A fold is worth what follows it

The axiom that reaches furthest wastes most counted what a round of folds delivers. Form every fold the axioms specify from what is on the sheet, intersect all of them, and count the crossings that land on the paper: a reference a folder can put a finger on. The share lost off the edge rose with every axiom added, to three quarters for the conic axiom in a single round.

It closed on the obvious objection. A folder does not make a round. A folder makes one fold, chooses it, and then makes another. The number a construction actually depends on is what one well-chosen fold produces, and the gap between that and the round’s average is the value of knowing what one is doing.

That gap turns out to have the wrong sign for the obvious reading, and the reason is the whole of this essay. A single fold, however well chosen, yields less than the round’s average — because most of what a round produces is not the yield of any one fold. It is crossings between folds, and a fold only collects those if the others are made too.

The menu the second round offers #

One round of the four linear axioms from a bare square — fold through two points, bring a point onto a point, bring a line onto a line, fold square to a line through a point, the four whose worth what each axiom is worth priced — leaves the sheet with nine references: the corners, the midpoints of the sides and the centre. It leaves twelve lines: the four edges, the two diagonals, the two midlines, and four lines that touch the square only at a corner — the outer bisectors of its corners, which the axiom bringing a line onto a line specifies alongside the diagonals and which no folder would make.

From those nine points and twelve lines the same four axioms specify ninety-two folds, of which twelve are the lines already there. The menu is the other eighty. Each is a fold a folder could make next, and each would add a reference wherever it crosses one of the twelve lines on the paper at a point that is not already one of the nine.

Made alone, the eighty add very different amounts. Sixteen add six references each, the most any fold can. Four add five, eight add four, eight add three, twelve add two and four add one. Twenty-eight add nothing at all, and every one of the twenty-eight misses the paper — a line the axioms specify that lies wholly off the square or only touches a corner, which is the per-fold form of the field having no edge. The average fold on the menu adds 2.50.

The fold worth making first #

The best first fold is a line through the point on the vertical midline a quarter of the way up, falling two units for every unit across. It crosses the top edge, one diagonal, the horizontal midline, the other diagonal, the vertical midline and the bottom edge, and not one of those six crossings was a reference before. Fifteen other folds on the menu tie it at six.

Six is the best a single fold can do, and the round averages 6.95. Made all together, the eighty folds put 556 new references on the paper, which is 6.95 a fold — more than the best fold manages alone. That is the fact the round-average reading hides. The round’s yield is not eighty yields added up; most of the 556 are crossings of one new fold with another, and a fold made on its own collects none of those.

A yield that grows with the folds before it #

A folder making folds one at a time therefore sees each fold’s yield rise, because each new fold has more lines to cross.

The chosen order’s first eight folds add six, seven, eight, nine, nine, ten, eleven and twelve references: each one crosses the first round’s lines and every chosen fold before it, and the later ones have more to cross. The ninth drops to eight. By then the folds that cross the sheet most usefully have been made, and the best remaining fold’s crossings begin to land on references already created by earlier pairs.

A fold’s yield is not a property of the fold. The same crease adds six references if it is the first fold made and could add twelve if it were the eighth, and the difference is entirely what else has been creased. So the question how much is one well-chosen fold worth has no answer until it says after which others. The honest measurement is a race: every fold on the menu made in some order, the references counted after each, and two orders compared.

The folds a chooser makes, and where they come from #

The first eight chosen folds fall into two kinds, four of each, and the second kind is the more interesting.

The first, second, fifth and sixth are the slope-two lines: through a point a quarter of the way along a midline, falling two units for every unit across, in the orientations that cross the most first-round lines. Their crossings are all rational — quarters, eighths and twelfths — and they are the folds anybody reaching for the obvious would find.

The third, fourth, seventh and eighth are not rational at all, and they are the folds the numbers a fold reaches is about: the first step off the rationals. The third chosen fold is a line falling $\sqrt{2} - 1$ for every unit across, crossing the left edge at $0.828$ and the right at $0.414$, and it is the first fold in the chosen order to put an irrational coordinate on the paper. It is specified in exactly one way from the first round’s twelve lines: by bisecting the angle between the bottom edge and the outer bisector at the top right corner, the line through that corner at a right angle to the diagonal. That outer bisector touches the sheet at a single point and adds nothing to any construction — it is one of the four lines the first round specifies that no folder would crease. One of the most productive folds on the menu is the child of one of the least.

