Generator

Twos and threes, and nothing else

A generator in the axioms and construction library, called 47 times across 8 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

number-tower is one function. Everything below was drawn by it when the site was last published, at settings taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

With nothing specified

`view: "chosen", show: "first"`

`view: "chosen", show: "best"`

`view: "chosen", show: "yields"`

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

7 of them have a degree that is a product of twos and threes and 2 do not, which is the line between what a fold reaches and what it does not ×4
every number named here satisfies the rational polynomial stated for it, and none of them satisfies one of lower degree with coefficients up to 6 — checked by exhaustion rather than by algebra ×3
4 of the 13 degrees are certified by Eisenstein at two and the rest by exhaustion over integer coefficients to 6 ×2
and the whole round, made at once by the closure itself, leaves the same 565 references the orders end at ×2
choosing each fold reaches half of what the whole round gives in 28 folds, and folding in random order takes 51.0 on average over 20 orders ×2
the 52 folds that are each the only second fold through some reference make every reference of the round between them, so 52 is the fewest folds the round's references need ×2
the share lost runs from 47% to 72% across 4 proportions, so what the edge costs is a fact about the shape of the sheet ×2
1 pair of these numbers put a larger degree on a shorter tower — 2^(1/9) is of higher degree than 2^(1/8) and reaches it in 2 steps against 3 ×1
11 of the 13 numbers have a degree made of twos and threes and the rest do not, so the boundary is still on the page ×1
after 29 chosen folds the paper holds 589 references and the second round 565, and only 15 are the same points — 9 of them the first round's own ×1
and the conic axiom loses more at one round — 76% — than the four linear ones lose at two, which is 72% ×1
and the eleven-sided polygon is out of reach altogether, its degree being five ×1
and the factor by which a fold beats a compass rises at every decade rather than settling, because one count grows as a logarithm squared and the other as the logarithm ×1
and the set that loses most cannot be run a second round at all: the folds it specifies outgrow the cap the closure is computed under ×1
at every share of every round, choosing each fold gets there in fewer folds than folding at random ×1
every one of the 30 chosen folds adds exactly as many references as the bound allows — the lines already through the paper's interior plus the two points where it meets the rim — which is 6 at the first fold and one more at each fold after, so after k folds the sheet holds 9 + k(k + 11)/2 ×1
every one of the six operations, applied to numbers a fold reaches, produces a number a fold reaches — which is why a construction may be built out of constructions ×1
every order of the same folds ends at the same references, so the orders differ only in how fast they get there ×1
no 28 linear folds of any kind can put the round's 565 references' worth on the paper — the most they can make is 555 — and the chooser passes 565 at fold 29, against the 52 folds the round itself needs ×1
on a square, 2 rounds put 1,440 crossings off the paper against 565 on it — the sheet's edge discards 72% of what the plane produces ×1
on the round's own menu no fold beats the bound either, and over its 52 folds the chooser falls 1082 references short of it, because the round's folds are specified through its own references and cross one another there ×1
the 100 folds on the conic menu that never cross the paper add nothing, first or last ×1
the 28 folds on the linear menu that never cross the paper add nothing, first or last ×1
the best single fold on the menu adds 6 references and the average fold 2.50, so choosing the first fold is worth 2.4 of an unchosen one ×1
the chosen folds add more as they go — 6, 7, 8, 9, 9, 10, 11, 12, 8, 9, 9, 8, 9, 10, 10, 11 — because each crosses the folds chosen before it as well as the first round's lines ×1
the crossings are counted rather than estimated: every pair of fold lines is intersected and the ones that miss the paper are tallied instead of dropped ×1
the first chosen fold crosses 6 of the first round's twelve lines on the paper, all at new points; the six it does not are two edges it misses and the 4 lines that touch the square only at a corner ×1
the heptagon costs one extension step and no compass reaches it; the seventeen-sided polygon costs three and a compass does — so the polygon a compass cannot draw is the cheaper of the two for a folder ×1
the loss is counted rather than estimated: every crossing of every pair of fold lines is formed, and the ones that miss the paper are tallied instead of dropped ×1
the menu grows from 80 folds at the first choice to 335,193 at the 30th, and at every choice at least one fold in eleven meets the bound ×1
the reachable degrees to 200 are exactly the 25 lattice points under the line, counted two ways ×1
the references the chooser makes crowd as it goes: after 29 folds the median reference is 0.014 mm from its nearest neighbour on a 150 mm sheet and 539 of 589 lie within a millimetre of another, against a median of 2.70 mm for the round's 565 ×1
the round's chooser meets the bound for its first 4 folds and never again, and no fold of the round adds more than 14 however many lines it has to cross ×1
the share of crossings that miss the paper rises with every axiom added — 0%, 0%, 57%, 76% at one round ×1
the share of degrees a fold reaches falls from 70% to 0.014% across the range drawn ×1
the tallest tower drawn is 4 steps, for 2^(1/16) at degree 16 ×1

