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>It seems there is no clear definition of "Fractal" to begin with. Yes there is: A fractal is a set for which the Hausdorff Besicovitch dimension strictly exc
by nabla9 2y ago
>It seems there is no clear definition of "Fractal" to begin with.
Yes there is: A fractal is a set for which the Hausdorff Besicovitch
dimension strictly exceeds the topological dimension.
(topological dimension means Lebesgue covering dimension)
For example: Mandelbrot set boundary has Hausdorff dimension 2 and topological dimension 1.
- emblaegh 2y agoThe Sierpinski pyramid has Hausdorff dimension 2 but topological dimension 3, or am I mistaken?
- nabla9 2y agoIts Hausdorff dimension is something like ~1.6 and topological dimension is 1.
- emblaegh 2y agoPyramid, not triangle. Maybe the topological dimension of that is not 3, I don't know, I'm not familiar with the concept. But the fractal dimension is definitely 2.
- nabla9 2y agoAh. Topological dimension is definitely not 3. It's almost certainly just 1. Hausdorff dimension probably 2 or 2ish.
- arutar 2y agoThis is a reasonable candidate for a notion of a "Fractal" but these days it is generally accepted that there is no universal definition. For instance, this definition would not include objects such as the Devil's staircase [1] or more generally images of the unit interval under a continuous monotonically increasing function. Some more exposition about attempts to rigorously define the notion of a fractal can be find in the introduction to Kenneth Falconer's book [2] [1] https://en.wikipedia.org/wiki/Cantor_function https://en.wikipedia.org/wiki/Cantor_function [2] https://zbmath.org/1285.28011 https://zbmath.org/1285.28011 Edit: Fixed incorrect zbmath link.
- dbqpdb 2y agoAnother issue is that the conventional usage of the term 'fractal' implies some degree of 'self-similarity' across scale, which is a very nebulous concept and is not at all captured by defining the term with respect to sets and various measures.