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mathematics, in pictures
Concept

Mandelbrot set

The parameters for which the orbit of zero under squaring-and-adding stays bounded. It is connected, its boundary has dimension two, and which parameters lie in it is decided by an escape-time test.

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

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 The logistic map applied 6 times at 3.5, 3.83, 4, and its folds. Side-by-side graphs of the logistic map composed with itself, one per parameter, each labelled with how many monotone pieces it has.

The folds that measure chaos

Apply the logistic map six times and its graph goes up and down 38 times at r = 3.5 and 64 times at r = 4. How fast that number of folds multiplies with each further step is the map's topological entropy: zero through the whole cascade of period doublings, log of the golden ratio in the window of three, log 2 at the top — and it never decreases as r rises.

dynamics · Symbolic dynamics

Named alongside it

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

Complex numbersConnectednessEscape-timeFractalIterationOrbitBasin of attractionBoundednessChaosGolden ratioJulia setLogistic map

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