- Reflection group
A

**reflection group**is agroup action , acting on a finite dimensionalvector space , which is generated by reflections: elements that fix ahyperplane 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 discretepoint groups in three dimensions with this property are "C_{nv}", "D_{nh}", and the together threesymmetry group s of the 5Platonic solid s (tetrahedron, cube, octahedron, dodecahedron, and icosahedron). Among thefrieze group s $*inftyinfty$ and $*22infty$ are reflection groups. Among thewallpaper group s 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 polytope s, Hecke algebras,Coxeter group s, Shephard groups, andbraid group s all play a prominent role in investigations on reflection groups. Reflection groups also appear incoding theory , physics, chemistry, and biology.**Kaleidoscopes**Roe Goodman's article on [

*http://www.math.rutgers.edu/pub/goodman/monthly.pdf The Mathematics of Mirrors and Kaleidoscopes (PDF)*] from the "American Mathematical Monthly" of April 2004 gives extensive background on the relationship between reflection groups andkaleidoscope s.The Goodman article discusses Coxeter groups -- reflection groups in Euclidean space. However, as [

*http://www.math.unt.edu/~ashepler/ a 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 offinite geometry .**See also***Euclidean plane isometries as reflection group

*Coxeter group

*Weyl group

*Complex reflection group

*Regular polytope

* [*http://www.kaleidica.com Digital Kaleidoscope*] the Kaleidica**External links*** [

*http://www.cms.math.ca/Publications/Reviews/2003/rev4.pdf Reflection groups and invariant theory*] , by Richard Kane (review, pdf)

*Reflection groups and Coxeter groups (ISBN 0-521-43613-3), by James E. Humphreys

* [*http://arxiv.org/abs/math/0405135 Jacobians of reflection groups*] , by Julia Hartmann and Anne V. Shepler

* [*http://www.math.rutgers.edu/~goodman/pub/monthly.pdf The Mathematics of Mirrors and Kaleidoscopes*] (pdf), by Roe Goodman

* [*http://arxiv.org/abs/math.AG/0610938 Reflection groups in algebraic geometry*] , by Igor V. Dolgachev

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