Fractals
Geometric shapes that are self-similar at every scale. The boundary between order and chaos — infinite complexity from a simple rule iterated endlessly.
Mandelbrot set
The set of complex numbers c for which the iteration zn+1 = zn² + c does not diverge to infinity, starting from z₀ = 0. Colour encodes how quickly each point escapes.
The iteration
``` z₀ = 0 z₁ = 0² + c = c z₂ = c² + c z₃ = (c² + c)² + c ... ```If |zn| stays ≤ 2 after n iterations, c is (probably) in the set.
Sierpiński triangle
A triangle subdivided recursively — remove the central triangle, repeat for each remaining triangle, forever. Fractal dimension: log(3)/log(2) ≈ 1.585.
Julia sets
For each fixed c, the Julia set is the boundary of points that don't escape under z → z² + c. Every point in the Mandelbrot set corresponds to a connected Julia set; outside → disconnected (Cantor dust).