Fractal

A fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually with a fractal dimension that strictly exceeds its topological dimension. Many fractals look similar across different scales, a property called self-similarity, and if the replication is exactly identical at every scale, as in the Menger sponge, the shape is called affine self-similar; fractal geometry connects to measure theory through the Hausdorff dimension. Fractals differ from ordinary geometric figures in how they scale: doubling the edge lengths of a filled polygon multiplies its area by four, two raised to its dimension of two, while doubling a fractal's one-dimensional lengths scales its spatial content by a power that need not be an integer, called the fractal dimension. Starting from notions of recursion in the seventeenth century, fractals developed through the nineteenth-century study of continuous but nowhere-differentiable functions by Bernard Bolzano, Bernhard Riemann and Karl Weierstrass, to the coining of the word fractal by Benoit Mandelbrot in the twentieth century, who in 1982 defined a fractal as a set whose Hausdorff-Besicovitch dimension strictly exceeds its topological dimension before later broadening the definition to a rough or fragmented shape splittable into parts that are each an approximate reduced copy of the whole.