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Geometrical entities characterized by basic patterns that are repeated
at ever decreasing sizes. For example, trees describe an approximate fractal
pattern, as the trunk divides into branches which further subdivide into
smaller branches which ultimately subdivide into twigs; at each stage of
division the pattern is a smaller version of the original.
Fractals are not able to fill spaces, and hence are described as having
fractional dimensions. They were devised in 1967 by
Benoit
Mandelbrot, during a study of the length of the coastline of Britain.
They are relevant to any system involving self-similarity repeated on diminishing
scales, such as in the study of chaos, fork ligihtning,
or the movement of oil through porous rock. They are also used in
computer
graphics.
The most famous fractal is the Mandelbrot set, the set of complex
numbers that describes the behaviour of points z under the iterative process
f(z) = z2 +c for different values of c. Its fractal nature
illustrates the sensitive way in which iterations depend on the choice of
c. It also occurs in other iterative processes.
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