Matpic
mathematics, in pictures
Concept

Arcsine distribution

The probability law with density proportional to 1/√(x(1 − x)) on an interval, piling up at both ends. It is where a steadily turning angle spends its time when only its sine is recorded, and it governs the last lead change in a fair coin game.

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

Where a long orbit of the logistic map spends its time. A histogram of 60000 iterates in 32 bins, with the density the map preserves drawn over it as the exact share each bin should hold.

The histogram an orbit leaves

When no single step of an orbit is worth reporting, what is left is where it spends its time. That distribution is not uniform, it does not depend on where the orbit started, and it can be computed in closed form.

dynamics · Iteration Lissajous figures for every coprime pair of frequencies up to 4. A 4 by 4 grid of Lissajous figures x = sin(pt + 0.3), y = sin(qt), with the crossing count 2pq − p − q under each and the non-coprime pairs left blank.

When two circular motions come home

Drive a point across with one sine wave and up and down with another. If the two frequencies are in a whole-number ratio the point retraces a closed figure whose crossings can be counted in advance — 2pq − p − q of them — and if they are not, it never comes back and fills the square, spending twenty times longer in the corners than in the middle.

analysis · Circular functions

Named alongside it

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

ConjugacyCounting argumentDense orbitEquidistributionErgodicityInvariant densityIrrational rotationIterationMeasurePeriodicitySineSpace average

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