The axiom that names two folds

There are exactly seven ways a single fold can be specified by alignments of points and lines already on the paper, and the list is complete: a fold line in the plane has two degrees of freedom, and enumerating the ways to spend two constraints is where the number seven comes from.

An axiom may name no fold is one thing that can go wrong with that picture. An alignment can have no solution — twenty-three point eight per cent of the fifth axiom’s inputs, in a square — or a solution whose crease misses the paper entirely, which happens to thirteen point three per cent of the third’s.

The other end of the same under-determination is less often mentioned and is much commoner.

Two folds, and the axiom does not choose #

The third axiom brings one line onto another. Two lines that meet have two bisectors of the angle between them, and folding along either one carries the first line onto the second. Both satisfy the axiom’s statement exactly, and the statement says nothing about which.

They are always perpendicular. That is not a coincidence of the example: the two bisectors of a pair of crossing lines are the internal and external ones, and a pair of angle bisectors is square by construction. The measurement checks it to machine precision on every input, and it has never been off.

So this is not a case of two nearly-identical answers where either would do. The two folds are as far apart in direction as two folds can be.

The parallel case, which is the exception #

Two lines that do not meet have no angle between them and no pair of bisectors. The third axiom on parallel lines names exactly one fold: the midline, half way between them and parallel to both.

That is the only input on which the third axiom is single-valued, and it is a set of measure zero — two lines drawn at random are never parallel. It matters anyway, because the lines a folder actually works with are the sheet’s own edges and its previous creases, and those are parallel far more often than a random draw would suggest. A square has two pairs of parallel edges; a grid of creases is parallel families.

So the axiom’s ambiguity is at its worst on general inputs and vanishes on a large share of the inputs anybody uses, which is a second reason the counts below are about a distribution rather than about folding.

How often both are available #

The sheet decides how often the ambiguity actually arises, because a fold that misses the paper is not a fold a folder can make.

Both land on the paper 2,965 times of 4,000, one lands 1,035 times, and neither lands never. So three quarters of the time the axiom hands a folder a choice, and a quarter of the time the sheet makes the choice for them by putting one of the answers off the edge.

The midpoints of the two creases are a third of a sheet apart at the median and nearly a whole sheet at the widest. Nobody is going to fold one intending the other.

Which of the two a construction takes #

There is a practical consequence for anything that computes with these, and it is the reason the multiplicity was noticed here at all.

A program handed an alignment has to return something. If it returns one fold it has made a choice the axiom did not make, and unless it says so the choice is invisible — the construction proceeds, the crease lands somewhere, and everything downstream is about one branch of a two-branched operation. If it returns both, everything downstream has to handle a set.

This collection returns both. The third axiom’s function hands back an array of two folds, or one where the lines are parallel and the only answer is the midline. What each axiom is worth counts what the operations reach, and the counting has to be over folds rather than over alignments, or the fifth axiom’s contribution is halved.

The fifth is worse #

The third’s ambiguity is at least regular: two answers, always at right angles, and knowing that is most of what a folder needs.

The fifth axiom folds a point onto a line through another point — a point placed on a line while a crease passes through a named point. Geometrically it is the intersection of a circle with a line, so it has two solutions, or one at tangency, or none.

Over four thousand random inputs in a square: two solutions 3,009 times, none 991 times, and exactly one never — tangency is a measure-zero coincidence and a random draw does not find it.

And the two solutions are at any angle at all. Half a degree apart at the closest of four thousand, ninety degrees at the widest, sixty-eight at the median. So there is no rule of thumb here: two folds that might be almost the same or might be square, with nothing in the statement to say which is meant or how far apart they will be.

The nine hundred and ninety-one failures are the other end of the same geometry and they are already counted: the point is further from the line than the crease through the named point can reach, so the circle misses. Two answers or none, with the boundary between them a coincidence.

What the seven actually are #

This changes what the list of seven is a list of, and the change is worth making explicitly.

An axiom is usually read as a fold specified by an alignment — hand it the alignment and get the crease. Read that way, an axiom is a function.

It is not a function. It is a relation: an alignment, and the set of folds satisfying it. That set has size nought, one, two or three depending on the axiom and the input, and only the first, second and fourth are single-valued for every input they accept.

The enumeration that produces the number seven is an enumeration of alignments, not of folds. That is the right thing to enumerate — it is what makes the list complete — and it means the seven cannot be read off as seven constructions.

Counted properly the seven alignments name rather more than seven folds. The first, second and fourth name one apiece; the third names two where the lines meet and one where they are parallel; the fifth names two or none; the sixth up to three; the seventh one. So an inventory of folds specifiable in one step is not seven of anything, and the number seven is doing a different job — it is the count of ways two constraints can be spent, which is what makes the list provably complete.

Why the list stops at seven is the argument for completeness, and it is an argument about alignments throughout. Nothing in it is disturbed by an alignment having several answers; the list would still be complete if every one of them had a hundred.

The answer count is the degree #

The essay sorts the seven into the single-valued ones with linear conditions and the multi-valued ones with curved conditions, and the sort is nearly right. Sharpened, it becomes an exact rule: an alignment names as many folds as the degree of the equation it produces.

Run the list. The first, second and fourth impose conditions that are linear in the fold line — through two points, the perpendicular bisector of two points, perpendicular to a line through a point — and a linear system has one solution. One fold each.

The third imposes an angle condition, which is quadratic, and names two. The fifth is a circle meeting a line, degree two, and names two. The sixth is the cubic, and names up to three. Every count matches its degree.

