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Dihedral angles of \(n\)-simplices. (English) Zbl 0766.51020

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 \(cos^{-1}\left({1\over n}\right)\).

MSC:

51M04 Elementary problems in Euclidean geometries
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References:

[1] Hopf, H., Differential geometry in the large, (Lecture Notes in Mathematics, Vol. 1000 (1989), Springer-Verlag: Springer-Verlag Berlin) · JFM 52.0571.01
[2] Cho, E. C., The generalized cross product and the volume of a simplex, Applied Mathematics Letters, 4, 6, 51-53 (1991) · Zbl 0751.51009
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