A fold needs something to align

The axioms are usually read as a list of things a fold can do, and read that way they sound unlimited. Bring two points together. Bring two lines together. Put a point on a line and pass the crease through another point.

Every one of those sentences names something that has to be there already. A fold is an alignment, and an alignment needs something to align — so the honest question is not what a fold can do but what a folder can point at.

That set starts as four corners and four edges, and it grows one fold at a time. It is finite at every depth, because each round can only combine what the last one left.

What an axiom needs before it can be used #

The list of seven is an enumeration rather than a collection: a crease is a line, a line is fixed by two conditions, and every way of imposing two conditions by aligning existing points and lines is on the list. That is why the list stops where it does, and it is also why the list is a statement about what is on the paper.

Read the sentences again with that in mind. Fold through two points needs two points. Fold square to a line through a given point needs a line and a point. Fold one line onto another needs two lines. Nothing in the list creates a reference out of nothing; each entry consumes what is available and returns a crease, and a crease becomes useful only where it crosses something else, because a crossing is a point a finger can be put on.

So the object worth building is the closure. Take the points and lines available, apply the axioms asked for, keep the distinct fold lines, cross them with one another and with what was already there, and keep the crossings that land on the paper. The result is the set of references available after one more fold, and the process repeats.

Two rules in that paragraph are doing real work. Folds are counted as lines rather than as constructions — two different alignments naming the same crease have made one crease. And only crossings on the sheet are kept, because a fold line extends forever and a piece of paper does not: a crossing off the edge is a point of the plane and not a reference a folder can use.

Nine, and every one of them a half #

The first round from a bare square is small enough to check by hand and surprising anyway.

Four corners and four edges specify thirty-eight folds by the four alignments a bare square admits, and twelve of them are different creases. Crossing those twelve with one another and with the edges, and discarding everything off the paper, leaves nine references: the four corners it started with, the four edge midpoints and the centre.

Nine is not a number anybody put in. It comes out of intersecting lines and throwing away what missed. And every coordinate among those nine is nought, a half or one — one fold reaches halves and nothing else, which is worth sitting with, because nothing anywhere in the axioms mentions a half. The half is what an alignment of two symmetric things produces, and symmetry is all a bare square has.

That is the first fold’s whole yield, and it is the reason a folder’s instinct is to halve. It is not a preference. Halves are the only thing available.

What two folds reach #

The second round is where the interest is, and where the usual telling of this subject turns out to be wrong.

Nine references and twelve lines specify three hundred folds by the four axioms, ninety-two of them different, and crossing all of those leaves 565 references on the paper. Restrict the round to the three alignments that are linear in the coordinates — through two points, point onto point, square to a line — and the same round leaves 133, every single one of which is a fraction.

Among those 133, the denominators are 1, 2, 3, 4, 5, 6, 8, 10, 12, 16 and 20 — and nothing else. Thirds are in the list. They arrive at the second fold, from the crossing of a diagonal with a line to an edge midpoint, and they need neither the fold that bisects an angle nor the fold that solves a cubic.

That is the opposite of how thirds are usually introduced. A third is presented as the awkward case: the thing a halving method only ever converges toward, or the thing a schoolteacher’s one-fold construction produces as a small miracle. It is neither exotic nor lucky. What one fold cannot do, two folds can, and the second fold does it with the plainest operations on the list.

The Haga case is the cleanest illustration of the accounting. The construction is famous for reaching a third in one fold, and it does; the reference it folds onto is an edge midpoint, and an edge midpoint is a thing the first round produced. Depth is measured from a bare square, and a construction that starts from a midpoint has already spent a fold.

The list repeats itself #

Something else falls out of the closure that nobody would set out to measure, and it changes how the axiom list should be read.

The first round specifies thirty-eight folds and makes twelve creases. The second specifies three hundred and makes ninety-two. So on both rounds, roughly seven folds in every ten that the axiom list names have already been made by some other alignment.

That is not a defect. It is what a list of ways to specify looks like on a configuration with symmetry in it: the fold that brings one corner onto its neighbour is also the fold that brings one edge onto another and the fold square to a third edge through the centre, and all three are separate entries producing one crease. The axioms are a vocabulary for describing folds a folder might want, not a generating set anybody would choose for enumeration — which is why thirty-eight named folds collapse to twelve on a blank square.

The wall inside the list #

Now the finding the whole closure exists to produce, and it sits between two adjacent entries in a list of seven.

Folding through two points, folding one point onto another, and folding square to a line are linear operations on the coordinates. Two rounds of them reach 133 references and every one is rational, and however long they ran they would stay that way — a linear map with rational data sends rationals to rationals, and there is no round at which that stops being so.

Allow the fold that brings one line onto another, and the same second round reaches 565 references of which 432 have a coordinate that is no fraction at all. Counted by coordinate rather than by point, 780 of the 1,130 are irrational — sixty-nine per cent of everything the round reaches, against a busiest single fraction sitting at under twenty.

The reason is one angle. Bisecting the angle between two edges of a square gives 45° and stays rational; bisecting the angle between an edge and the diagonal gives 22.5°, and 22.5° is the first angle this axiom produces that the other three cannot. Everything irrational in the round is built from it.

The two halves of that one angle are 2 − √2 ≈ 0.5858 along the edge and √2 − 1 ≈ 0.4142 beyond it, and they are the two most-carried irrational values in the whole set, at sixteen coordinates each — more than any other. That is the measurement worth having. It is not that the bisector produces one strange number; it is that one angle accounts for the busiest part of everything past the fractions, and the rest of the irrational half is that angle’s arithmetic playing out.

