I can only find definitions of spaces, $\mathbb{R}^p$ where $p\in\mathbb{N}$ yet I cannot convince myself that having fractions of dimensions or even irrational amount of dimensions in a space would make sense or not. I.e how a sphere would look like in $2.5$ dimensions, how would we define distances, etc. Fractal dimensions do not seem to help, because as I see, they have a very narrow usage.(cannot define shapes or angles on them, or can we?)

negative integer dimensions may be interesting too.

  • $\begingroup$ What is your definition of dimension? $\endgroup$ – Bruno Joyal Jan 14 '14 at 1:32
  • 1
    $\begingroup$ See Hausdorff dimension, or just Google fractional number of dimensions. $\endgroup$ – Lucian Jan 14 '14 at 3:31
  • $\begingroup$ Also see Dragon Scales by Vi Hart for an entertaining introduction to fractional dimensions. Won't be much use if you are looking for defining shapes or angles, so I won't post this as an answer. $\endgroup$ – MvG Jan 14 '14 at 10:39

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