Convex regular 4-polytope
The English used in this article may not be easy for everybody to understand. (December 2011)
In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional (4D) polytope which is both regular and convex. These are the four-dimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions).
These polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. Schläfli discovered that there are precisely six such figures. Five of these may be thought of as higher dimensional analogs of the Platonic solids. There is one additional figure (the 24-cell) which has no three-dimensional equivalent.
Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces in a regular fashion.
The following tables lists some properties of the six convex regular polychora. The symmetry groups of these polychora are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group.
|Vertices||Edges||Faces||Cells||Vertex figures||Dual polytope||Symmetry group|
- <math>N_0 - N_1 + N_2 - N_3 = 0\,</math>
where Nk denotes the number of k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).
The following table shows some 2 dimensional projections of these polytopes. Various other visualizations can be found in the other websites below. The Coxeter-Dynkin diagram graphs are also given below the Schläfli symbol.
- H. S. M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons Inc., 1969. Template:Catalog lookup linkScript error: No such module "check isxn"..
- H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Template:Catalog lookup linkScript error: No such module "check isxn"..
- D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
- Eric W. Weisstein, Regular polychoron at MathWorld.
- Jonathan Bowers, 16 regular polychora
- Regular 4D Polytope Foldouts
- Catalog of Polytope Images A collection of stereographic projections of 4-polytopes.
- A Catalog of Uniform Polytopes
- Dimensions 2 hour film about the fourth dimension (contains stereographic projections of all regular polychrons)