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.
A finite reflection group is a subgroup of the general linear group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin.
The corresponding notions can be defined over other fields, leading to complex reflection groups and analogues of reflection groups over a finite field.
If the angle between two lines is an irrational multiple of pi, the group generated by reflections in these lines is infinite and non-discrete, hence, it is not a reflection group.
Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups.
A reflection group W admits a presentation of a special kind discovered and studied by H. S. M.
[1] The reflections in the faces of a fixed fundamental "chamber" are generators ri of W of order 2.
All relations between them formally follow from the relations expressing the fact that the product of the reflections ri and rj in two hyperplanes Hi and Hj meeting at an angle
When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane.
Geometrically, this amounts to including shears in a hyperplane.
Reflection groups over finite fields of characteristic not 2 were classified by Zalesskiĭ & Serežkin (1981).
Discrete isometry groups of more general Riemannian manifolds generated by reflections have also been considered.
The most important class arises from Riemannian symmetric 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 hyperbolic reflection groups.