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

A fold is worth what follows it

After one round of folds from a bare square the sheet carries nine references and twelve lines, and a second round of the four linear axioms offers eighty new folds. Made alone, the best of them adds six references and the average two and a half. Made all together they add 556 — nearly seven a fold — because a fold's yield is mostly its crossings with the folds that come after it. So the value of choosing has to be measured along an order: always making the most productive fold next reaches half the round in 28 folds where a random order takes 51, and all of it in 52, which is the fewest folds that can. The conic axiom's 305 extra folds leave the first eight choices exactly as they were.

Each new fold can cross every old one

A folder choosing each fold from the second round's fixed menu needs fifty-two folds for the round's 565 references, and no fewer. Let the menu grow — after every fold, every fold the references now on the paper specify — and the count obeys a bound that has nothing to do with axioms: a new line can cross each line already through the paper once, and the rim twice. After k folds that is at most 9 + k(k + 11)/2 references. The greedy folder meets the bound exactly at every one of thirty folds, passes 565 at the twenty-ninth, and no twenty-eight folds of any kind can. But of its 589 references only fifteen are the round's, and they are nearly useless: the median lies a seventieth of a millimetre from its nearest neighbour.

Gauss's polygon is the expensive one

Which regular polygons a fold reaches is a condition on the factorisation of Euler's totient, and every polygon that passes it also has a height — the number of extension steps the shortest tower to it takes. Read that column instead of the verdict and the field inverts: the heptagon, which no compass reaches, costs one step; the seventeen-sided polygon that made Gauss famous costs three, the most on the list; and the polygons a compass finds easy are the ones a folder pays most for.

Reachable is not cheap

The closure is what makes folding a theory rather than a bag of tricks: constructions can be built out of constructions. What that also means is that constructions have lengths and the lengths compose, so every reachable number has a height as well as a degree — the number of extension steps the shortest tower to it must take. The two orderings disagree, and a ninth root is a shorter tower than an eighth.

The axiom that reaches furthest wastes most

The field of origami numbers is defined on an unbounded plane and a folder has a square. Counted axiom set by axiom set on the same sheet, the share of crossings that land off the paper rises with every axiom added: nothing at all from the first two, fifty-seven per cent from the four linear ones at a single round, and seventy-six per cent from the conic axiom at a single round — more, in one round, than the linear four lose in two. The instrument that reaches furthest into the field delivers the smallest share of what it specifies.

The field has no edge

Origami numbers are a field on the unbounded plane and a folder has a piece of paper. A fold line runs forever; a crossing of two of them is a number in the field wherever it lands, and it is a reference somebody can put a finger on only where there is paper under it. Counted rather than assumed, two rounds of the four linear axioms on a square put seventy-two per cent of their crossings off the sheet.

The numbers a fold reaches

Folding solves cubics, which is one fact about one fold. The reason the subject has a theory rather than a bag of tricks is a second fact about all of them: the lengths a folder can mark are closed under addition, subtraction, multiplication, division, square roots and cube roots. Constructions can therefore be built out of constructions — and no tower of them ever arrives at a fifth root.

Twos and threes run out

A fold reaches a number exactly when the degree of its equation is a product of twos and threes, which sounds like a large set because it is infinite and because it is so much larger than the compass's. Counted, the reachable degrees are the lattice points under a straight line, so there are about half a log-squared of them: twenty of the first hundred, a hundred and forty-two of the first million. The share falls from a fifth to one part in seven thousand, and the factor by which folding beats the compass rises at every decade without ever settling.

All the figures · The axioms and construction field · The patterns a reader can fold