Origami & Folding
what one sheet can do
Concept

Origami number — where it appears

A length a folder can mark on a sheet by a sequence of axiom folds. The set of them is closed under addition, subtraction, multiplication, division, square roots and cube roots, which makes it a field and makes constructions composable — and a number is in it exactly when the degree of its minimal equation is a product of twos and threes.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

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 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 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 after one round from a bare square, the 80 new folds a second round offers, by what each adds alonenine references and twelve lines on the sheet; a fold adds a reference wherever it crosses a line already there, on the paper, somewhere newadds 61620 per cent of the menuadds 545 per cent of the menuadds 4810 per cent of the menuadds 3810 per cent of the menuadds 21215 per cent of the menuadds 145 per cent of the menuadds 02835 per cent of the menuthe average fold on the menu adds 2.50; 16 folds tie for the most, 6; of the 28 that add nothing, 28 never cross the paper at all

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.

construction · Origami numbers references on the paper, fold by fold, choosing from a growing menu and from the round's ownthe bound is what k folds can make at most, whatever they are; the growing menu's curve lies on it020040060002040folds madereferences on the paperthe round, 565growing menuthe boundthe round's menuthe round's 565 references take its own menu 52 folds; a growing menu passes 565 at fold 29, and no 28 folds of any kind can

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.

construction · Origami numbers

Named alongside it

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

The axiomsField extensionReachable setReference pointConstructionCrossingCube rootThe Huzita–Hatori axiomsClosureConstructible numberConstructible polygonCounting

All concepts