The Mandelbrot Set
The endless world of a simple formula
Summary
z<sub>n+1</sub> = z<sub>n</sub>² + c: a fractal of endless complexity from a formula this simple. For every pixel, run this iteration and measure whether it escapes. The boundary holds infinite detail; every zoom reveals new mini-Mandelbrots.
In depth
When Benoît Mandelbrot studied this set in 1980, computer graphics were new, and seeing an image of the set for the first time astonished the mathematical world. The definition is strikingly simple: for a complex number c, start from z₀ = 0 and run the iteration z<sub>n+1</sub> = z<sub>n</sub>² + c. If the iteration stays bounded (for example, |z| never exceeds 2), c is in the Mandelbrot set. If it escapes, c is outside the set, and how quickly it escapes is shown with colour. The boundary of the set is fractal (its Hausdorff dimension is 2): an infinitely long boundary around a finite area. Even more surprising: wherever you look along the boundary, you find tiny copies of the whole set (self-similarity). Computer zoom has its limits, but in theory you could keep going deeper forever; new mathematical landscapes appear at every level.
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