A convex polytope may be defined as the Convex Hull of a finite set of points (which are always bounded), or as the
intersection of a finite set of half-spaces. Explicitly, a -dimensional polytope may be specified as the set of solutions
to a system of linear inequalities

where is a real Matrix and is a real -Vector. The positions of the vertices given by the above equations may be found using a process called Vertex Enumeration.

A regular polytope is a generalization of the Platonic Solids to an arbitrary Dimension. The Necessary condition for the figure with Schläfli Symbol to be a finite polytope is

Sufficiency can be established by consideration of the six figures satisfying this condition. The table below enumerates the six regular polytopes in 4-D (Coxeter 1969, p. 414).

Name | Schläfli Symbol | ||||

Regular Simplex | 5 | 10 | 10 | 5 | |

Hypercube | 16 | 32 | 24 | 8 | |

16-Cell | 8 | 24 | 32 | 16 | |

24-Cell | 24 | 96 | 96 | 24 | |

120-Cell | 600 | 1200 | 720 | 120 | |

600-Cell | 120 | 720 | 1200 | 600 |

Here, is the number of Vertices, the number of Edges,
the number of Faces, and the number of cells. These quantities satisfy the identity

which is a version of the Polyhedral Formula.

For -D with , there are only three regular polytopes, the Measure Polytope, Cross Polytope, and regular Simplex (which are analogs of the Cube, Octahedron, and Tetrahedron).

**References**

Coxeter, H. S. M. ``Regular and Semi-Regular Polytopes I.'' *Math. Z.* **46**, 380-407, 1940.

Coxeter, H. S. M. *Introduction to Geometry, 2nd ed.* New York: Wiley, 1969.

Eppstein, D. ``Polyhedra and Polytopes.'' http://www.ics.uci.edu/~eppstein/junkyard/polytope.html.

© 1996-9

1999-05-26