5000 x 3715 px | 42,3 x 31,5 cm | 16,7 x 12,4 inches | 300dpi
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In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole". The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning broken or fractured. A fractal as a geometric object generally has the following features: fine structure at arbitrarily small scales is too irregular to be easily described in traditional Euclidean geometric language is self-similar (at least approximatively or stochastically) has a Hausdorff dimension that is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve) has a simple and recursive definition Because they appear similar at all levels of magnification, fractals are often considered to be 'infinitely complex'. Obvious examples include clouds, mountain ranges and lightning bolts. However, not all self-similar objects are fractals — for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics. A relatively simple class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, Koch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can be deterministic (all the above) or stochastic (that is, non-deterministic). For example the trajectories of the Brownian motion in the plane have Hausdorff dimension 2. Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (see attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has the Hausdorff dimension equal to its topological dimension of 2 —but what is truly surprising is that the
