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>There is a common misconception, repeated here, that fractals are defined by their self-similarity. This is not a misconception. Self-similarity at all length
by oarabbus_ 7y ago
>There is a common misconception, repeated here, that fractals are defined by their self-similarity.
This is not a misconception. Self-similarity at all length scales is indeed a defining factor of fractals.
From that fact it _does_ follow that _most_ fractals have a non-integer fractal dimension. However it is absolutely not true that fractals have non-integral dimensionalities; for example the most famous fractal of them all, the Mandelbrot set, has an integer Hausdorff dimension of 2.
[0] https://arxiv.org/abs/math/9201282 https://arxiv.org/abs/math/9201282
- JadeNB 7y agoThere'll doubtless be more argument about this, but I think that, before disagreeing with you, people should note that 'fractal' is not a disembodied term, but the creation of one person who has the right to define it—but who unfortunately can't make up his mind. Per Wikipedia (https://en.wikipedia.org/wiki/Fractal https://en.wikipedia.org/wiki/Fractal): > Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot settled on this use of the language: "… to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". (Note that the mention of dimension only requires one notion of dimension to exceed another, not that either dimension be non-integral!)