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There are several ways of defining a fractal, and a reader will need to reference their source to see which definition is being used.
Perhaps the simplest definition is to define a fractal to be a subset of $\mathbb{R}^n$ with Hausdorff dimension greater than its Lebesgue covering dimension. It is worth noting that typically (but not always), fractals have non-integer Hausdorff dimension. See, for example, the Koch snowflake and the Mandelbrot set (named after Benoit Mandelbrot, who also coined the term ``fractal'' for these objects).
A looser definition of a fractal is a ``self-similar object''. That is, a subset or $\R^n$ which can be covered by copies of itself using a set of (usually two or more) transformation mappings. Another way to say this would be ``an object with a discrete approximate scaling symmetry''.
See also the discussion near the end of the entry Hausdorff dimension.
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"fractal" is owned by mathcam. [ full author list (3) | owner history (2) ]
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Cross-references: near, mappings, transformation, objects, term, Benoit Mandelbrot, Mandelbrot set, Koch snowflake, dimension, covering, Hausdorff dimension, subset, source, reference
There are 13 references to this entry.
This is version 14 of fractal, born on 2002-05-31, modified 2006-07-07.
Object id is 2979, canonical name is Fractal.
Accessed 6004 times total.
Classification:
| AMS MSC: | 28A80 (Measure and integration :: Classical measure theory :: Fractals) |
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Pending Errata and Addenda
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