# Regular Polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ jn, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n.

Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians.

Classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes.

A regular polytope can be represented by a Schläfli symbol of the form {a, b, c, ...., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}.

### Other articles related to "regular polytope, polytopes, polytope, regular":

Regular Polytopes in Nature
... Higher polytopes can obviously not exist in a three-dimensional world ... that the Universe itself has the form of some higher polytope, regular or otherwise ... the sky in the last few years, for tell-tale signs of a few regular candidates, so far without definite results ...

### Famous quotes containing the word regular:

While you’re playing cards with a regular guy or having a bite to eat with him, he seems a peaceable, good-humoured and not entirely dense person. But just begin a conversation with him about something inedible, politics or science, for instance, and he ends up in a deadend or starts in on such an obtuse and base philosophy that you can only wave your hand and leave.
Anton Pavlovich Chekhov (1860–1904)