Concept

Julia set

The boundary between the starting points whose orbits under a fixed polynomial map escape to infinity and those whose orbits stay bounded. For z² + c it is connected exactly when c lies in the Mandelbrot set, and otherwise it is a dust of points.

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

A Julia set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region.

One c, one picture

The same iteration, with the parameter held still and the starting point varied instead. Every complex number gets its own picture, and moving the parameter a hair can shatter it into dust.

dynamics · Complex numbers $Every fraction has a bulb. The cardioid with 277 computed bulbs of period 2 to 30; the bulbs of period up to five labelled 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5.$

The fraction written on every bulb

Round the edge of the Mandelbrot set's cardioid hangs one bulb for every fraction between nought and one. The fraction is the bulb's address, its period, and the turn its cycle makes round a fixed point, all at once, and it sets the bulb's size as well: sin(πp/q)/q², which 245 computed bulbs obey to within fourteen per cent.

algebra · Complex numbers

Named alongside it

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

Mandelbrot set Basin of attraction Complex numbers Connectedness Escape-time Farey sequence Fractal Iteration Mediant Orbit Periodic orbit Roots of unity

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