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).