Chebyshev polynomial
Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.
A solvable chaos of every degree
The logistic map at four is chaotic and, through a change of coordinates, completely solvable: its orbits are cosines of doubling angles. The trick is not a one-off. For every whole number n there is a polynomial of degree n that multiplies angles by n instead of 2, and every one of them is exactly as solvable, has exactly nᵏ points of period k, and preserves the same distribution — and any two of them commute, which almost no two polynomials do.
The error that keeps coming back to its worst
Judge a polynomial by its largest error on an interval and there is exactly one best one of each degree. It is recognised without comparing it to anything else — its error rises to the same largest size, alternately above and below, one more time than there are coefficients.
An ellipse, not a disc
A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.
Two constants that do not walk at random
Pollard's factoring method trusts x² + c modulo a prime to repeat as soon as a random function would, after about √p steps. For c = 1 and c = 3 it does. For c = 0 and c = −2 it runs twelve to eighteen times longer on average, with almost no tail and enormous cycles, because those two maps are multiplication in disguise and their cycles are set by the order of 2 rather than by chance. Every other constant walks at random — including in the one respect in which x² + c is plainly not random, that it is two-to-one.
The boards on which every light goes out
In Lights Out, pressing a light toggles it and its neighbours, and the goal is to turn every light off. Over the field with two elements the puzzle is a system of linear equations, and whether every pattern can be cleared depends only on whether one matrix is invertible. On a 3 × 3 board it is; on 5 × 5 a quarter of patterns can be cleared; on 39 × 39 one in four billion. Which sizes fail is decided by a common factor of two polynomials.
The cosines of the powers of two
Start at cos 1 and apply x ↦ 2x² − 1 over and over. The map is chaotic and exactly solvable: the k-th point is cos(2ᵏ), the cosine of two to the k radians. Iterate it in floating point and the computed orbit leaves the true one after 51 steps. Read the binary digits of 1/(2π) instead and the true orbit can be followed for as long as digits can be computed — here 262,080 steps — and at that length it is as fair as any orbit can look. Whether it stays fair is a question about those digits, and nobody can answer it.
Named alongside it
The objects these essays reach for when they reach for this one.
ApproximationInvariant measureBinary expansionBirthday problemBoundChaosCollisionComplex planeConjugacyConvergenceEllipseErgodic theorem