Mandelbrot set

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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

Multibrot sets

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

See also


Original Page by Claude