Reflection group

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A reflection group is a group action, acting on a finite dimensional vector space, which is generated by reflections: elements that fix a hyperplane in space pointwise.

For example, with regard to ordinary reflections in planes in 3D, a reflection group is an isometry group generated by these reflections. The discrete point groups in three dimensions with this property are C_nv, D_nh, and the together three symmetry groups of the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron). Among the frieze groups $*\infty\infty$ and $*22\infty$ are reflection groups. Among the wallpaper groups we have pmm, p3m1, p4m, and p6m.

Each reflecting hyperplane acts as a mirror for the reflection. Reflection groups include Weyl and Coxeter groups, complex (or pseudo) reflection groups, and groups defined over arbitrary fields. Mathematical tools from geometry, topology, algebra, combinatorics, and representation theory are used to study reflection groups. For example, invariant theory (including modular), arrangements of hyperplanes, regular polytopes, Hecke algebras, Coxeter groups, Shephard groups, and braid groups all play a prominent role in investigations on reflection groups. Reflection groups also appear in coding theory, physics, chemistry, and biology.

[edit] Kaleidoscopes

Roe Goodman's article on The Mathematics of Mirrors and Kaleidoscopes (PDF) from the American Mathematical Monthly of April 2004 gives extensive background on the relationship between reflection groups and kaleidoscopes.

The Goodman article discusses Coxeter groups -- reflection groups in Euclidean space. However, as the above definition by Anne V. Shepler states, reflection groups may be defined over arbitrary fields (including Galois, or finite, fields). Such fields underlie the study of Galois geometry, a part of finite geometry.