# fractal

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 ${\mathbb{R}}^{n}$ 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 (http://planetmath.org/HausdorffDimension).

Title | fractal |
---|---|

Canonical name | Fractal |

Date of creation | 2013-03-22 12:41:51 |

Last modified on | 2013-03-22 12:41:51 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 17 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 28A80 |

Related topic | MengerSponge |