# Mandelbrot set

The Mandelbrot set is a subset of the complex numbers such that iteration of [math]f_c(z) = z^2 + c[/math] remains bounded.

[math]M = \{ c \in \mathbb{C} : f_c^n(0) \not \to \infty \text{ as } n \to \infty \}[/math]

## Escape-time

The exterior of the Mandelbrot set can be coloured using escape-time methods, which reveals intricate shapes that a binary membership colouring misses.

It can be proven that [math]2[/math] is a sufficiently large escape radius.

## Distance estimation

Analytic distance estimation formulas (both exterior and interior) exist for the Mandelbrot set, and also reveal the fine structure of the filaments.

### Pictures

- Mandelbrot set (detail).png
Detail of the Mandelbrot set, showing approximate self-similarity: a

minibrot

decorated with filaments

## Multibrot sets

[math]f_{d,c}(z) = z^d + c[/math]

## See also

Original Page by Claude