In article <AtTX1.664$XN5.firstname.lastname@example.org>, M. Veve <email@example.com> wrote: >By computation, one can show that the dihedral angle between any two sides >of a regular tetrahedron is ArcCos[1/3] which is approximately 70.5288 >degrees. Is this known to be a rational or irrational multiple of Pi? Answer >plus reference would be most welcome. Thanks in advance.
In fact, the only cases where t is a rational multiple of Pi and cos(t) is rational are when 2 cos(t) is an integer. To prove this, note that 2 cos(n t) = exp(int) + exp(-int) is a monic polynomial in 2 cos(t) = exp(it) + exp(-it) with integer coefficients. This is easy to show by induction, using