Matpic
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 5 essays across 2 fields — 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 The Mandelbrot set counted cell by cell. A 96-cell-wide grid over the Mandelbrot set: 2034 cells certainly inside (1.3794), 290 undecided after 60 steps (0.1967).

The area of the Mandelbrot set, from two sides

The Mandelbrot set has an area, and nobody knows it. Counting grid cells closes in on 1.5066 from both sides, but every count judges a cell by a single point, and points on the boundary never declare themselves. An exact formula exists — π times one minus a sum over the coefficients of the map onto the set's outside — and every partial sum is a guaranteed upper bound; after sixteen thousand coefficients the bound is still 1.786, and at the rate it falls it would need something like 10²⁵ more.

algebra · Complex numbers Where the free critical point goes, for every cubic in a family. The parameter plane of the cubics (z − 1)(z² + z + a) from −2 to 3 across and −2.5i to 2.5i up, coloured by the root Newton's method takes 0 to, with small dark regions where 0 is trapped in a cycle.

Small Mandelbrot sets among the cubics

For most cubics Newton's method finds a root from almost every start. For some it does not: a whole region of starting points falls into a cycle and never finishes. Which cubics are the bad ones is decided by a single starting point — the one critical point that is not a root — and when the cubics are laid out in a plane and coloured by where that point goes, the bad ones form small, exact copies of the Mandelbrot set.

dynamics · Newton basins

Named alongside it

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

Basin of attractionComplex numbersConnectednessEscape-timeFractalIterationOrbitPeriod-doublingPeriodic orbitAreaBoundednessChaos

All concepts