According to
Benoit Mandelbrot, "A
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
is by definition a set for which the
Hausdorff-Besicovitch dimension strictly exceeds the
topological dimension."
Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.
Deterministic fractals
Random and natural fractals
See also
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Fractal dimension
In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured.
It ...
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Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
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Scale invariance
Notes and references
Further reading
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External links
The fractals on MathworldOther fractals on Paul Bourke's websiteFractals on mathcurve.com*
ttps://web.archive.org/web/20060923100014/http://library.thinkquest.org/26242/full/index.html Fractals unleashedIFStile - software that computes the dimension of the boundary of self-affine tiles
{{DEFAULTSORT:Fractals By Hausdorff Dimension
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
Mathematics-related lists