Origami & Folding
what one sheet can do
Concept

Constructible number — where it appears

A number a stated set of operations can produce. Compass and straightedge reach a tower of square roots, folding reaches cube roots as well, and the two are separated by exactly one degree.

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

axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

One fold at a time, and there are exactly seven of them

A fold is specified by bringing points and lines into coincidence. There are seven ways to do that, the list is provably complete, and one of the seven does something no compass can.

construction · The axioms hh/263°42.0°21.0°The foldone fold, made so that the cornerreaches the lower crease at the sameinstant as the point above it reachesthe ray — two conditions, one creaseWhat it produces63° divided into21.00° and 42.00°a third of 63° is 21.00°— measured off the fold, not drawnA cubic, so no compass reaches it.mountainvalley

Folding beats the compass, by exactly one degree

Straightedge and compass solve quadratics. A single fold solves cubics. That one-step difference settles two problems Greek geometry could not, and leaves a third exactly as impossible as it was.

construction · The axioms pp − 1, factoredcompassfoldingthe regular p-gon32both tools52 · 2both tools72 · 3folding only112 · 5neither132 · 2 · 3folding only172 · 2 · 2 · 2both tools192 · 3 · 3folding only232 · 11neither292 · 2 · 7neither312 · 3 · 5neither372 · 2 · 3 · 3folding only11 is the first prime out of a fold's reach — 11 − 1 = 2 · 5the factor of five is the obstruction, and no arrangement of folds produces onea compass needs a power of two; a fold needs nothing above three

The eleven-sided one nobody can fold

Folding reaches the heptagon, which a compass cannot. It does not reach the hendecagon, and the obstruction is a single prime factor: ten has a five in it, a fold solves cubics, and no arrangement of cubics produces a five.

construction · Constructible polygons the bar is how many folds the alignment names, averaged over the trialsa1 — the fold through two points1.000.0% none · 0.0% twoa2 — one point onto another1.000.0% none · 0.0% twoa3 — one line onto another2.000.0% none · 100.0% twoa4 — a perpendicular through a point1.000.0% none · 0.0% twoa5 — a point onto a line, through a point1.5223.8% none · 76.2% twoan alignment with no fold is not a failed construction; it is an alignment the paper cannot make

An axiom may name no fold

The seven operations are stated about points and lines in a plane, and a plane has no edges. On a square, three of them always name exactly one fold and always land it on the paper; placing a line on a line names two, and 13.3% of the folds it specifies are creases the sheet never reaches; and placing a point on a line through a second point names two folds, one, or — 23.8% of the time — none at all.

construction · The axioms 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 the proportion at which the hexagon passes each polygonmeasured on the share curves, and computed as √3⁄2 + ½√(8K⁄3√3 − 1) with K the polygon's constantsidessearchedclosed formdegreeodd primesthe tool it needs71.0696431.069643243one fold81.1284411.1284418nonea compass91.0802961.080296123one fold101.1163331.11633316nonea compass111.0852961.085296405two folds at once121.1097491.1097494nonea compass131.0880641.088064483one fold141.1057721.105772243one fold151.0897621.08976216nonea compass161.1031871.10318716nonea compassthe sheet is one wide; each proportion is where the hexagon's share equals the polygon's, found two ways

The crossing is as hard as the polygon

Lengthen a square sheet and the largest hexagon it holds turns, pressed against all four edges, until it overtakes the polygons held by the short side alone. Every one of those overtakings happens at a proportion with a closed form, √3⁄2 + ½√(8K⁄3√3 − 1), and the number that comes out is exactly as hard to mark as the polygon being overtaken is to build. The octagon's 1.1284 is a compass number of degree eight. The heptagon's 1.0696 has degree twenty-four and needs a fold. The hendecagon's 1.0853 has degree forty and needs two folds at once.

construction · Optimal constructions

Named alongside it

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

Field extensionReachable setThe axiomsCube rootReference pointSheet shapeCompletenessFermat primesThe Huzita–Hatori axiomsPierpont primesTotientClosure

All concepts