Origami & Folding
what one sheet can do
Concept

Reachable set — where it appears

The points or numbers a stated set of operations can produce from a stated start. For folding it is what the axioms build from a bare square, and its growth is much faster than a folder can use.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

the bare sheet4 references · 4 linesnothing has been foldedafter 1 fold9 references · 12 lineshalves, and nothing elseafter 2 folds565 references · 92 lineshalves, thirds, fifths — and worsea fold is an alignment, and an alignment needs something already on the paper to align565 references after 2 folds, and the count is finite however many folds are allowed

A fold needs something to align

Every axiom names things that must already be on the paper — a point to fold onto a point, a line to bring to a line. So what a folder can build is bounded by what they can refer to, and that set is finite at every depth: nine references after one fold, several hundred after two, and every one of them computable in advance.

construction · Reference points distinct fold lines this axiom specifies and no other doesaxiomafter 4 pointsafter 9 pointsafter 565 pointsA1 — through two points08121054A2 — one point onto another08142649A3 — one line onto another4564994A4 — perpendicular through a point001661distinct lines in all1292274300the four operations name 38 folds at the first round and draw 12 lines with them

What each axiom is worth

The list of seven folds is complete, and the proof of that says nothing at all about whether its members are independent or equal. Measured on a bare square, one of the four elementary axioms supplies every fold the others cannot and the other three supply nothing. Two rounds later the ranking has inverted, and the one that carried the first round is the least productive of the four.

construction · The axioms the degree of the equation, and what it is made ofnumberdegreemade ofwhere it comes from½11a fold in half√222^1the diagonal of the squareφ22^1the silver rectangle's cousin∛233^1doubling the cube2 cos(2π/7)33^1the regular heptagon∜242^2a square root of a square root∛2 · √262^1 · 3^1a product of two of them2^(1/5)5not twos and threesa fifth root2 cos(2π/11)5not twos and threesthe regular hendecagonchecked by exhaustion: no number here satisfies a rational equation of lower degree with coefficients up to 6

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.

construction · Origami numbers the bar is the largest gap between anywhere on the sheet and a referencea third fold specifies more folds than can be listed, so a sample of them is taken insteadnone of them0.089565 references, from two folds50 of them0.0763,498 references · 0.02% of the round100 of them0.0508,056 references · 0.04% of the round200 of them0.04723,480 references · 0.07% of the round400 of them0.02274,694 references · 0.15% of the round800 of them0.014270,882 references · 0.29% of the roundevery row is a lower bound on what the whole round would buy, because leaving folds out can only make the gap larger

The third fold cannot be listed

Two folds from a bare square reach five hundred and sixty-five reference points. The third round specifies three hundred and seventy-eight thousand folds, of which two hundred and seventy-four thousand are distinct — and the crossings of those with each other run to the tens of billions. The closure stops being computable at exactly the depth a folder starts working at, and what can be said instead is a bound rather than a list.

construction · Reference points degree is how far out the number is; height is what the construction costsnumberdegreesteps in the towerwhat the steps are½10 stepsa fold in half√221 stepthe diagonal of the squareφ21 stepthe silver rectangle's cousin∛231 stepdoubling the cube2 cos(2π/7)31 stepthe regular heptagon∜242 stepsa square root of a square root∛2 · √262 stepsa product of two of them2^(1/5)5no tower reaches ita fifth root2 cos(2π/11)5no tower reaches itthe regular hendecagon2^(1/8)83 stepsthree square roots2^(1/9)92 stepstwo cube roots2^(1/12)123 stepstwo squares and a cube2^(1/16)164 stepsfour square roots1 pairs invert: 2^(1/9) is of degree 9 and 2 steps, 2^(1/8) of degree 8 and 3a degree 2^a · 3^b is a steps of square root and b of cube root, so the height is a + b and nothing else

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.

construction · Origami numbers what the plane specifies, and what the paper carries2 rounds from a bare sheet, every proportion at the same rulesheeton the paperlost to the edgesquare1 × 1.000565 references72%1,440 off the paperA-series1 × 1.414114,927 references47%102,162 off the paper3 : 21 × 1.500135,281 references48%123,872 off the paperdouble square1 × 2.00044,673 references59%65,294 off the paper2 rounds of a1, a2, a3, a4 · a crossing is a reference only where the paper is · square loses the largest share, at 72%

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.

construction · Origami numbers 050100150200250300350400020406080100120140160sides, up to npolygons reachablea fold — 155a compass — 40both sets computed by division to 400 · 4 Fermat primes and 13 Pierpont primes below it

How many polygons a fold reaches

The heptagon is what the extra axiom buys and it is one polygon. What it actually buys is a density: to a thousand sides a compass reaches fifty-two regular polygons and a fold reaches two hundred and seventy-five, and the ratio between them is still widening. The compass has five usable primes in the whole of arithmetic and may use each once; a fold keeps acquiring new ones and may repeat the factor of three as often as it likes.

construction · Constructible polygons how squarely the folds that fix a reference crossand what a crossing at that angle does to an error in the foldingthe four linear axioms565 references · worst 36.9°with the conic axiom16,890 references · worst 6.3°60° to 90°error × 1.264.6%54.4%45° to 60°error × 1.418.2%21.8%30° to 45°error × 2.017.2%17.1%15° to 30°error × 3.90.0%5.2%8° to 15°error × 7.20.0%1.2%under 8°error × 14.30.0%0.3%the linear axioms bottom out at 36.9° — a multiplier of 1.67 — and the conic axiom reaches 6.3°, a multiplier of 9.12 rounds on a square · a reference priced at 1 ⁄ sin of the widest angle its own folds make · shares, so the two sets are comparable

What buys the reach costs the accuracy

A reference is a crossing of two creases, and a crossing transmits a folding error multiplied by one over the sine of the angle the creases make. Measured across the whole closure on a square, the four linear axioms never produce a crossing shallower than thirty-seven degrees — and the conic axiom, the one that sends a point onto a line and reaches the heptagon, produces crossings under seven.

construction · Constructible polygons every degree a fold reaches up to 200, as a square root count against a cube root counta point at (a, b) is the degree 2 to the a times 3 to the b, and a tower to it takes a + b steps0123456701234square rootscube roots13927812618541624123610882472164814432966419212825 of the first 200 degrees, and they are the lattice points under a line of slope minus log 2 over log 3the pale points are the degrees a compass reaches as well

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.

construction · Origami numbers what each axiom set specifies, and what of it the sheet carrieson a unit square, from its four corners and four edgesaxiomshow manyroundsfold lineson the paperoff itlosta line through two points116500.0%a line through two points126500.0%and a point onto a point218900.0%and a point onto a point22321336432.5%and the other two linear ones411291257.1%and the other two linear ones42925651,44071.8%and a point onto a line through a point51283310475.9%and a point onto a line through a point52———past the capthe conic axiom loses three quarters of its crossings at one round, and cannot be run a second

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.

construction · Origami numbers

Named alongside it

The objects these essays reach for when they reach for this one.

The axiomsReference pointConstructible numberClosureCube rootField extensionOrigami numberEnumerationThe Huzita–Hatori axiomsSheet shapeAxiomCombinatorial explosion

All concepts