## Polytope

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).

See also 16-Cell, 24-Cell, 120-Cell, 600-Cell, Cross Polytope, Edge (Polytope), Face, Facet, Hypercube, Incidence Matrix, Measure Polytope, Ridge, Simplex, Tesseract, Vertex (Polyhedron)

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.