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This title may not explain my question right but I could not think of any better short explanation. My question is, if there is a possibility of a structure (or space) with the dimension $Dim = 3,5$, $Dim = 4,8...$ or any other real numbers value? It's seems pretty obvious that there is no such thing like a $4,5$ dimensional sphere but I'm interested in a proof that no such structure can exist. Does a real dimensional space make any physical sense? Thanks for your help.

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    $\begingroup$ Depends on how you define the word “dimension”. What do you have in mind? $\endgroup$ – knzhou Apr 11 at 9:58
  • $\begingroup$ I think of dimensions, like the euklidien space. $\endgroup$ – gamma Apr 11 at 11:11
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    $\begingroup$ What you're looking for are fractal dimensions. Take a look at this video for a nice introduction youtube.com/watch?v=gB9n2gHsHN4 $\endgroup$ – Halbeard Apr 11 at 11:20

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