The axiom that reaches furthest wastes most

The field has no edge and the sheet does counts what a square throws away: two rounds of the four linear axioms put seventy-two per cent of their crossings outside the paper, so most of what the plane specifies is a number in the field and not a reference anybody can use. That measurement holds the axiom set fixed and varies the proportion of the sheet.

This one holds the sheet fixed and varies the axiom set, and the answer runs the wrong way.

An axiom is a way of spending a fold line’s two degrees of freedom on alignments. More axioms means more lines, more lines means more crossings, and more crossings ought to mean more references — which it does. What also rises, and rises faster, is the share of those crossings that land on the table instead of on the paper.

Nothing, then more than half, then three quarters #

The first axiom loses nothing. A line through two of the four corners is a diagonal or an edge, there are six of them, and every crossing of two of them is a corner or the centre. Five references, none off the paper, and a second round adds nothing at all because the configuration is closed.

The first two lose nothing at one round. Adding the perpendicular bisector of two points gives eight lines and nine references — the corners, the edge midpoints and the centre — all on the sheet. It is at the second round that the edge starts to cost: thirty-two lines, a hundred and thirty-three references kept, sixty-four crossings discarded, a third of the total.

The four linear axioms lose fifty-seven per cent at one round. Twelve lines, nine references kept, twelve crossings thrown away — and this is the first round, from a bare square. More than half of what a single round of folding specifies is already off the paper before anything has been built on it.

And the conic axiom loses seventy-six per cent at one round. Twenty-eight lines, thirty-three references kept, a hundred and four crossings discarded.

Line those four up and the share lost is 0, 0, 57, 76 per cent at a single round. The share is monotone in the axiom set, and the last entry is the point: the conic axiom at one round wastes a larger share than the four linear axioms waste at two.

Why more axioms means steeper lines #

The mechanism is geometric and it is the reason the result is a rule rather than an accident of the square.

The first axiom’s lines are determined by pairs of corners, so they run along the edges and the diagonals — at nought, forty-five and ninety degrees to the sheet’s own boundary, which is to say parallel to it or symmetric within it. Two such lines cross inside the square or not at all, because both of them pass through the square’s interior along its own directions.

Each further axiom specifies lines at angles the sheet’s boundary does not supply. A perpendicular bisector of two points that are not symmetric is at some angle; a line bringing a point onto a line is at another; the conic alignment, which brings a point onto a line while passing through a second point, is at an angle that varies continuously with the configuration. A pair of lines at a general angle crosses at a general point, and a general point of the plane is not on the square.

So the count of crossings rises quadratically with the number of lines, and the count of on-sheet crossings rises much more slowly, because the square occupies a bounded region of a plane in which the crossings are spreading. Twelve lines give sixty-six pairs; twenty-eight give three hundred and seventy-eight; and the square does not get any larger.

The first axiom is the odd one #

The zero in the first row deserves a paragraph of its own, because a measurement that begins at exactly nothing usually means the first case is degenerate, and here it means something else.

A line through two of the square’s corners is one of six lines: four edges and two diagonals. Intersect them in pairs and every crossing is a corner or the centre — five points, all on the paper. Run a second round and nothing changes at all, because the new points are points the six lines already pass through, so no new line is specified. The configuration is closed after one fold.

That is the only axiom set in this collection with that property, and it is why the compass-and-straightedge comparison has to be made carefully. A folder with only the first axiom has a finite world: five references, six creases, and no way to reach a sixth. Adding the second axiom opens it — nine references at one round, a hundred and thirty-three at two — and the price of opening it is that the closure stops being closed and starts spilling off the paper.

So the sequence 0, 0, 57, 76 is not a smooth degradation of something that started perfect. It is a boundary being crossed: the sets that lose nothing are the sets that reach almost nothing, and the first reference a folder gains beyond the trivial five is already paid for in crossings that miss the sheet.

The set that cannot be run twice #

The conic axiom’s row for two rounds is a dash, and the dash is not a shortcut.

From thirty-three points and twenty-eight lines, a second round of the five axioms specifies 16,762 distinct fold lines — past the cap the closure is computed under, and past any cap that would let the crossings be enumerated, since sixteen thousand lines have a hundred and forty million pairs. The axiom that brings a point onto a line through a second point takes three arguments rather than two, so its count is cubic in the configuration where the others are quadratic, and the configuration it is run on is the one the first round already enlarged.

That is a second cost with the same shape as the first, and it is worth naming separately. The first cost is that a larger share of the output falls off the paper. The second is that the output is unenumerable one round sooner — so the axiom that reaches furthest is the one whose closure cannot be computed far enough to see what it reaches.

Both costs are invisible in the field. The field is a set of numbers closed under six operations; it does not have a round structure, it does not have a sheet, and every one of the sixteen thousand lines is as good as any other in it.

What this does to the reach argument #

Twos and threes run out counts the degrees each instrument settles and finds folding beating the compass by a factor that rises without bound — and finds the whole of that advantage coming from the one axiom that produces a cubic. This essay prices the same axiom on paper.

Put the two together and the reading is not that the reach is illusory. The references really do rise: nine on the sheet from the four linear axioms at one round, thirty-three from five. Adding the conic axiom nearly quadruples what a folder can put a finger on after one fold, which is a large gain by any measure.

What falls is the efficiency of specification. A crossing is something a folder has to form, look at, and decide about; of the twenty-one crossings the linear axioms produce at one round, nine are usable, and of the hundred and thirty-seven the five axioms produce, thirty-three are. The yield per crossing falls from forty-three per cent to twenty-four.

