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For which value of the integer $n$ does there exist a partition of $S^{n-1}$, the unit sphere of $\mathbb{R}^n$ for the euclidean norm, by a family of images of the standard hypercube $C=\{ (e_1, ..., e_n)|\forall i, e_i = \pm 1/\sqrt{n}\}$ by isometries of $\mathbb{R}^n$ ?

$n=1, 2,4,8$ can be shown to work with the help of complex numbers/quaternions/octonions (for instance, for $n=2$, the set of cubes $\{C_{\theta}, 0<\theta\leq \pi/2\}$ for $C_{\theta}=e^{i\theta}\{1,i,-1,-i\}$ is a solution to the problem); I am interested in the general behavior, and in computable solutions if they exist.

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  • $\begingroup$ Could you be more specific about what you call an isometric image of the hypercube? In particular, are the cube and the sphere endowed with the restriction of the distance, or the corresponding intrinsic metrics? Is the hypercube convex, or do you mean just the boundary? could you explain at least the case $n=2$ to clarify these points? $\endgroup$
    – Benoît Kloeckner
    Commented Jan 14, 2011 at 13:43
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    $\begingroup$ I edited the question to answer your comment $\endgroup$
    – Sébastien Kunz-Jacques
    Commented Jan 14, 2011 at 13:59
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    $\begingroup$ "isometries of R^n" isn't very clear. If you want isometric maps from the hypercube to the sphere, there are none because a sphere has nonzero curvature and the hypercube is flat. $\endgroup$
    – Dmitry Vaintrob
    Commented Feb 22, 2016 at 21:35
  • $\begingroup$ @DmitryVaintrob: it appears Sebastien considered the cube to be a finite metric space -- just the vertices. The partition of the sphere is just a topological or PL partition, and the metric criterion appears to be only for the vertices of each "face" of the partition. $\endgroup$
    – Ryan Budney
    Commented Feb 22, 2016 at 22:11

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