That is the reason the closure keeps lines it cannot use as references. The lines that miss the paper cost nothing to record and a great deal to discard, because the axiom that brings a line onto a line reads them as inputs; a folder specifying folds only from lines that cross the sheet would not find the third fold on this menu at all, and would lose the angle of 22.5 degrees that the reference closure identifies as the first irrational the linear axioms produce. The chosen order arrives at it by the third fold because it crosses more of the sheet’s existing lines at new points than any rational fold remaining.

Chosen against random #

The chosen order is greedy: at every step, make the fold that adds most references now. The random orders make the same eighty folds shuffled, twenty times with a fixed seed, and the curve is their mean. Both end at the same place — the whole round’s 565 references, nine old and 556 new, the same set the closure computes when it makes the round at once — so they differ only in how fast they get there.

The chosen order reaches a quarter of the round in 16 folds, against 31.3 at random; half in 28, against 51.0; nine tenths in 45, against 74.5. Everything is there after 52 chosen folds, and the last twenty-eight add nothing, because they are the twenty-eight that miss the paper.

The saving is largest at the start and shrinks toward the end. A chosen order is 1.95 times faster to a quarter of the round, 1.82 to a half, 1.66 to nine tenths and 1.53 to the whole. That is what choosing is for: at the beginning a random fold is often one of the twenty-eight that cannot add anything, or one that crosses the sheet near a corner and meets two lines, while the chosen fold meets six. At the end both orders are finishing the same set and there is little left to choose between.

Fifty-two, and no fewer #

That the chosen order finishes in fifty-two folds invites a sharper question. Is fifty-two the fewest folds that make every reference the round makes, or merely what a greedy folder happens to need?

It is the fewest, and the argument is short enough to check on the numbers. Take every reference the round makes and count the lines through it. Each of the fifty-two folds that cross the paper passes through at least one reference that it and exactly one other line make between them. Leave that fold out and that reference is gone, since a point needs two lines. So every complete set of folds contains all fifty-two; and the fifty-two together do make every reference, which is what the greedy order found. The minimum is exactly fifty-two, and the only folds a round could spare are the twenty-eight that never touch the paper.

That makes the round’s yield into a clean pair of numbers. A round of the four linear axioms from a bare square specifies eighty new folds, and fifty-two of them are the construction; twenty-eight of them are fold lines that exist in the plane and not on the sheet. A folder who folded all eighty would have folded twenty-eight creases into the air.

The conic axiom’s menu #

The same race can be run with the conic axiom on the menu: the fold that puts one point onto a line while passing through another, the alignment that solves cubics. From the same nine references and twelve lines, the five axioms specify 385 new folds.

The round is large enough to count exactly and still large. Made all together the 385 folds put 16,881 new references on the paper, thirty times what the linear round makes — and lose 42,843 crossings off it, 71.7 per cent of what they specify, almost exactly the linear round’s 72.1. The conic axiom’s first-round loss of three quarters, which the earlier count measured, is matched rather than exceeded here, where its second round is made from the linear round’s references.

The first-fold picture barely moves. The best single fold still adds six, and it is the same fold. The chosen order’s first eight folds are the same eight folds, in the same order, with the same yields, whether the conic axiom is on the menu or not. The conic axiom’s lines are not what a folder should reach for first; they begin to win at the ninth fold, where one of them adds eleven against the linear menu’s best of eight, and they win more and more from there, because they are many and they cross the growing web of creases at new points.

Choosing is worth somewhat less here, proportionally: 1.71 times faster to a quarter, 1.63 to a half, 1.43 to nine tenths. A hundred of the 385 never cross the paper. And here the greedy order is not quite optimal. It completes the round in 285 folds, while the fewest that can is 284: one of its early choices, the best fold at the moment it was made, passes only through references that later folds also make, and could have been left out. Choosing by immediate yield is not the same as choosing the smallest set, and the difference, on this menu, is exactly one fold in 284.

