Dihedral angles of \(n\)-simplices (Q1195661)

It is proved that if \(\{v_ 1,\dots,v_ n\}\) is a set of orthogonal nonzero vectors in \(E^ n\), if \(F_ i\) is an \((n-1)\)-dimensional face of the \(n\)-simplex \([v_ 0,v_ 1,\dots,v_ n]\) for \(i=1,2,\dots,n\) and if \(\theta_{kl}\) is the dihedral angle between \(F_ k\) and \(F_ l\), then \(\Sigma\cos^ 2\theta_{kl}=1\). Also in the case of a regular \(n\)- simplex, it is proved that the dihedral angle \(\theta_{kl}\) between the faces \(F_ k\) and \(F_ l\) is equal to \(cos^{-1}\left({1\over n}\right)\).