So the honest statement is a trade rather than a refutation, and it has the same shape as what the accuracy costs and as what the polygons cost. The conic axiom buys degree three, which is most of the reachable degrees; it costs accuracy, it costs a larger share of its own output to the edge, and it costs the ability to compute its own closure past one round. Every one of those is a cost the characterisation of the field does not have a place for, because the characterisation is about which numbers exist and all three are about what happens on a piece of paper.

Two prices for one axiom #

It is worth putting the two costs of the conic axiom side by side, because they are independent and they compound.

On the plane it is the cheapest thing in the subject. One application multiplies the reachable degree by three, where every other axiom multiplies it by two — which is why a cube root is one step and an eighth root is three, and why the heptagon costs less than the polygon Gauss proved.

On the paper it is the most wasteful. Three quarters of its crossings miss the sheet at the first round, against a little over half for the linear four, and its closure cannot be computed to a second round at all.

Neither of those is a correction to the other. They are measurements of different quantities that a construction has to satisfy at once, and a folder choosing between a cheap tower and a route that stays on the paper is choosing between them with no exchange rate. That is the same predicament the accuracy result leaves, arriving from a third direction — and three independent costs all attaching to the same axiom is the reason to stop calling it the axiom that makes folding powerful and start calling it the axiom that makes folding expensive.

What is being counted #

Every crossing is formed, and the ones that miss are tallied rather than dropped. The closure forms every fold line the current points and lines specify, intersects every pair of them, and keeps the intersections that land on the sheet. The discarded ones used to vanish silently, which is the right thing to do with a point a folder cannot use and the wrong thing to do with its number: the count of them is exactly the difference between the field and the sheet.

Lines are deduplicated and crossings are deduplicated. Two axioms reaching the same crease have produced one crease, and two pairs of lines meeting at the same point have produced one reference. Without both, the “specified” counts would be counts of constructions rather than of objects, and the share lost would be a share of something nobody could act on.

The sheet is a unit square with its four corners and four edges. That is the configuration every measurement in this collection starts from, and the proportion matters — the figure above shows the same loss running from a third to most of the total across four proportions — so the numbers here are the square’s numbers rather than folding’s.

The cap is a refusal, not a truncation. A closure asked for at a depth it cannot be computed at throws rather than returning a partial answer, because a partial closure reported as a closure is a count of references that is wrong in the direction nobody would check.

What the count does not show #

A crossing is not the only kind of reference. A point where a fold line meets the sheet’s edge is a reference a folder can use, and this count does not include those — it counts crossings of pairs of fold lines. Including them would raise every on-sheet figure and would not change the ordering, since a line at a general angle meets the boundary in two points however many axioms produced it.

Two rounds is not a construction. Real constructions are performed in sequence, each fold chosen for a purpose, and nobody forms every line an axiom set specifies. The closure is a measure of what is available, not of what is done, and the share lost is a property of the availability.

And nothing here says a discarded crossing is useless. A number in the field is a number, and a construction can reach it on a larger sheet, or by a different route whose intermediate points all land. What the count says is that on this sheet, at this round, that particular crossing is not a reference — which is a statement about a procedure and not about a number.

The proportion is one sheet. A square is a choice and a costly one; the axiom-by-axiom comparison has not been run on the other proportions, so whether the ordering survives a sheet with no symmetry is not established here.

What a folder does instead #

Nothing in the practice of folding looks like this, and the difference is the reason the loss has gone uncounted.

Every axiom names things that must already be on the paper, and a folder performing a construction picks the alignment that produces the mark they want. They never form the other eleven lines, so they never see eleven-twelfths of the crossings, so the waste is not something anybody has had occasion to notice. It is visible only to a closure, which forms everything and is therefore the only instrument that can be asked how much of everything is usable.

And the practical constructions dodge it. Dividing a square into any number of equal parts works along the sheet’s own diagonals and edges — lines of the first axiom’s kind, whose crossings are guaranteed to land — which is why it is exact, why it needs no search, and why its references never fall off. The constructions that are taught are the ones living in the part of the closure that loses nothing, and the selection is invisible from inside because nobody teaching them chose on that basis.

That is worth separating from the arithmetic above. The measurement says a large share of what the axioms specify is unusable; the practice says the usable part is where everything anybody does already is. Both are true, and what sits between them is the question of whether the unusable share is where the unfound constructions are — which is not a question a count of crossings can answer.

Still open: the yield of a fold rather than of a round #

The measurement is per round, and a folder does not make a round.

What a folder makes is one fold, chosen. The quantity that would actually inform a construction is the yield of a single well-chosen fold — how many usable references one new crease produces, given the references already on the sheet — and that is a different number from the round’s average because the folder is choosing. A round forms every specified line including the ones at absurd angles; a folder forms the one that crosses the sheet usefully. The gap between the round’s yield and the chosen fold’s yield is the value of knowing what one is doing, and it is computable: form every line the axioms specify, rank them by how many on-sheet crossings each contributes, and read the top of the list against the mean.

And the conic axiom’s second round is the one measurement missing. Sixteen thousand lines is too many to intersect but not too many to sample: drawing a random subset of the pairs and counting the share on the sheet would estimate the loss to whatever precision the sample supports, which is the standard answer to an enumeration that has outgrown itself and is the one used on crease-pattern populations. It would say whether the conic axiom’s seventy-six per cent at one round keeps rising or settles.

The habit worth carrying is about instruments compared in the space they are defined on. An axiom’s reach is a statement about the plane and its yield is a statement about the sheet, and adding axioms moves the two in opposite directions. A comparison made only in the first space will keep saying that more is better, and the direction it is wrong in is the one a folder is standing in.