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Re: dihedral angle of regular tetrahedron
Posted:
Oct 25, 1998 1:41 AM


In article <AtTX1.664$XN5.2186@news9.ispnews.com>, M. Veve <mikev@hunterlab.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
(exp(it) + exp(it))^n = exp(int) + exp(int) + sum_{1<=k<n/2} (n choose k) (exp(i(n2k)t) + exp(i(n2k)t)) + (if n even) (n choose n/2)
If t = m Pi/n, then cos(n t) is an integer, so 2 cos(t) is an algebraic integer. But the only algebraic integers that are rational numbers are rational integers.
Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2



