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Not a mathematician, and the question probably does not make sense but asking it anyway:

In usual euclidian spaces, central inversion through O(0,...,0) of point M(x1,...,xN) gives you one and only one point M'(-x1,...,-xN). I would write this $$I_O(M)=M'$$ Do you know of any bizarre space where this wouldn't be the case ? Where : $$I_O(M)=M' \text{or } M''$$

DarkBulle
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    if we want the reflection to be inverse to itself ($I_O \circ I_O = \operatorname{id}$) then it would certainly need to be bijective... the question would be what the precise meaning of "reflection" in "bizarre" spaces should be... – hgmath Feb 19 '21 at 16:23
  • Maybe something like : there exist some integer $k$ such as $I_O^k=Id$, without $I_O$ being a rotation, I actually don't know what really characterizes point symmetries except for the condition you mentionned – DarkBulle Feb 19 '21 at 16:37

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