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Topic: dihedral angle of regular tetrahedron
Replies: 6   Last Post: Oct 27, 1998 12:39 AM

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Robert Israel

Posts: 11,902
Registered: 12/6/04
Re: dihedral angle of regular tetrahedron
Posted: Oct 25, 1998 1:41 AM
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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(n-2k)t) + exp(-i(n-2k)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








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