Polytope

The word polytope is used to mean a number of related, but slightly different mathematical objects. A convex polytope may be defined as the convex hull of a finite set of points (which are always bounded), or as a bounded intersection of a finite set of half-spaces. Coxeter (1973, p. 118) defines polytope as the general term of the sequence "point, line segment, polygon, polyhedron, ...," or more specifically as a finite region of -dimensional space enclosed by a finite number of hyperplanes. The special name polychoron is sometimes given to a four-dimensional polytope. However, in algebraic topology, the underlying space of a simplicial complex is sometimes called a polytope (Munkres 1991, p. 8). The word "polytope" was introduced by Alicia Boole Stott, the somewhat colorful daughter of logician George Boole (MacHale 1985).

The part of the polytope that lies in one of the bounding hyperplanes is called a cell.

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 regular polytopes were discovered before 1852 by the Swiss mathematician Ludwig Schläfli. For dimensions with , there are only three regular convex polytopes: the hypercube, cross polytope, and regular simplex, which are analogs of the cube, octahedron, and tetrahedron (Coxeter 1969; Wells 1991, p. 210).