Generalization of sum of angles to polyhedra? The sum of interior angles of a polygon is (n-2)*180. Is there a similar statement for the sum of the solid angles of a polyhedra?
Is there any non-trivial relationship, $f(\alpha,\beta,...)=0$, between the solid angles at all? If not, can we prove it? 
If not, is there another generalization of this formula for polyhedra or higher dimensions?
 A: There is a beautiful formula for $d$-dimensional convex polytopes $P$, called Brianchon-Euler-Gram theorem (see for instance Theorem 22 here)  which generalizes the classical formula for the angle sum of polygons:
$$
\sum_{i=0}^d (-1)^i \sum_{F, \ dim(F)=i}\angle_F(P)=0. 
$$
Here $\angle_F(P)$ is the solid angle of $P$ at the face $F$ (of dimension $i$). In order to understand this formula, let's consider the case $d=2$. Then $P$ will have faces of dimension $0$ (vertices), dimension $1$ (edges) and dimension $2$ (the single facet, $P$ itself). If $F$ is an edge, then $\angle(F)=\pi$, if $F=P$ then $\angle(F)=2\pi$, while at the vertices we get the usual angles. 
(In order to understand this, you note that $\angle(F)$ is defined by picking a generic point $x\in F$ and computing the total solid angle occupied by $P$ in the small sphere centered at $x$.) If $P$ has $n$ vertices (and, hence, edges), the B-E-G formula then reads
$$
(\sum_{v\in Vert(P)} \angle_v(P) )- n\pi  + 2\pi=0,
$$
i.e.,
$$
\sum_{v\in Vert(P)} \angle_v(P)= (n-2)\pi. 
$$
If you want to see some identities involving only solid angles at the vertices, consider reading for instance the paper 
"Generalized Solid Angle Theory for Real
Polytopes", by DeSario and Robbins, 
and papers cited there, although, personally, I find this staff impenetrable. 
A: The sum of the solid angles of polyhedradoes not follow such a simple law. For example, the eight solid angles of a cube add up to one full angle, i.e.  $4\pi$ sr, but for a very flat quadratic frustrum the sum is approximately twice as much.