It is worth saying what is not claimed, because the easy sentence here is false. 2 − √2 is not the smallest irrational the round reaches: (3 − 2√2)/4 is about 0.0429 and several dozen others are smaller. It is the first one the closure meets and the one it lands on most often, and those are the properties that were checked.

Which claim was checked, and how #

The closure is a search, and a search that grades its own homework proves nothing, so each claim is put against something that has never heard of the construction.

The count of nine is not written anywhere in the code that produces it. It is what survives intersecting the twelve creases and discarding the crossings that miss the paper, and the check is that it comes out at nine and that every coordinate is nought, a half or one.

Rationality is recovered from a floating-point number by a continued fraction, which knows nothing about squares, folds or axioms. It is given each coordinate of each of the 133 references reached without the bisector and required to name a fraction for every one; a single failure would mean a linear operation had left the rationals, which cannot happen, and would therefore mean the closure was not doing what it says. The denominators it finds are then required to include a three, which is the third’s arrival stated as a test rather than as an observation.

The irrational reference is checked against Math.sqrt, which the closure never calls, and against the tangent of the diagonal’s half-angle, which it has never heard of either — what 2 − √2 leaves beyond it on the edge is tan 22.5°, and the two are required to agree to a part in a million million.

And the refusal is provoked. Asked for a round it cannot compute, the closure has to throw rather than return the part of the answer it managed — because a truncated reachable set is indistinguishable from a small one, and a small one is exactly what this essay would read as a result. That refusal is exercised deliberately, since an assertion that has never rejected anything has established nothing.

Where the closure stops #

Three limits, and the first is arithmetic rather than principle.

There is no third round here. A third round would ask the axioms for a fold from every pair of 565 references, and the branching is savage enough that the computation refuses. So every number above is about depth one and depth two, and the shape of the growth beyond them is not something these figures show.

The fifth alignment — put a point on a line with the crease passing through another point — makes the cost visible at depth one. It offers a fold for every point, line and second point together, so adding it to the other four takes the first round from thirty-eight named folds to ninety-four, from twelve creases to twenty-eight, and from nine references to thirty-three. A second round of that asks for over a thousand creases and a million crossings, which is why the closure is drawn to two.

The second limit is that the closure describes what is reachable, not what is convenient. A folder does not build the crossing of the ninetieth crease with the twelfth; a folder guesses and corrects, which is an algorithm rather than a construction and lands on points the closure never enumerates. The two answers do not contradict each other — they are answers to different questions, and the reason a folder needs the second is that getting close is often the whole of what is available.

The third is the idealisation, and it is the usual one. Every point in these figures is exact, every alignment is perfect, and a crossing is a crossing to a part in ten million. A real folder aligns by eye, with paper that has a thickness and a crease that has a width, so the reference set on a real sheet is a set of small regions rather than points — and two references a thousandth of a sheet apart are, in the hand, one place. The closure says what is available in principle; it does not say what is distinguishable.

What the compass has to do with it #

A reader arriving from the other tradition will want the comparison, and it is worth being exact about which part of it belongs here.

That straightedge and compass reach exactly the numbers built by repeated square roots, and that this is settled by a theorem about the degree of a field extension, is classical and belongs elsewhere in this fleet — it is quoted here as the baseline and nothing about it is derived. That folding reaches one degree further is this site’s subject, and where the cubic comes from is where the reason is set out.

What the closure adds is easy to miss. The reach of a tool is stated as a class of numbers, which is a statement about the limit of arbitrarily many operations. The closure is a statement about few operations: what is available after one fold, or two, from a specific starting object. A number can be perfectly reachable in the limit and eleven folds away, and a folder with a sheet of paper cares which.

Who noticed it, and when #

The axioms were assembled late and in pieces. Humiaki Huzita catalogued six of them in 1991; Koshiro Hatori added the seventh in 2001; Robert Lang later established that there is no eighth, by the enumeration the list’s completeness rests on. Jacques Justin had published closely related work at the end of the 1980s, and the attribution is genuinely divided rather than merely disputed.

What nobody assembled in the same way is the reachable set, and the reason is instructive. The axioms were catalogued by people asking what a fold can do, which is the natural question for somebody proving that paper reaches a cubic. The reachable set is the answer to what a fold can be given, which is the natural question for somebody sitting in front of a square with no marks on it — and that is a folder’s question rather than a geometer’s.

The practical form of it is old and unwritten. Every folding sequence in the traditional repertoire is a sequence of references — fold in half to get a midpoint, fold the corner to the midpoint to get a third — each step consuming what the last one made. So the closure is not a new idea about folding so much as an arithmetic for something folders have always done, and the number nine is a thing a folder’s hands have known for a very long time.

Where the ladder goes next #

The immediate direction is depth, and the obstacle is cost rather than principle. What the third round holds is a real question with a real answer, and reaching it needs a closure that does not intersect every pair of lines with every other.

The other direction is the operation rather than the depth. Two creases made at once is a different alignment vocabulary with a different reach, and it settles the eleven-sided polygon that one fold at a time cannot — so the same question about references can be asked again with a different list, and the answer will not be the same set grown faster.

And there is the question the closure makes askable: whether a published crease pattern’s own vertices are reachable at a stated depth. A pattern is published as a drawing rather than as a sequence, so what a reader is handed is a set of points with no account of how to reach any of them. A pattern whose vertices are all at depth three is a different object from one whose vertices need eleven folds each, and nothing on the drawing says which it is. The heptagon a compass cannot reach and an exact division of the square are both reachable by paper; how far in either lies is a question about references, and it now has a way of being answered.