Reflection group

The symmetry groups of the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) are generated by reflections and rotations in space. Since any rotation in 3D can be written as a combination of two reflections, the symmetry groups are also generated by just reflections. These symmetry groups are therefore examples of reflection groups. A reflection group acts on a finite dimensional vector space and is generated by reflections: elements that fix a hyperplane in space pointwise. 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.

Related Topics:
Symmetry groups - Platonic solids - Reflections - Rotations - Vector space - Hyperplane - Weyl - Coxeter - Groups - Fields - Representation theory - Invariant theory - Regular polytopes - Hecke algebras - Braid groups - Coding theory

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