Origami & Folding
what one sheet can do
Generator

Twos and threes, and nothing else

A generator in the axioms and construction library, called 34 times across 6 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

number-tower is one function. Everything below came out of it during this build, at arguments 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.

At its defaults

Twos and threes, and nothing elseEach number with the degree of the simplest rational equation it satisfies. A fold reaches a number exactly when that degree is a product of twos and threes — a compass supplies the twos and one fold supplies the threes — so a fifth root is out of reach at any number of folds, and saying so needs no new argument once the reachable set is known to be a field.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

view: "degrees", show: "polycost"

What each polygon costs, rather than whether it is possibleFor every regular polygon up to 40 sides that a folded construction reaches, the number of extension steps the shortest tower to it takes. The polygons a compass also reaches are marked, and they include the most expensive ones on the list while the cheapest include several a compass cannot draw at all.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

view: "degrees", show: "polygons"

Every polygon, by what its tower costsFor every regular polygon up to 40 sides: the degree of the equation its cosine satisfies, whether a folded construction reaches it, how many extension steps the shortest tower takes, whether a compass reaches it, and the factorisation the last two columns come from.every regular polygon to 40 sides, by degree and by costthe degree is Euler's totient of n, halved; the cost is how many twos and threes it is made ofsidesdegreea fold reachesstepsa compass reachesmade of31yes0yes2^0 · 3^041yes0yes2^0 · 3^052yes1yes2^1 · 3^061yes0yes2^0 · 3^073yes1no2^0 · 3^182yes1yes2^1 · 3^093yes1no2^0 · 3^1102yes1yes2^1 · 3^0115no—nonot twos and threes122yes1yes2^1 · 3^0136yes2no2^1 · 3^1143yes1no2^0 · 3^1154yes2yes2^2 · 3^0164yes2yes2^2 · 3^0178yes3yes2^3 · 3^0183yes1no2^0 · 3^1199yes2no2^0 · 3^2204yes2yes2^2 · 3^0216yes2no2^1 · 3^1225no—nonot twos and threes2311no—nonot twos and threes244yes2yes2^2 · 3^02510no—nonot twos and threes266yes2no2^1 · 3^1279yes2no2^0 · 3^2286yes2no2^1 · 3^12914no—nonot twos and threes304yes2yes2^2 · 3^03115no—nonot twos and threes328yes3yes2^3 · 3^03310no—nonot twos and threes348yes3yes2^3 · 3^03512yes3no2^2 · 3^1366yes2no2^1 · 3^13718yes3no2^1 · 3^2389yes2no2^0 · 3^23912yes3no2^2 · 3^1408yes3yes2^3 · 3^0the heptagon costs one step and the 17-gon three, and only the 17-gon is a compass construction

view: "degrees", show: "counts"

How many degrees there are to reachFor each decade, how many degrees of algebraic equation a folded construction can settle, how many a compass and straightedge can, the ratio between them, and the share of all degrees the first is. The ratio rises at every decade and the share falls at every decade.how many degrees each instrument settles, and what share of all degrees that isthe third column is the area of the triangle the lattice points sit in, computed from the logarithms aloneup toa foldthe trianglea compassratioshare of all1075.941.7570.0%1002018.472.8620.0%1,0004037.9104.004.00%10,0006764.4144.790.670%100,00010197.8175.940.101%1,000,000142138.2207.100.0142%one count grows as the square of a logarithm and the other as the logarithm, so the ratio rises without bound and both shares fall to nothing

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

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.

Every generator · The axioms and construction field · The patterns a reader can fold