Polytope

A convex polytope may be defined as the Convex Hull of a finite set of points (which are always bounded), or as theintersection of a finite set of half-spaces. Explicitly, a -dimensional polytope may be specified as the set of solutionsto a system of linear inequalities

A regular polytope is a generalization of the Platonic Solids to an arbitrary Dimension. TheNecessary condition for the figure with Schläfli Symbol to be a finitepolytope is

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

For -D with , there are only three regular polytopes, the Measure Polytope, Cross Polytope, andregular Simplex (which are analogs of the Cube, Octahedron, and Tetrahedron).

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.

• Side
• Sidon Sequence
• Siegel Disk Fractal
• Siegel Modular Function
• Siegel's Paradox
• Siegel's Theorem
• Sierpinski Arrowhead Curve
• Sierpinski Carpet
• Sierpinski Constant
• Sierpinski Curve
• Sierpinski Gasket
• Sierpinski-Menger Sponge
• Sierpinski Number of the First Kind
• Sierpinski Number of the Second Kind
• Sierpinski's Composite Number Theorem
• Sierpinski Sieve
• Sierpinski Sponge
• Sierpinski's Prime Sequence Theorem
• Sierpinski's Theorem
• Sierpinski Tetrahedron
• Sierpinski Triangle
• Sieve
• Sieve of Eratosthenes
• Sievert Integral
• Sievert's Surface