Each new fold can cross every old one

A fold is worth what follows it raced a folder against a round. After one round of the four linear axioms from a bare square the paper carries nine references and twelve lines, and the second round offers eighty new folds. Made one at a time, always the fold that adds most next, they reach half the round’s 565 references in twenty-eight folds and all of them in fifty-two; and fifty-two is the fewest, because each of those folds is the only second fold through some reference the round makes.

That essay ended by naming what its race held fixed. The menu was the round’s, and a folder’s menu is not. After a fold the paper has new references, and the new references specify new folds — the menu the third fold cannot be listed found running to hundreds of thousands by the third round. A folder who chooses from all of it is doing something the round never does: using what the last fold made to choose the next.

The answer turns out to be governed by a bound that mentions no axiom at all, and the greedy folder meets it exactly. After k folds from the first round’s sheet there can be at most $9 + k(k+11)/2$ references on the paper, whatever the folds are and however they were specified; the folder choosing from the growing menu has exactly that many after every one of thirty folds; so it passes the round’s 565 at the twenty-ninth fold, and no twenty-eight folds of any kind can. And the references it makes are almost none of them the round’s.

What one fold can add #

A reference is a crossing of two creases on the paper. A new fold is a straight line, and a straight line meets another straight line once at most, so a new fold can add at most one reference for each line already there. That is the crude bound, and it is not reached, because some lines can never be crossed on the paper.

The rim is crossed twice and no more. A line that passes through the square’s interior leaves it at two points, one on each of two edges, and those are its only crossings with the four edges that land on the paper; the other two edges it meets, if at all, beyond the corners. A line that does not pass through the interior can be crossed on the paper only where it touches it. The first round has four such lines, each through a corner of the square at forty-five degrees to its edges, running outside the paper everywhere else. Any crossing with them is off the paper or at a corner, which is already a reference.

So a fold adds at most the number of lines already through the paper’s interior, plus two for the rim. The first round has four such lines — the two diagonals and the two midlines — and every fold that adds anything passes through the interior and becomes one more. The $k$-th fold can therefore add at most $k + 5$ references, and $k$ folds at most

$$6 + 7 + \cdots + (k+5) = \frac{k(k+11)}{2}$$

on top of the first round’s nine. The argument uses nothing about where the folds came from. It holds for the four linear axioms, for the conic axiom, for a fold placed by eye.

The growing menu meets the bound #

The folder choosing from the growing menu adds six references with its first fold, seven with its second, eight with its third, and so on to thirty-five with its thirtieth. Every fold adds exactly the bound: each crosses every line already through the paper, inside the paper, at a point that was not a reference before, and meets the rim at two more new points. The count after $k$ folds is exactly $9 + k(k+11)/2$ — fifteen after one fold, 114 after ten, 589 after twenty-nine.

Twenty-nine folds, and no fewer. The bound at twenty-eight is 555, ten short of the round’s 565, so no sequence of twenty-eight folds of any kind can put 565 references on the paper; the growing menu passes it at the twenty-ninth. The round’s own menu needed fifty-two. A folder who reads the whole sheet before each fold needs fewer than three fifths of the folds for more references.

That is the fifty-two-and-no-fewer argument read the other way. Fifty-two was the minimum for the round’s references, a particular set of 565 points; twenty-nine is the minimum for any 565, and the folder that achieves it makes a different set.

Where the round’s crossings go #

The round’s folds obey the same bound, and fall far below it. Over its fifty-two folds the bound allows 1,638 new references and the round makes 556. Two in every three of the crossings a round’s folds could have made land on points already made.

The round keeps pace for four folds. Its chooser’s first four folds add six, seven, eight and nine references, each exactly the bound, and they are the same kind of fold the growing menu makes: the first choice is made from the same eighty folds on both menus, and sixteen of the eighty meet the bound. From the fifth fold on the round’s best falls short — nine where ten were possible, then ten where eleven were — and from the ninth it falls away. No fold of the round ever adds more than fourteen references, at the twenty-seventh fold when the bound allows thirty-two, or at the thirty-sixth when it allows forty-one. The last four add six each against a bound of fifty-four to fifty-seven. A fold in the round’s last stretch has more lines to cross than any before it, and adds less.

That is the opposite of the trend a fold is worth what follows it reported, and the two are consistent. That essay found the round’s chosen folds adding more as they go over the first sixteen, because each crosses more lines than the one before; the growth was real, and it was the bound’s growth leaking through. As the round fills, its menu runs out of folds whose crossings are new, and its chooser is left finishing a set rather than extending one.

The reason is in how the round’s folds are specified. Every one of them is defined by the first round’s references: a line through two of them, the fold that brings one onto another, the fold that brings one line onto another, the perpendicular through a reference. A fold through a reference crosses every line through that reference at a point that is already there. And the round’s folds are specified by the same nine points, so they pass through those nine points over and over, and through one another’s crossings near them. A round is concurrent by construction: its lines meet in crowds because they were all drawn from the same few places.

