Origami & Folding
what one sheet can do
Concept

Field extension — where it appears

Enlarging a set of numbers by adjoining a root of an equation. The reach of a construction tool is a statement about which extensions it can climb, which is why constructibility questions have algebraic answers.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

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 compassfolding3242546276the first gap8496104111012413121461581681716186191820821122210232224825202612n across the top, φ(n) underneathcompass: φ(n) a power of two — Gauss, and Wantzel's proof that nothing else worksfolding: φ(n) with no factor above three, because one fold solves a cubic

The heptagon a compass cannot reach

Which regular polygons a tool can build is a condition on a single number. The compass needs it to be a power of two; a fold needs only that it has no factor above three — and seven is the first place the two answers differ.

construction · Constructible polygons startend8x³ + 4x² − 4x − 1legs 1.000, 0.500, -0.500, -0.125each turn a right angle3 real rootsx = -0.900969x = -0.222521x = 0.623490each ray lands on the end to 1.1e-16the launch angle's negative tangentis the root — which is the folda right-angled bounce off two linesat once is exactly what one fold does

Where the cubic comes from

Folding solves cubics, and the usual explanation stops at the sixth axiom. The reason is older and better: a right-angled bounce along a path of coefficients is a root-finder, and one fold is exactly such a bounce.

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 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 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 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 how many extension steps the shortest tower to each polygon takesa polygon of n sides needs the degree of two cosine of a turn over n, which is Euler's totient halved3 sides0degree 1 · square roots only, so a compass reaches it4 sides0degree 1 · square roots only, so a compass reaches it5 sides1degree 2 · square roots only, so a compass reaches it6 sides0degree 1 · square roots only, so a compass reaches it7 sides1degree 3 · 1 cube root8 sides1degree 2 · square roots only, so a compass reaches it9 sides1degree 3 · 1 cube root10 sides1degree 2 · square roots only, so a compass reaches it12 sides1degree 2 · square roots only, so a compass reaches it13 sides2degree 6 · 1 square root and 1 cube root14 sides1degree 3 · 1 cube root15 sides2degree 4 · square roots only, so a compass reaches it16 sides2degree 4 · square roots only, so a compass reaches it17 sides3degree 8 · square roots only, so a compass reaches it18 sides1degree 3 · 1 cube root19 sides2degree 9 · 2 cube roots20 sides2degree 4 · square roots only, so a compass reaches it21 sides2degree 6 · 1 square root and 1 cube root24 sides2degree 4 · square roots only, so a compass reaches it26 sides2degree 6 · 1 square root and 1 cube root27 sides2degree 9 · 2 cube roots28 sides2degree 6 · 1 square root and 1 cube root30 sides2degree 4 · square roots only, so a compass reaches it32 sides3degree 8 · square roots only, so a compass reaches it34 sides3degree 8 · square roots only, so a compass reaches it35 sides3degree 12 · 2 square roots and 1 cube root36 sides2degree 6 · 1 square root and 1 cube root37 sides3degree 18 · 1 square root and 2 cube roots38 sides2degree 9 · 2 cube roots39 sides3degree 12 · 2 square roots and 1 cube root40 sides3degree 8 · square roots only, so a compass reaches itthe pale bars are the polygons a compass reaches, and they are not the cheap ones — 4 of the one-step polygons need a cube root

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.

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.

Constructible numberCube rootOrigami numberReachable setThe axiomsTotientClosureConstructible polygonCubicFermat primesPierpont primesBeloch's fold

All concepts