# Spatial angle at a simplex vertex

I asked yesterday a question about spatial angles in the following post:

Spatial angles in higher dimensions

I'm still stuck with the same problem: what is the spatial fraction at a hyper-tetrahedron edge at the edge in origo?

http://www.math.cornell.edu/~hatcher/Other/hopf-samelson.pdf , p. 1 has a recursive equation ,which maybe holds only on the step $R^3$ --> $R^3$ :

$\Omega= 2\sum_{i=1}^3{\phi_i} -\pi$

There $\phi_i$ is the "edge angle", the fraction of the unit sphere surface the two planes defining the edge cut from the unit sphere. This edge angle can be solved in the projection plane $R^2$. (See an edge angle visualization in Fig. 4 of Heinz Hopf)

1) Is there any general recursive step (which would produce the exact same relation in case $d: 3 \rightarrow 2$ ?

2) Is there any direct vector definition, a la Wikipedia 3-D tetrahedron case:

$\tan{\Omega/2} =\frac{ \left|\mathbf{a}_1\, \mathbf{a}_2\,\mathbf{a}_3 \right| }{a_1 a_2 a_3 +\sum_i^3{\mathbf{a}_i \cdot \mathbf{a}_j a_k} } ,\; i,j,k\,\text{ all different}$

And here $\Omega$ refers to the spatial angle but don't worry, the spatial fraction is just an inch away...

All help greatly appreciated :)

• I spent a considerable amount of time in the higher dimensions, and could come no further than $S_n/(n-1)!$ solid radians, where $1 <S_{n-1} < S_n < \sqrt{n/4}$. Some else might do better. I have exact values to $N=4$, that is dihedral-angle less 1/5 (as a fraction of space). – wendy.krieger Aug 16 '13 at 8:02