The growing menu escapes this by having more places to draw from. Plenty of folds on it still pass through references — every fold through two points does, by definition — and those fall short of the bound exactly as the round’s do. But among the hundreds of thousands there are always folds that pass through none, cross every existing line inside the paper, and avoid every existing crossing. Not every axiom is tied to the points that specify it: the fold that brings one reference onto another is the perpendicular bisector of the two, and passes through neither.

The square of the folds #

The bound turns the count into a law, and the law has a shape worth naming. References after $k$ folds grow as $k^2/2$ at leading order, because the $k$-th fold’s yield grows by one with every fold before it: each fold is worth one more than the last because it is one more line for the next to cross. That is the arithmetic of lines in general position — $n$ lines, no two parallel and no three through a point, cross in exactly $n(n-1)/2$ points — confined to a square, with the first round’s lines as a head start and its four corner lines as dead weight.

A round has no such law. Its count is fixed when the round is chosen: eighty folds and 565 references, whatever order they are made in, and the only question an order answers is how soon. The growing menu has no final set. Its count after thirty folds is 624, after a hundred it would be $9 + 100 \cdot 111/2 = 5{,}559$ if the bound kept being met, and nothing about the menu suggests it stops being met: the share of folds meeting it was as large at the thirtieth choice as at the first. Whether it keeps being met past thirty was not measured, because each choice costs more than the last; the bound itself holds at every $k$.

So the comparison between a round and a folder is not a race between two speeds toward one target. It is a fixed quantity against a growing one. The axiom that reaches furthest wastes most counted what a round delivers and found most of it lost off the paper; a folder who chooses loses, at every fold, only the six crossings the first round’s sheet makes unusable — two edges and four corner lines — and loses nothing to concurrence at all.

Twenty-nine folds, drawn #

Watching the folds accumulate shows what the bound’s proof says in words. To meet the bound a fold has to do two things. It has to cross every line already through the paper before it leaves the paper, so its crossings with all of them fall inside the square rather than beyond an edge; a short line clipping a corner cannot, because most lines through the paper never come near that corner. And it has to miss every crossing already made, so that each of its crossings is a new point.

The second requirement has a consequence that can be read straight off the count. Since every one of the thirty chosen folds met the bound, none of them passed through a reference already on the paper, and so no three of the creases meet at any point the folds made: every reference after the first round’s nine lies on exactly two lines. The arrangement is what geometers call general position, and the greedy folder keeps it from the first fold to the last. The round’s arrangement is its opposite, crowded with concurrences, which is the 1,082 references it never makes.

What the drawing shows that the count does not is where the folds go. They do not spread over the square. After the first few they fall into two fans — one family running steeply from the top edge to the bottom, one shallowly from the left edge to the right — and each new fold joins a fan a fraction of a degree from the fold before it. The angles of the first twelve run 117, 173, 111, 169, 108, 167, 107, 166, 106, 166, 106 and 165 degrees, alternating between the families and converging in each. Two nearly parallel lines that both run the full height of the square do cross inside it, which is what the bound needs; they cross at a shallow angle, near the other fan, and close to every other crossing there.

Two sets with fifteen points in common #

The two sets share fifteen points. Nine are the first round’s own, which both start from; the other six are the first chosen fold’s crossings, which the round also makes. After that the two diverge completely. The round’s references are its eighty folds’ crossings with one another and the first round’s lines: points two folds deep, specified from the nine. The chooser’s are crossings of folds each specified from the references the previous folds made, so a reference created at the $k$-th fold is specified from references that are themselves up to $k$ folds deep.

So the race was never a race to the same set. The round and the chooser make different numbers, and neither set contains the other. A folder who wanted one particular reference that only the round makes — a third, say, read off a crossing two folds deep — gains nothing from the growing menu’s speed; the 589 points include no promise of any particular one. What the growing menu buys is count, and count is a currency only for a folder who needs many references and does not mind which.

That is the distinction reachable is not cheap drew between whether a number can be folded and what it costs, and what buys the reach costs the accuracy between reaching a number and reaching it well. Here it is drawn between reaching many numbers and reaching the ones a design needs.

The count is paid for in spacing #

The chooser’s references crowd, and they crowd faster than they multiply. After five folds the median reference is four millimetres from its nearest neighbour on a 150-millimetre sheet, which a hand can find. After ten it is a millimetre. After twenty it is under a tenth of a millimetre, and after twenty-nine it is 0.014 millimetres — a seventieth of a millimetre — with 539 of the 589 lying within a millimetre of another and the closest pair less than a thousandth of a millimetre apart. The round’s 565 references, made by folds that waste two thirds of their crossings, have a median spacing of 2.70 millimetres; only 32 of them lie within a millimetre of another, and its closest pair is half a millimetre apart.

