Reflection Group

In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately.

Read more about Reflection Group:  Definition, Kaleidoscopes, Relation With Coxeter Groups, Finite Fields, Generalizations

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Reflection Group - Generalizations
... Discrete isometry groups of more general Riemannian manifolds generated by reflections have also been considered ... spaces of rank 1 the n-sphere Sn, corresponding to finite reflection groups, the Euclidean space Rn, corresponding to affine reflection groups, and the hyperbolic space Hn, where the corresponding groups are called ... In two dimensions, triangle groups include reflection groups of all three kinds ...

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