An axiom may name no fold

The Huzita–Hatori operations are the definition of what a single fold can construct, and they are stated the way geometry is usually stated: given two points, fold through them; given a point and a line, fold the point onto the line; and so on. Seven of them, and the list stops there for a reason.

Every one of those sentences is about a plane. A plane has no edges, every line in it is infinite, and a construction either exists or does not. A folder has a square, every line is a segment, and there are two further ways for an operation to fail: it can name a fold that does not exist, and it can name one whose crease is nowhere on the paper.

Three that always work #

Fold through two points, fold one point onto another, fold through a point perpendicular to a line: over four thousand random alignments each, all three name exactly one fold every time, and in every case the crease crosses the paper.

That is not luck and it is worth saying why, because the reason is what the other two lack. Each of those three is determined by objects that are on the sheet. The fold through two points is the line through them, and two points inside a square have a line between them that is inside the square. The fold placing one point on another is their perpendicular bisector, which passes through their midpoint — also inside the square. The perpendicular through a point passes through that point.

In every case the construction produces a line that is guaranteed to touch the paper, because it is guaranteed to pass through somewhere the paper is.

The one that names two, and misses #

Placing one line onto another names the bisectors of the angle they make — two of them, always, at right angles to each other, meeting where the two lines cross.

Where they cross is the difficulty. Two lines each specified by two points inside the square can meet far outside it, and when they do, one of their two bisectors runs off across the plane and never comes back. Counted: 13.3% of the folds this operation specifies are creases the paper never reaches.

The pair is not symmetric in this. The bisector that runs “between” the two lines in the direction of the paper is nearly always available; it is the other one, at right angles to it, that leaves. So a folder given this operation has, on average, 1.73 usable folds rather than two — and no way to know which without drawing the line.

That is the first of the two failures a plane cannot have: an operation that names a fold nobody can make, because the paper has run out.

The one that names none #

The fifth operation is the one that behaves differently, and this collection has said so before: it returns two folds, one, or nothing, depending on a distance, which is the reason a compass-and-straightedge argument does not transfer.

Now with a number on it. Over four thousand random alignments, placing a point onto a line through a second point names no fold at all 23.8% of the time, exactly one 0.0% of the time to within the resolution of the sweep, and two folds 76.2% of the time.

The mechanism is a circle meeting a line. The point being placed must land somewhere on the target line, and it must stay the same distance from the point the fold passes through — so its image lies on a circle, and the fold exists exactly when that circle reaches the line. Two crossings give two folds; a tangent gives one, which is a coincidence of measure zero and never appears in a random sweep; no crossing gives nothing.

So nearly a quarter of the alignments a folder might specify with this operation cannot be made. That is the second failure a plane does not have: an operation whose input is legal and whose output does not exist.

How far outside the paper the missing folds are #

A fold that misses the sheet misses it by an amount, and the distribution says whether this is a near thing or not.

The bisector that leaves does so because the two lines cross outside the square, and the further outside they cross, the further the offending bisector runs. Two lines each drawn through two uniform points in a unit square cross outside it more often than not — nearly parallel lines cross very far away indeed — so the misses are not clustered just past the edge. Most of them are gross: the crease is not a centimetre off the paper, it is several sheet widths off.

That matters for what a folder should do about it. A near miss could be rescued by a slightly larger sheet or by translating the construction; a gross one cannot. The operation has simply named a fold belonging to a different piece of paper.

It also explains the shape of the 13.3%. If the misses were near ones, the share would depend strongly on the sheet’s size relative to the points; because they are gross, it depends mostly on how often two random lines cross outside the square at all, which is a property of the configuration and not of the paper.

The test a folder can run first #

The fifth operation’s failure has a criterion, and it is two distances a folder can compare on the sheet before committing to anything.

The fold must carry the point $P$ onto the line $\ell$ while passing through the point $Q$. A fold through $Q$ is a reflection fixing $Q$, so it preserves the distance from $Q$: the image of $P$ lies on the circle centred at $Q$ of radius $|PQ|$.

