I grasp how fractals can be self-similar and have other weird properties. But I don’t quite get how they can have fractional dimensionality, even though that’s the property they’re named after.
How can a shape have a dimensionality *between*, say, two and three?
In: Mathematics
While the other responses about doubling size are all true, I think they’re all missing the fact that there are different ways to use the word “dimension” in mathematics, and that’s what really causing your problems.
The everyday usage of “dimension” that we use in relation to height, width, depth is the Euclidean dimension. Or, fancier, the topological dimension of a Euclidean space. It’s always an integer.
The dimension the others have been describing is the Hausdorff dimension. It doesn’t have to be an integer, and it doesn’t need to correspond to the topological dimension of the space it’s in. It also doesn’t need to correspond to its own TD. The TD for the Sierpinksi triangle is 1.
There are other fractional dimensions, too. The box counting dimension, for instance, is distinct from the Hausdorff dimension.
You can have infinite dimensions, in some settings.
If you applied the doubling principle to the entire 3D space, you’d have the same 3D space. So… is the dimension of a three dimensional space 0? No, of course not.
Bottom line, this whole “fractals have fractional dimension” thing has lost some of it’s rigor as it’s been translated into popular science, so that’s why you have trouble making sense of it. That isn’t to dismiss the other answers — the doubling thing is intuitive and a useful measure. But I wouldn’t say that’s the whole picture.
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