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 Cartanâ€“DieudonnĂ© 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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### Famous quotes containing the words group and/or reflection:

“Stripped of ethical rationalizations and philosophical pretensions, a crime is anything that a *group* in power chooses to prohibit.”

—Freda Adler (b. 1934)

“There are three principal means of acquiring knowledge available to us: observation of nature, *reflection*, and experimentation. Observation collects facts; *reflection* combines them; experimentation verifies the result of that combination. Our observation of nature must be diligent, our *reflection* profound, and our experiments exact. We rarely see these three means combined; and for this reason, creative geniuses are not common.”

—Denis Diderot (1713–1784)