# Mandelbrot

Complex · w = f(z, t)

`w = z² + c`

[Open in the app](https://www.wavelace.com/app#p=47) · [This page](https://www.wavelace.com/presets/mandelbrot)

### What it draws

`w = z² + c` is applied over and over with `c` held at the point being drawn, and the points whose orbits stay bounded are the Mandelbrot set. `Iterations` sets how many steps a surviving orbit has to take. `Span of z` is 2, so the square on show runs from `−2` to `2` along both axes. That holds the whole set: no `c` with `|c| > 2` stays bounded.

Starting the orbit at `z = c` instead of the textbook `z = 0` costs nothing: the first step from zero is `0² + c = c` anyway, so this is the standard orbit from its second step on.

### Landmarks

The blunt point on the right, at `c = 0.25`, is the cusp of the main cardioid, where the orbit settles onto a fixed point. In the disc attached at the left, centred on `−1` with radius `0.25`, it settles into a two-step cycle instead. Because the orbit alternates there, the hue of that disc swaps when `Iterations` changes by one. Along the real axis the set is exactly the interval from `−2` to `0.25`. The boundary between kept and escaped is as crumpled as a curve can be: Mitsuhiro Shishikura proved in 1998 that it has Hausdorff dimension 2.

### History

Pierre Fatou and Gaston Julia worked out the iteration theory around 1917 to 1920 with no way to see it. Robert Brooks and Peter Matelski printed the first crude picture of this set in 1978. Benoit Mandelbrot, with IBM's plotters, made the detailed images from 1980. Adrien Douady and John Hubbard proved the set connected in 1982 and named it after him.

### Try

- `Top` in the deck for the picture as it is usually printed, straight down on the plane.
- Raise `Iterations` toward 300: the filaments round the edge thicken and reach further out.
- Drop `Iterations` to 8 instead, and only the coarsest terraces are left.
- Type `z³ + c`: the degree-3 multibrot, with two lobes and two-fold symmetry instead of one lobe.
- Raise `Span of z` to 4: the set shrinks to the middle of a wide plain of escape terraces.

### Read more

- [Mandelbrot set](https://en.wikipedia.org/wiki/Mandelbrot_set)
- [Complex quadratic polynomial](https://en.wikipedia.org/wiki/Complex_quadratic_polynomial)
- [Julia set](https://en.wikipedia.org/wiki/Julia_set)
- [Domain coloring](https://en.wikipedia.org/wiki/Domain_coloring)