In a kaleidoscope

In the 30-ies of XX century geometer Harold Coxeter developed the theory of groups of reflections in Euclidean spaces and on spheres. Such
groups can be obtained as the group generated by reflections in faces of a polytope of any dimension with dihedral
angles of the form π/n. These polyhedra were called
Coxeter polyhedra. Images
Coxeter polyhedron with a consecutive reflection in their faces fill
all of the space or sphere, like a pattern in a kaleidoscope.

Kokster
in 1934 he obtained a complete classification of such polytopes (and even
most of Coxeter groups) in Euclidean spaces and on spheres of any
dimension. Classification is based on the language of Coxeter diagrams is a special kind
graph describing a Coxeter polyhedra.

Vinberg
summarized and developed this theory in the case of the Lobachevsky spaces. In 1967 he gave
the exact wording of the task. Despite the fact that the Lobachevsky space
is the third type of spaces of constant curvature Euclidean and after
spherical, even the formulation of the problem in this case requires a significant
the refiners.

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