2: What is a fractal? What are some examples of fractals?
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This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.
2: What is a fractal? What are some examples of fractals?
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. Fractals are generally self-similar
and independent of scale.
There are many mathematical structures that are fractals; e.g. Sierpinski
triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz
attractor. Fractals also describe many real-world objects, such as clouds,
mountains, turbulence, and coastlines, that do not correspond to simple
geometric shapes.
Benoit Mandelbrot gives a mathematical definition of a fractal as a set for
which the Hausdorff Besicovich dimension strictly exceeds the topological
dimension. However, he is not satisfied with this definition as it excludes
sets one would consider fractals.
According to Mandelbrot, who invented the word: "I coined _fractal_ from
the Latin adjective _fractus_. The corresponding Latin verb _frangere_
means "to break:" to create irregular fragents. It is therefore sensible -
and how appropriate for our needs! - that, in addition to "fragmented" (as in
_fraction_ or _refraction_), _fractus_ should also mean "irregular," both
meanings being preserved in _fragment_." (_The Fractal Geometry of
Nature_, page 4.)
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