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We know that fermions and bosons are the only two (indistinguishable) particle statistics for $d\geq 3$, and that anyons are for $d=2.$

What if the space were a fractal? Like the Sierpinski gasket, with $d=1.585$?

EDIT for clarity, to try and keep the questio open:

Here they have electrons in a Sierpinski gasket shaped potential.
The Sierpinski gasket has a fractal dimension.
If I want to permute the positions of indistinguishable particles, I have to do this in this fractal dimension. So this neither $SO(2)$ ($d=2$) not $SO(1)$.
The first homotopy (fundamental) group of such group, $\pi_1(SO(d))$ would give the number of distinct statistics: $\mathbb{Z}$ in $d=2$ (anyons) and $\mathbb{Z_2}$ in $d\geq3$ (fermions, bosons).

How would I go on about it here, in $d = 1.58$?

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    $\begingroup$ I think the main confusion I have is that fractal dimensions are not dimensions, but a measure of the complexity of the shape/curve. $\endgroup$ – Kyle Kanos Dec 19 '18 at 11:00