A guided tour of the Mandelbrot set—its history, the core iterative rule Z_{n+1} = Z_n^2 + C, and how escape-time coloring reveals its beauty. We explore self-similarity and Misiurewicz points, the connected boundary with its fractal dimension, and the deep connections to Julia sets. Along the way we meet the key mathematicians and consider what fractal geometry can teach us about complex systems in nature.

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