formally defined mandelbrot summarized beautiful damn hard increasingly useful 's fractals formally mandelbrot defined ''fractal'' follows fractal definition set hausdorff besicovitch dimension strictly exceeds topological dimension later seeing restrictive simplified expanded definition fractal shape made parts similar whole way still later mandelbrot proposed use ''fractal'' without pedantic definition use ''fractal dimension'' generic term applicable ''all'' variants consensus among mathematicians theoretical
Formal science Mathematics Applied mathematics Dynamical systems Fractal geometry