Editing Regular polytope (section)

===Elements and symmetry groups===

At the start of the 20th century, the definition of a regular polytope was as follows.
*A regular polygon is a polygon whose edges are all equal and whose angles are all equal.
*A regular polyhedron is a polyhedron whose faces are all congruent regular polygons, and whose [[vertex figure]]s are all congruent and regular.
*And so on, a regular ''n''-polytope is an ''n''-dimensional polytope whose (''n'' &minus; 1)-dimensional faces are all regular and congruent, and whose vertex figures are all regular and congruent.

This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry.

* An ''n''-polytope is regular if any set consisting of a vertex, an edge containing it, a 2-dimensional face containing the edge, and so on up to ''n''&minus;1 dimensions, can be mapped to any other such set by a symmetry of the polytope.

So for example, the cube is regular because if we choose a vertex of the cube, and one of the three edges it is on, and one of the two faces containing the edge, then this triplet, known as a '''[[Flag (geometry)|flag]]''', (vertex, edge, face) can be mapped to any other such flag by a suitable symmetry of the cube. Thus we can define a regular polytope very succinctly:
*A regular polytope is one whose symmetry group is transitive on its flags.

In the 20th century, some important developments were made. The [[symmetry]] [[group (mathematics)|group]]s of the classical regular polytopes were generalised into what are now called [[Coxeter group]]s. Coxeter groups also include the symmetry groups of regular [[tessellation]]s of space or of the plane. For example, the symmetry group of an infinite [[chessboard]] would be the Coxeter group [4,4].