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 $d$-dimensional polytope may be specified as the set of solutions to a system of linear inequalities

$${\hbox{\sf m}}{\bf x} \leq {\bf b},$$

where ${\hbox{\sf m}}$ is a real $s\times d$ Matrix and ${\bf b}$ is a real $s$-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 $\{p, q, r\}$ to be a finite polytope is

$$\cos\left({\pi\over q}\right)<\sin\left({\pi\over p}\right)\sin\left({\pi\over r}\right).$$

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	$N_0$	$N_1$	$N_2$	$N_3$
Regular Simplex	$\{3, 3, 3\}$	5	10	10	5
Hypercube	$\{4, 3, 3\}$	16	32	24	8
16-Cell	$\{3, 3, 4\}$	8	24	32	16
24-Cell	$\{3, 4, 3\}$	24	96	96	24
120-Cell	$\{5, 3, 3\}$	600	1200	720	120
600-Cell	$\{3, 3, 5\}$	120	720	1200	600

Here, $N_0$ is the number of Vertices, $N_1$ the number of Edges, $N_2$ the number of Faces, and $N_3$ the number of cells. These quantities satisfy the identity

$$N_0-N_1+N_2-N_3=0,$$

which is a version of the Polyhedral Formula.

For $n$-D with $n\geq 5$, 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

Solid Geometry

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.