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Why are all symmetry groups of regular polytopes are finite Coxeter groups?

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up vote 1 down vote accepted

Because they are reflection groups, and Coxeter proved in

H.S.M. Coxeter, "Discrete groups generated by reflections" Ann. of Math. , 35 (1934) pp. 588–621

and in this paper in the following year that symmetries of polytopes are among the finite reflection groups.

Check out either of those two articles: Coxeter is generally easy to read.

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