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Aug 8 at 14:06 comment added Abdelmalek Abdesselam +1 for the nice answer. Another funny thing is that the fundamental identity satisfied by $\sigma$ holds and looks exactly the same for $p$-adics if one defines the norm of a vector in $\mathbb{Q}_p^d$ by $\|x\|=\max_i|x_i|_p$, and unit sphere inversion by $\sigma(x)=\|x\|^{2}x$. The $+2$ instead of $-2$ as exponent is not a typo. This is elaborated upon in my youtube talk mentioned in my answer.
S Aug 8 at 2:23 review First answers
Aug 8 at 5:59
S Aug 8 at 2:23 history edited Alex Elzenaar CC BY-SA 4.0
Correct the hyperbolic metric which I copied and pasted without thinking before...
Aug 7 at 22:49 history edited Alex Elzenaar CC BY-SA 4.0
Added link
S Aug 7 at 22:35 review First answers
Aug 7 at 23:55
S Aug 7 at 22:35 history answered Alex Elzenaar CC BY-SA 4.0