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Q: What are fractional dimensions? Can space have a fractional dimension?

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Posted in Strange

Physicist: There are a couple of different contexts in which the word “dimension” comes up. In the case of fractals the number of dimensions has to do (this is a little hand-wavy) with the way points in the fractal are distributed. For example, if you have points distributed at random in space you’d say you have a three-dimensional set of points, and if they’re all arranged on a flat sheet you’d say you have two-dimensional set of points. Way back in the day mathematicians figured that a good way to determine the “dimensionality” of a set is to pick a point in the set, and then put progressively larger and larger spheres of radius R around it. If the number of points contained in the sphere is proportional to Rd, then the set is d-dimensional.

However, there’s a problem with this technique. You can have a set that’s really d-dimensional, but on a large scale it appears to be a different dimension. For example, a piece of paper is basically 2-D, but if you crumple it up into a ball it seems 3-D on a large enough scale. A hairball or bundle of cables seems 3-D (by the “sphere test”), but they’re really 1-D (Ideally at least. Every physical object is always 3-D).

This whole “look at the number of points inside of tiny spheres and see how that number scales with size” thing works great for every half-way reasonable set. However, fractal sets can be “infinitely crumpled”, so no matter how small a sphere you use, you still get a dimension larger than you might expect.

When the “sphere trick” is applied to tangled messes it doesn’t necessarily have to give you integer numbers until the spheres are small enough. With fractals there is no “small enough” (that should totally be a terrible movie tag line), and you find that they have a dimension that’s often a fraction. The dimension of the Mandelbrot’s boundary (picture above) is 2, which is the highest it can be, but there are more interesting (but less pretty) fractals out there with genuinely fractional dimensions, like the “Koch snowflake” which has a dimension of approximately 1.262.

That all said, when somebody (looking at you, all mathematicians) talks about fractional dimensions, they’re really talking about a weird, abstract, and not at all physical notion of dimension. There’s no such thing as “2.5 dimensional universe”. When we talk about the “dimension of space” we’re talking about the number of completely different directions that are available, not the whole “sphere thing”. The dimensions of space are either there or not, so while you could have 4 dimensions, you couldn’t have 3.5.