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After reading these very interesting questions, I came up with another one:

  • Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting at a common vertex are rational multiples of $\pi$?
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    $\begingroup$ In view of Mnev' universality theorem, there are rigid convex polytopes (in sufficiently high dimension) which do not have angles equal to rational multiples of $\pi$. $\endgroup$
    – Misha
    Commented Nov 19, 2013 at 8:12

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