What these counts rest on #

The sheet is the unit square and a reference is a crossing on it. A crossing a hair outside the edge is off; a crossing exactly on the edge is on. That is the convention the closure has always used, and the counts are exact under it.

Two references are the same when they agree to seven decimal places, the rounding the closure itself deduplicates by. With the conic axiom, square roots make nearly coincident points that a finer rounding would split, and doing so adds a hundred and twenty-three spurious references to a round of sixteen thousand; the coarser rounding is the one the closure’s own total agrees with, and every figure here checks that the orders end at the closure’s total.

A fold is a line, and making it is free. A real fold has an error, and what buys the reach costs the accuracy found the conic alignment the least accurate of the axioms; a construction that chose by yield alone would reach for exactly the folds that are hardest to make well. The race counts references and not their quality.

The menu is fixed at the second round. A folder who has made a few folds can specify new ones from the references those folds created, and the menu grows as the folding proceeds. The race here keeps the menu to what the first round’s nine references specify, so that the chosen and random orders are choosing from the same list and end at the same set. A folder allowed the growing menu would reach more references in fewer folds than either curve, and how many more is not measured.

And the random orders are twenty. Their mean is steady to a fold at the shares reported; the spread between orders is wide, and a lucky random folder can do much better than the mean.

The difference between a round and a folder #

What connects this to the rest of the subject is the direction of the surprise. A round of folds is how the theory of origami numbers is built — the reachable set is the closure under rounds, and reachable is not cheap measured how many rounds a number needs. A folder’s sequence is how a construction is actually made. The two were always going to differ, and the natural guess is that a folder does better than a round, because a folder chooses.

Per fold, a folder does worse than a round, because a round is quadratic. Eighty folds make on the order of eighty squared crossings, and the round’s average credits each fold with its share of all of them; the folder’s first fold can only cross the twelve lines already there. What choosing buys is not a better fold than the round’s average. It is getting to any given share of the round in fewer folds, and on the linear menu that is 28 folds rather than 51 for half of everything a second round can make.

That also reframes the loss off the edge. The earlier essay found the conic axiom losing three quarters of what it specifies, and read it as waste. Per fold, the waste is concentrated: a hundred of its 385 folds lose everything, because they never meet the paper, and the rest lose some crossings and keep others. A folder who never makes a fold that misses the paper has already recovered most of what the edge takes, and the remaining loss is crossings between folds that do meet the paper, landing beyond it.

Still open: the growing menu, and the price of a fold #

The race holds the menu fixed, and a folder does not. The natural next measurement lets the menu grow, which is the menu the third fold cannot be listed found running to hundreds of thousands of folds by the third round: after each fold, add every fold the new references specify, and choose among all of them. That is a search whose cost grows with every fold made, since every new reference multiplies the menu, and the greedy version of it is cheap enough to run for tens of folds; whether a growing menu reaches the fixed round’s 565 references in far fewer than 52 folds, or reaches a different and larger set, would say how much of the round’s structure is an artefact of doing everything in rounds.

The other direction is to charge folds unequally. A square-root step is a bisection and a cube-root step is a conic alignment, and twos and threes run out asked for exactly this weighting. A race in which each fold costs its axiom’s error, and the chosen order maximises references per unit of accumulated error rather than per fold, would put the conic folds that win from the ninth fold onwards against their own imprecision — and would say whether the folds a greedy folder reaches for are the ones a careful folder should.

Sideways from here, the one-fold gap between the greedy order and the minimum on the conic menu is a small instance of a large fact: covering a set of points with lines, two lines a point, is a covering problem, and greedy covers are not optimal in general. A fold needs something to align is the reminder that a folder’s references are a resource spent by every fold; and this is the first measurement of them in which the order of spending has a measurable best.

The habit worth carrying is about averages over a whole. When a total is made of interactions between parts, the average part is credited with interactions it cannot have on its own. A round’s yield per fold was higher than any single fold’s, and the reason was not that the round’s folds were better but that the round had made the others.