This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Zooming into the boundary of the Mandelbrot set
(Note that the colored sections of the image are not actually part of the Mandelbrot Set, but rather they are based on how quickly the function that produces it diverges.) Mandelbrot set with 12 encirclements Mandelbrot set at the cardioid left boundary The Mandelbrot set: its boundary is a fractal curve with Hausdorff dimension 2.
For other uses, see Fractal (disambiguation).