That is the bound’s price, and it follows from the two requirements a fold must meet. To cross every line through the paper inside the paper, a new fold must run nearly the full height or width of the square in the direction the lines already there leave room for; to miss every existing crossing, it need only miss them by a ten-millionth of the sheet. Nothing rewards a fold for missing a crossing by a centimetre rather than by a hair, and the folds the chooser takes are, fold after fold, the ones squeezed in beside the last fold of their fan. The count says a folder choosing from the whole menu makes the most references possible; the spacing says almost none of them are references a folder could use. A crease is a few tenths of a millimetre wide, and a paper limits spacing, not density found that the spacing, not the count, is where paper sets its limits.

So the comparison with the round reverses once spacing is counted, and the measurements suggest a reading of why, though they do not prove it. The round is slow because its folds meet one another at its own references, and those meetings are what keep its references apart: a crowd of lines through nine points is a crowd that leaves the space between the points alone. The chooser is fast because it never meets an existing crossing, and the fastest way never to meet one is to run alongside the last fold.

The menu, and how much of it is optimal #

The menu explodes, as the third fold cannot be listed found that it would: eighty folds at the first choice, 9,855 at the tenth, 86,602 at the twentieth, 335,193 at the thirtieth. A folder could not survey it.

But a folder would not need to. At every choice, at least one fold in eleven on the menu meets the bound — sixteen of the first eighty, 1,740 of the 9,855 at the tenth, 78,028 at the thirtieth. The greedy folder’s advantage is not a needle found in a haystack. It is a rule a folder can check by eye: a fold that crosses every crease already through the paper inside the paper, and passes through no point where two creases already meet. Between about a tenth and a quarter of all the folds the axioms specify pass that check at every choice measured, and every one of them is as good as the best.

What the count does not measure #

A reference here is a point, exact. Two crossings within a ten-millionth of the sheet are treated as one, and every other pair as two, which is the convention the round’s own count used; the section above is what that convention hides.

A fold is a line, made exactly. The bound is a fact about straight lines in a square, so it holds for any folding error that keeps a crease straight. What error does is move the crossings, and a fold that crosses everything is crossing at whatever angles the arrangement happens to give; what buys the reach costs the accuracy found that shallow crossings amplify error by one over the sine of their angle, and nothing in the greedy rule prefers a steep crossing to a shallow one.

Ties are broken by the menu’s order. At each choice tens of thousands of folds meet the bound, and the folder takes the first one listed. A different rule would make different folds and different references in the same number; the count, being the bound, would not change.

The sheet is the first round’s. The race starts from nine references and twelve lines, the same start the round had, so that the two can be compared; a folder starting from a bare square would face a different first step.

How it was counted #

The menu is generated, not sampled. At each choice, every fold the four linear axioms specify from every reference and line on the paper is listed, lines already creased are removed, and each remaining fold’s new on-paper crossings are counted against the current references, with crossings merged at a ten-millionth of the sheet as the closure does. The fold with the largest count is made.

The bound is checked against every fold, not only the chosen one. For every fold on every menu the count of new references is required to be at most the number of lines then through the paper’s interior plus two, and the chosen fold’s count is required to equal it at every one of thirty choices. The round’s own race is the one a fold is worth what follows it made, measured the same way.

The shared references are compared point by point against the second round of the closure itself, made in one step from the first round’s references, which is the independent count of the round’s 565.

Still open: the conic menu, and a race for usable references #

The bound mentions no axiom, so the conic axiom cannot beat it: a fold that brings a point onto a line while another point lands on another line is still a straight line. Whether the five-axiom menu also meets the bound at every fold, or whether its extra folds change which references a greedy folder makes without changing how many, is the same measurement on the axiom that reaches furthest wastes most’s menu, and it is a direct test of what each axiom is worth to a folder rather than to the field.

The count should be replaced by a usable count. A reference within a fold’s width of another adds nothing a hand can use, so the honest race maximises references at least some spacing apart — half a millimetre, say, on a 150-millimetre sheet. The spacing table says that race cannot meet the bound for long, since the bound is met here by crowding; how far below the bound a spacing rule forces the count, and whether its folds still fall into fans or spread over the square, would turn this essay’s arithmetic into advice. Twos and threes run out and a fold needs something to align both point at references as a resource; the spacing race is the first that would price them as one.

The habit worth carrying is about records set inside a format, and about the record that beats them. Before treating a minimum as a fact about the task, ask what the format fixed; and before treating the new record as better, ask what the format was protecting. Fifty-two was the fewest folds for a round’s references, and the round had fixed both the menu and the set; with the menu free and the set unspecified, the minimum is a bound on lines in a square, and twenty-nine. The round’s slowness was also its spacing, and the record that beat it gave the spacing up.