UP

To generalize to n dimensions we consider a small regular n-simplex at the origin and a large sphere about the origin. For the plane of each of the n+1 faces of the n-simplex (each itself an (n−1)-simplex) we consider the half-space of all points on the same side of that plane as the simplex. The simplex itself is the intersection of these n+1 half-spaces. For any k from 1 to n, the intersection of k of these half-spaces is congruent to the intersection of any other set of k half-spaces, since the group of the regular simplex is the group of permutations of its vertices. Let Ank denote the fraction of the sphere that intersects the intersection of some particular set of k half-spaces. (For irregular simplexes there are 2n+1 angles, one for each subset of half-spaces, including the trivial angles for the null set (=1) and that for the set of the n+1 half-spaces (=0).) This definition is equivalent to David Eppstein’s.

Angles are conventionally measured in radians, not circles and the conversion factor is 2π. Solid angles are usually measured in steradians, not spheres and the conversion factor is 4π. There are good reasons for this convention and the general conversion factor is cn = πn/2n/(n/2)! which is the content (measure) of Sn−1 which is the boundary of the n dimensional unit ball. We will write αnk = ckAnk for the conventional magnitude of k-hedral angle in the regular n-simplex. For half-integral values of n we say that
n! = Γ(n+1) or merely that n! = n(n−1)!, 0!=1 and (−1/2)! = √π.

Here is an equivalent definition that might be simpler for some purposes.