There are several ways of defining a fractalMathworldPlanetmath, 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 n with Hausdorff dimensionMathworldPlanetmath 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 snowflakeMathworldPlanetmath and the Mandelbrot setMathworldPlanetmath (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 n which can be covered by copies of itself using a set of (usually two or more) transformationMathworldPlanetmath mappings. Another way to say this would be “an object with a discrete approximate scalingMathworldPlanetmath symmetryMathworldPlanetmathPlanetmath”.

See also the discussion near the end of the entry Hausdorff dimension (

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