Coxeter Groups as Automorphism Groups of Solid Transitive 3-simplex Tilings
2014
In the papers of I.K. Zhuk, then more completely of E. Molnar, I. Prok, J.
Szirmai all simplicial 3-tilings have been classified, where a symmetry
group acts transitively on the simplex tiles. The involved spaces depends on
some rotational order parameters. When a vertex of a such simplex lies out
of the absolute, e.g. in hyperbolic space H3, then truncation with its polar
plane gives a truncated simplex or simply, trunc-simplex. Looking for
symmetries of these tilings by simplex or trunc-simplex domains, with their
side face pairings, it is possible to find all their group extensions,
especially Coxeter’s reflection groups, if they exist. So here, connections
between isometry groups and their supergroups is given by expressing the
generators and the corresponding parameters. There are investigated
simplices in families F3, F4, F6 and appropriate series of trunc-simplices.
In all cases the Coxeter groups are the maximal ones.
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