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Concept

Cubic — where it appears

An equation of degree three, whose roots are the numbers a compass and straightedge cannot reach and an angle trisector can. Adding any instrument that solves one — a marked ruler, a drawn conic, a paper fold — extends the reachable numbers by exactly that degree.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

An angle of 60° cut in three with one mark. A circle with a marked point on it, a straightedge laid through that point so the segment between the extended diameter and the circle equals the radius, and the third-angle it makes.

The mark that changes what is reachable

Two thousand years of failure to trisect an angle with compass and straightedge was failure at a stated set of operations. Scratch two marks on the straightedge and Archimedes trisects any angle in four steps — because the new operation solves a cubic, and the old ones could only ever solve quadratics.

computation · Neusis Which polygons two instrument sets reach, up to 24. A strip of the polygons from 3 to 24 sides, each marked according to whether compass and straightedge reach it and whether a conic or a trisector does, with the degree of its cosine beneath.

Two instruments with one reach

Allow every conic to be drawn at will, or allow an angle to be cut in three. The two permissions look nothing alike and reach exactly the same numbers — because what an operation buys is a degree, and both of these buy three.

computation · Neusis The polynomial the 11-sided polygon needs. The minimal polynomial of twice the cosine of the central angle of an 11-sided polygon, with its degree, the rational root test applied to it, and whether that degree is reachable by cubic steps.

A quintic a sliding mark reaches

The eleven-sided polygon needs a number of degree five, and five is not a product of twos and threes — so no conic and no angle trisector reaches it. A ruler with two scratches does, which places the marked ruler strictly above the conics and leaves its exact reach unknown.

computation · Neusis Every quadratic is a point. The plane of monic quadratics x² + px + q with p across and q up. The parabola q = p²/4 divides it: the region below, shaded, holds the equations with two real roots, the curve itself the ones with a repeated root, and the region above the ones with none. 5 equations are marked and labelled.

Where two roots run into each other

Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.

algebra · Completing the square The first 600 terms of the continued fraction of ∛2. Partial quotients of ∛2 computed exactly: 1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3 …; the largest in the first 600 is 7451.

A cube root that looks like chance

The continued fraction of √61 repeats after eleven terms, and every square root's must. The cube root of 2 obeys no such law. Computed exactly, by a method of Lagrange's that never leaves the whole numbers, its terms run 1, 3, 1, 5, 1, 1, 4 … and throw up a 534 at the thirty-fifth place and a 7,451 at the 571st — and in every statistic anyone has measured, they look like the terms of a number picked at random.

number · Pell

Named alongside it

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

Constructible numberDegreeField extensionMarked straightedgeNeusisOperation setTrisectionAlgebraic numberAlgorithmCompleting the squareContinued fractionsDiscriminant

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