measured values — 0.707, 0.354, 0.089, and at most 0.014 — fall fast enough to suggest one. Four points is not a curve, and the fourth is a bound rather than a value, so nothing is fitted here. What can be said is that each fold has so far bought between a factor of two and a factor of six, and that the factor is not obviously settling.