That circle meets $\ell$ exactly when

$$\operatorname{dist}(Q,\ell) \le |PQ|$$

So the operation returns nothing precisely when the through-point is further from the target line than it is from the point being placed, and both quantities are lengths on the paper.

Which turns a failure into a measurement #

That is a considerably more useful statement than a percentage, because it can be checked before the fold rather than discovered during it.

A folder about to slide a corner along an edge can look at the distance from the pivot to that edge and at the distance from the pivot to the corner, and know at once whether the alignment is available. No sliding, no guessing, no discovering after thirty seconds that nothing lines up.

It also says how a near miss is repaired. The shortfall is $\operatorname{dist}(Q,\ell) - |PQ|$, and it can be closed from either end: move the pivot toward the line, or choose a point further from the pivot. Both are moves within the construction rather than changes to the sheet, which puts this failure in a different class from the third operation’s — where the fold exists and only a larger sheet reaches it.

So the three kinds of nothing separate further. The fifth’s failure is repairable by moving the inputs, the third’s by changing the paper, and the placement failure by neither.

And it explains the 0.0 per cent #

The tangent case falls out of the same inequality and confirms the sweep’s most suspicious entry.

Exactly one fold requires $\operatorname{dist}(Q,\ell) = |PQ|$ — an equality between two continuous quantities, which a random draw meets with probability zero. Four thousand trials returning 0.0 per cent is not a rounding of something small; it is the correct answer.

The operation returns two folds or none, and never one, except on a set of configurations of measure zero that a folder aiming by eye will nevertheless land on constantly — because a folder’s pivots are corners and their lines are edges, and a corner of a square is exactly as far from one edge as from the adjacent corner.

Three different kinds of nothing #

The counts separate three things that all look like the fold cannot be made and are not the same.

No solution. The circle misses the line; there is no fold with the required property anywhere in the plane. Nothing about the paper is involved and a larger sheet would not help.

A solution off the paper. The fold exists as a line in the plane and the crease does not touch the sheet. A larger sheet would help — which makes this the only one of the three that is about the paper’s size rather than about the construction.

A solution the folder cannot place. The crease exists and crosses the paper and its position is decided by references that are closer together than a crease is wide. This one is invisible to the counts here, because a sweep over exact coordinates never meets it, and it is the one that actually stops people.

Three failures, three different repairs, and only the middle one has anything to do with the sheet’s edges. The habit of describing all three as “the axiom fails” hides which repair applies.

The conditional ones are the powerful ones #

There is a pattern in which operations fail, and it is not a coincidence.

The three that never fail are linear: each of them produces a line determined by incidences among the points and lines given. Between them they reach only what a straightedge reaches, and the marks they produce on a square are all rational.

The two that can fail are quadratic and worse: placing a line on a line involves choosing between two bisectors, and placing a point on a line through a point solves a circle against a line. A quadratic has two roots or none, and that is the whole of why these operations are conditional — the same algebra that gives them their reach gives them their failures.

Push it one step further and the sixth operation solves a cubic, with one, two or three roots depending on the configuration. It is the operation that makes folding stronger than the compass, and it is the most conditional of all.

So the ranking by power and the ranking by reliability are the same list read backwards. An operation that always works is an operation that adds nothing new, and one that sometimes returns nothing is one that reaches numbers a compass cannot. That is a considerably better reason for the seven to behave as they do than “some of them have edge cases”.

Why this is not what an axiom is worth #

What each axiom is worth is a different measurement and the two are easy to run together. That one counts what the operations reach — which points become available, which numbers become constructible, how much of the closure each one contributes. This one counts whether the fold an operation names exists at all.

An operation can be enormously valuable and frequently empty. The fifth is exactly that: it is the one that makes the Beloch fold possible, which is the reason folding solves cubics and a compass cannot — and it comes back with nothing a quarter of the time. Value and availability are independent, and the list of seven says nothing about the second.