The seventh is the case that makes the rule better than the linear-versus-curved reading. It brings a point onto a line and is perpendicular to another line — one curved condition and one linear one — and it names exactly one fold. The reason is that the perpendicularity fixes the fold’s direction first, and with the direction fixed, placing a point on a line is one equation in one remaining unknown and is linear in it. A curved condition can be single-valued once something else has pinned down the freedom the curvature lived in.

So the rule is not about whether a condition is curved. It is about the degree of the system the two conditions form together, and the seven read off as 1, 1, 2, 1, 2, 3, 1.

Which says how many folds the seven really name #

That list also answers the question the essay poses and leaves as a remark — how many folds are specifiable in one step, as against how many alignments there are.

On the measured distributions the count is not the sum of those degrees, because some solutions miss the paper. The third names two folds that both land three quarters of the time and one the rest, which averages 1.74 usable folds per input. The fifth names two three quarters of the time and none otherwise, averaging 1.50. The four single-valued alignments contribute one apiece.

Those six together average 7.25 usable folds per set of inputs, before the sixth axiom is counted at all. So the seven alignments already name more than seven folds, and the sixth — with up to three solutions of its own — pushes it higher.

The number seven is therefore not a count of folds and is not even close to one by accident. It is a count of ways to spend two constraints, which is what makes the list provably complete, and the fold count is a different and larger quantity that depends on the sheet.

The two ends of the same defect #

Read together, the two under-determinations are one fact seen from opposite sides, and the fact is that an alignment is a condition rather than an instruction.

A condition on a fold line carves out a subset of the two-dimensional space of lines. Two independent conditions carve out an intersection, and an intersection of two curves in a plane is a finite set of points — usually more than one, sometimes empty, and only in special cases exactly one. Everything counted here is that generic behaviour showing through.

Seen that way, the surprise is not that the third and fifth axioms have two answers. It is that the first, second and fourth have exactly one, and they do because their conditions are linear rather than curved: a fold through two points is a line through two points, and two lines meet once. The axioms with multiple answers are the ones whose conditions involve a parabola or a circle, and those are precisely the ones that reach past the compass.

So the ambiguity and the power arrive together. An axiom that names one fold per alignment is an axiom whose conditions are linear, and linear conditions do not solve cubics.

What a folder does about it #

In practice the ambiguity is resolved by the sheet, by convention, or by looking.

The sheet resolves a quarter of the third axiom’s cases outright, by putting one answer off the paper. That is the commonest resolution and it is invisible: a folder who has never seen the other answer does not know a choice was made.

Convention resolves most of the rest. The internal bisector is the one somebody drawing a diagram means, and it is the one that stays inside the region between the lines. Nothing in the axiom says so.

And looking resolves the fifth, which has no convention available because its two answers are not related by any symmetry. A folder given the alignment has to place the point on the line and see where the crease falls, twice, and pick.

The one axiom whose answers are truly interchangeable #

There is a case where the two answers really do not matter, and separating it out is worth doing because it is the only one.

Two parallel lines have one bisector, the midline, and the third axiom is single-valued on them. That is not the interchangeable case; it is the case with no choice.

The interchangeable case is the third axiom on two lines meeting at a right angle. There the two bisectors are the two diagonals of the right angle, and they are related by the reflection that swaps the two given lines — so any construction symmetric in its two inputs gets the same answer either way, up to that symmetry. Every other angle breaks it: the internal bisector lies between the lines and the external one does not, and they do genuinely different things to the paper.

That is a narrow exception, and the reason to state it is that right angles are exactly what a square sheet’s edges provide. A folder working from the corners of the paper meets the interchangeable case far more often than a uniform draw suggests, which is one more reason the counts above are about a distribution nobody folds in.

Why this matters for the arithmetic #

The cubic comes from the sixth axiom, and the connection between folding and root-finding runs through exactly this multiplicity.

A cubic has up to three real roots, and the sixth axiom has up to three solutions, and those are the same three: the fold that solves the cubic is the root, and an equation with three roots is an alignment satisfied by three folds. The multiplicity is not a defect in the statement of the axiom. It is the arithmetic showing through.

So the same fact is a nuisance for a folder and the entire point for a mathematician. An axiom that named exactly one fold per alignment could not solve a cubic, because a cubic does not have exactly one root.

Where the ambiguity has already been paid for #

This collection computes with the axioms in two places, and the two handle the multiplicity differently for reasons worth separating.

The reachable set takes every solution of every alignment. That is the right choice there, because the question is which points a folder can reach and a fold that reaches a point is a fold whether or not another fold satisfies the same alignment. Dropping one branch would have shrunk the set by a quarter with nothing to say it had happened.

The constructions — trisection, the cube root, an exact seventh — name their branch. Each is a sequence of specific folds arriving at a specific point, and a construction that said bring this line onto that one without saying which bisector would be a procedure a reader could follow to the wrong answer half the time. So the diagrams show the crease and the text names the alignment, and the crease is what disambiguates.

That division is the standard one in the literature too, and it is why the ambiguity is invisible: nobody meets it, because every published construction has already resolved it by drawing the answer.

What is being counted, and what is not #

Two limits on these numbers.

They are over inputs drawn uniformly in a square, which is a stated distribution and not a natural one. A folder does not choose lines at random; they choose lines that are already on the paper, which are the edges and previous creases, and those are far from uniform. How far from the nearest reference measures the real population and it is nothing like this one.

And a fold that lands on the paper is not the same as a fold anybody can make. The crease has to be reachable by actually bringing the alignment together, which for a fold near the edge of the sheet means holding very little paper. Closer than a crease is wide is the account of where that limit sits, and it is a limit these counts do not apply.

What the numbers do establish is comparative and robust to both. The third axiom’s ambiguity is available on most inputs, its two answers are perpendicular, and the fifth’s are anywhere — and none of that depends on which distribution the inputs were drawn from.