Origami & Folding
what one sheet can do
Concept

Beloch's fold — where it appears

The fold that aligns two points onto two lines at once, solving a cubic. Margherita Beloch published it in 1936, fifty-five years before the axiom list it belongs to was assembled.

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

Also named here as lill's method — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

CubicLill's methodAttributionCommon tangentField extensionThe Huzita–Hatori axiomsParabolaPriority

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