Origami & Folding
what one sheet can do
Generator

Which primes each tool reaches

A generator in the axioms and construction library, called 20 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.

pierpont-primes 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

Which primes each tool reachesFor each small prime p, the factorisation of p − 1 and whether it clears the bar each tool sets. A compass needs p − 1 to be a power of two; a fold needs it to have no prime factor above three. Eleven is the first prime a fold cannot reach, and it is the first place folding runs out.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

sub: "density", upTo: 400, marks: [10, 50, 100, 200, 400]

How many polygons each tool reachesThe number of regular polygons with at most n sides that each tool can construct, counted to any bound by arithmetic rather than read off a list. A compass needs n's odd part to be a product of distinct Fermat primes and only five are known; a fold needs distinct Pierpont primes and those keep arriving. The heptagon is one point on the gap, and the gap widens at every bound.050100150200250300350400020406080100120140160sides, up to npolygons reachablea fold — 155a compass — 40both sets computed by division to 400 · 4 Fermat primes and 13 Pierpont primes below it

sub: "density", upTo: 1000, marks: [100, 400, 700, 1000]

How many polygons each tool reachesThe number of regular polygons with at most n sides that each tool can construct, counted to any bound by arithmetic rather than read off a list. A compass needs n's odd part to be a product of distinct Fermat primes and only five are known; a fold needs distinct Pierpont primes and those keep arriving. The heptagon is one point on the gap, and the gap widens at every bound.02004006008001000050100150200250300sides, up to npolygons reachablea fold — 275a compass — 52both sets computed by division to 1000 · 4 Fermat primes and 17 Pierpont primes below it

upTo: 40

Which primes each tool reachesFor each small prime p, the factorisation of p − 1 and whether it clears the bar each tool sets. A compass needs p − 1 to be a power of two; a fold needs it to have no prime factor above three. Eleven is the first prime a fold cannot reach, and it is the first place folding runs out.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

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.

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.

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.

Seven, and then twenty-two

The seven axioms are not seven useful folds somebody collected; they are the number of ways to spend a fold line's two degrees of freedom, and the count can be derived. Run the same derivation for two folds made at once and it gives twenty-two, for three fifty, for five a hundred and sixty-one — while the number of coincidences a pair of hands has to achieve in the same instant goes two, four, six, ten.

The biggest one that can also be folded

Which regular polygon uses a square sheet best, and which of them a fold can actually construct, are two questions with completely different pedigrees. Answered side by side over sixteen polygons, they turn out to agree — and the reason is that both are questions about the arithmetic of the same number.

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.

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.

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