Do there exist formulae relating the n-th dimensional solid angles of an n-simplex to either the n-th order dihedral angles, the volume of the n-1 dimensional facets, or the side lengths of the simplex?
2026-03-13 09:00:14.1773392414
solid angles of an n-simplex
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On Wolfram Mathworld you'll find a page on hypersine. (In order to grasp the there provided definition wrt. parallelotopes, you best would apply it to a 2D parallelogram first. Then it becomes nothing but the area formula thereof, which has been resolved to the "normal" sine of the dihedral angle.) There too is provided the searched for formula, relating the hypersine of a hypersolid angle $V_0$ of a hypersimplex quite nicely to the set of its dihedral angles $\alpha_{i,j}$:
(In fact there was a recent quest of mine within MO about the hypersolid angles of the 3 regular polytopes, which you might be interested in too. That question there indeed had been occurred to me by a comment to my other answer in the post, mentioned by @G.Cab.)
--- rk