Origami & Folding
what one sheet can do
Concept

Pierpont primes — where it appears

A prime one more than a product of powers of two and three. A regular polygon is foldable exactly when its side count is built from these, which is the folding analogue of the compass's Fermat primes.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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 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 sidesshare of the sheet the largest one usestilt346.4%15.00°4100.0%45.00°567.4%9.00°669.6%15.00°772.9%6.43°882.8%22.50°975.1%5.00°1075.3%9.00°1176.3%4.09°1280.4%15.00°1376.9%17.31°1476.9%6.43°1577.3%3.00°1679.6%11.25°the 8-sided polygon is the peak, and every one of the 8 polygons after it does worse

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.

construction · Optimal constructions 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 which tool the best polygon on each sheet needseach polygon at its own best rotation; the rank counts every polygon from three sides to twenty-foursheetbest polygonbest a compass buildsbest only a fold buildsbest no fold buildssquare8-gon · 82.8%8-gon21-gon · 7th · 77.9%23-gon · 6th11 : 108-gon · 75.3%8-gon14-gon · 4th · 72.6%22-gon · 9th6 : 56-gon · 72.2%6-gon14-gon · 5th · 66.6%22-gon · 9thA series6-gon · 61.2%6-gon14-gon · 5th · 56.5%22-gon · 9th3 : 26-gon · 57.7%6-gon14-gon · 5th · 53.3%22-gon · 9th2 : 16-gon · 43.3%6-gon14-gon · 5th · 39.9%22-gon · 9th3 : 16-gon · 28.9%6-gon14-gon · 5th · 26.6%22-gon · 9ththe square itself is left out; a fold builds an n-gon when the totient of n has no prime factor above three

Every even polygon beats every odd one

Crossed with what each tool can build, the census of the largest regular polygon a sheet holds gives the same verdict on every proportion from a square to three to one: the best polygon is one a compass already builds, and the best polygon only a fold can build places fourth at best. The ranking itself stops moving at a proportion of 1.1284, where the hexagon overtakes the octagon. On every longer sheet only the short side holds a polygon, each polygon's share is a fixed constant divided by the length, and the constant — its area over the square of its least width — comes down to the circle's π⁄4 for even polygons and climbs up to it for odd ones. So every even polygon beats every odd one.

construction · Optimal constructions

Named alongside it

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

TotientFermat primesConstructibilityConstructible numberConstructible polygonField extensionInscribed polygonOptimalityCompletenessReachable setSheet shapeSymmetry

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