Matpic
mathematics, in pictures
Concept

Escape-time

The number of steps an orbit takes to leave a bounded region, used to shade the plane around a set. It is what colours a picture of the Mandelbrot set or of a Newton basin, and the boundary is where the count stops being stable.

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

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

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

The shape in every picture of itself

One line of arithmetic, repeated, with a single complex number as its only input. Sort the numbers by whether the result stays bounded and the boundary between the two answers is the most complicated object anyone draws from a rule this short.

dynamics · Complex numbers 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

Named alongside it

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

Complex numbersConnectednessFractalIterationMandelbrot setOrbitBasin of attractionBoundednessJulia setSelf-similarity

All concepts