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 four-dimensional polytope is sometimes called a polychoron. 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 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).
Bisztriczky, T.; McMullen, P., Schneider, R.; and Weiss, A. W. (Eds.). Polytopes:
Abstract, Convex, and Computational. Dordrecht, Netherlands: Kluwer, 1994.
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.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 45,
1973.
Emmer, M. (Ed.). The Visual Mind: Art and Mathematics. Cambridge, MA: MIT
Press, 1993.
Eppstein, D. "Polyhedra and Polytopes." http://www.ics.uci.edu/~eppstein/junkyard/polytope.html.
Fukuda, K. "Polytope Movie Page." http://www.ifor.math.ethz.ch/~fukuda/polymovie/polymovie.html.
MacHale, D. George Boole: His Life and Work. Dublin, Ireland: Boole,
1985.
Munkres, J. R. Analysis on Manifolds. Reading, MA: Addison-Wesley, 1991.
Sullivan, J. "Generating and Rendering Four-Dimensional Polytopes." Mathematica
J. 1, 76-85, 1991.
Weisstein, E. W. "Books about Polyhedra." http://www.ericweisstein.com/encyclopedias/books/Polyhedra.html.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, 1991.
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