Origami & Folding
what one sheet can do
Concept

Quadratic — where it appears

A polynomial equation of degree two. It is what a compass and straightedge can solve, and the whole difference between the two classical tools is that a fold reaches one degree further than this.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

tilted 15.00°side 1.0352846.41% of the sheetand the same answer twicea corner construction givesside 1.03528from a quadratic, sharing no codethe width maximisation and the closed form agree to nine figures

The largest triangle in a square

The biggest equilateral triangle a square sheet holds is tilted by exactly fifteen degrees and uses 46.4% of the paper. Both numbers come out of a quadratic — which means a compass reaches this optimum too, and folding's advantage is not needed here at all.

construction · Optimal constructions 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

Named alongside it

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

AxiomClosureConstructibilityCube rootCubicField extensionInscribed polygonOptimalityOrigami numberReachable setSupport widthThe axioms

All concepts