The seventh operation is conditional twice over #

The two operations not swept here are the two that take the most inputs, and the shape of their failures follows the same rule with one addition.

The sixth places two points onto two lines with a single fold. It solves a cubic, so it names one, two or three folds — never none, since a real cubic always has at least one root — and the folds it names can miss the paper for the same reason the third’s can. It is the operation that makes folding stronger than the compass, and it is the least reliable of the seven in the sense that a folder can least easily predict what it will return.

The seventh places a point onto a line with a fold perpendicular to a second line. It names one fold or none: the perpendicular direction is fixed by the second line, and whether a fold in that direction can carry the point onto the first line depends on whether the two lines are parallel. Parallel is a coincidence of measure zero, so the seventh almost always works — and the case where it does not is a case a folder meets constantly, because folders align to edges and a square’s edges are parallel in pairs.

That last point is the one general caution the counts here cannot supply. A random sweep never meets the degenerate cases and a folder meets nothing else: the alignments people actually make are between edges, creases and corners of a square, which are parallel, perpendicular and coincident far more often than any random model allows. The measure-zero cases are the working repertoire.

What a folder does about it #

The practical answers are old and were arrived at without any of these counts, which is the usual relationship between craft and measurement here.

Specify from things that are already on the paper. A folder’s alignments are corners onto creases, edges onto edges, existing marks onto existing marks — and all of those are the first three operations, the ones that never fail. The whole traditional repertoire is built out of the operations that always work.

Use the fifth deliberately and check. Placing a point onto a line through a point is what a folder does when they slide a corner along an edge until something lines up, and the sliding is the check: if nothing lines up, the fold does not exist, and the hand finds that out in a second.

And choose the sheet. Which references a sheet admits depends on its proportion, and the same is true of which folds an operation can name: a longer sheet catches more of the lines that a square lets escape. That is a design decision taken before any fold is made, and it is the one the counts here bear on most directly.

What this does to a search over constructions #

The counts have a consequence for anything that enumerates folds rather than making them, which is what the reference closure does: apply every operation to every pair of things on the sheet, collect the folds, intersect them, and repeat.

A quarter of the fifth operation’s applications produce nothing, so an enumeration that assumes one fold per alignment overcounts its own reach by a factor that depends on which operations it uses. The closure computed here has never used the fifth for exactly that reason — it is the linear operations only, deliberately, so that the count is of what the edges are worth rather than of what the conditional operations add.

And 13.3% of the third operation’s folds are unusable, which is a smaller error and a subtler one: those folds exist, they are distinct, and a closure that counts lines rather than creases will count them. What it must not do is count the points they make, since a line off the paper crosses other lines off the paper and the crossings are not references a folder can put a finger on.

That second rule is already in the machinery — the closure counts only points that land on paper — and it is worth pointing at, because it is the kind of thing that is obviously right once stated and easy to omit. A crossing inside a hole is specified and is not a reference; a crossing off the sheet is the same fact one step further out.

What the sweep measures #

Uniform points and lines on a square, four thousand of each, with a line specified by two uniform points. That is one model of an alignment somebody might ask for and it is a generous one: a folder’s alignments are not uniform, they are drawn from the marks already on the sheet, which cluster.

The counts are of folds, not of usable folds. A crease that clips the very corner of the paper counts as being on the sheet here, and a folder would not use it. The 13.3% is therefore a lower bound on how often the second operation disappoints.

The fifth’s inputs are three independent uniform objects, which is the most generous reading of “an alignment somebody might ask for” and the least like practice: a folder choosing that operation has usually already seen that the point can reach the line. The 23.8% is what the operation does when nobody is looking at the paper first.

And two of the seven are not measured. The sixth and seventh operations take more inputs and, in the sixth’s case, solve a cubic with one, two or three roots — so it has the same conditional character as the fifth, more sharply. It is not in the sweep because this collection does not implement it, and saying so is better than implying the list of five is the list of seven.