Origami & Folding
what one sheet can do
Concept

Cubic — where it appears

A polynomial equation of degree three. A single fold can solve one, which is what puts trisecting an angle and doubling the cube inside folding's reach and leaves both of them outside the compass's.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

startend8x³ + 4x² − 4x − 1legs 1.000, 0.500, -0.500, -0.125each turn a right angle3 real rootsx = -0.900969x = -0.222521x = 0.623490each ray lands on the end to 1.1e-16the launch angle's negative tangentis the root — which is the folda right-angled bounce off two linesat once is exactly what one fold does

Where the cubic comes from

Folding solves cubics, and the usual explanation stops at the sixth axiom. The reason is older and better: a right-angled bounce along a path of coefficients is a root-finder, and one fold is exactly such a bounce.

construction · The axioms the cubic8x³ + 4x² − 4x − 1its roots-0.900969-0.2225210.6234903 real common tangentsone fold for each rootand the fold gives cos 2π/7each curve is the set of folds that puts one point on its line; a line touching both does the two at oncea compass intersects circles and gets two answers; a fold touches parabolas and gets up to three

Fifty years in the wrong language

Margherita Beloch showed in 1936 that one fold solves a general cubic. The result was correct, published, and in a mathematics journal — and the subject that needed it did not find it until 1991. The cost of a paper nobody reads is measurable, and it is most of a century.

history · Attribution nφ(n)its prime factorscompassone foldtwo at once322422542 · 2622762 · 3842 · 2962 · 31042 · 211102 · 51242 · 213122 · 2 · 31462 · 31582 · 2 · 21682 · 2 · 217162 · 2 · 2 · 21862 · 319182 · 3 · 32082 · 2 · 221122 · 2 · 322102 · 523222 · 112482 · 2 · 2the 11-gon is the first a single fold misses, and two simultaneous folds reach itthe 23-gon is the first that needs more than two, because 22 has an 11 in it

Two creases at once

The seven axioms describe what one fold can do, and the restriction to one fold is a rule somebody imposed rather than a property of paper. Allow two creases to be made simultaneously and the reachable degree rises — and the hendecagon nobody could fold becomes foldable.

construction · Multifold 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 first pointsecond pointthe sixth axiom, at its full countthe first foldthe second foldthe third foldthree folds, each carrying one point onto one line and the other point onto the other — the most any single fold can offer

Counting operations is not counting power

The catalogue of simultaneous-fold operations runs from seven to nearly ten million between one fold and five. What a construction can reach does not: each fold admits at most three lines, because two parabolas have three proper common tangents and not four, so m folds admit at most three to the m — and the largest polynomial degree they actually settle is smaller again, at twice m plus one. Three counts of the same subject, growing at three speeds.

construction · Multifold

Named alongside it

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

Beloch's foldConstructibilityField extensionThe Huzita–Hatori axiomsLill's methodMultifoldQuinticAttributionClosureCommon tangentCube rootEnumeration

All concepts