Convex regular 4-polytope

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.

## Properties

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.

|Names||Family||Schläfli symbol || Vertices || Edges || Faces || Cells || Vertex figures || Dual polytope|colspan=2|Symmetry group|
| --- | --- | --- | --- | --- | --- | --- |
|Pentachoron 5-cell pentatope hyperpyramid hypertetrahedron 4-simplex || simplex (n-simplex) || {3,3,3} || 5 || 10 || 10 triangles || 5 tetrahedra || tetrahedra || (self-dual) || _A_<sub>4</sub> || 120|||||||
|Tesseract octachoron 8-cell hypercube 4-cube || hypercube (n-cube) || {4,3,3} || 16 || 32 || 24 squares || 8 cubes || tetrahedra || 16-cell || _B_<sub>4</sub> || 384|||||||
|Hexadecachoron 16-cell orthoplex hyperoctahedron 4-orthoplex || cross-polytope (n-orthoplex) || {3,3,4} || 8 || 24 || 32 triangles || 16 tetrahedra || octahedra || tesseract || _B_<sub>4</sub> || 384|||||||
|Icositetrachoron 24-cell octaplex polyoctahedron || || {3,4,3} || 24 || 96 || 96 triangles || 24 octahedra || cubes || (self-dual) || _F_<sub>4</sub> || 1152|||||||
|Hecatonicosachoron 120-cell dodecaplex hyperdodecahedron polydodecahedron || || {5,3,3} || 600 || 1200 || 720 pentagons || 120 dodecahedra || tetrahedra || 600-cell || _H_<sub>4</sub> || 14400|||||||
|Hexacosichoron 600-cell tetraplex hypericosahedron polytetrahedron || || {3,3,5} || 120 || 720 || 1200 triangles || 600 tetrahedra || icosahedra || 120-cell || _H_<sub>4</sub> || 14400|||||||

Since the boundaries of each of these figures is topologically equivalent to a 3-sphere, whose Euler characteristic is zero, we have the 4-dimensional analog of Euler's polyhedral formula:
where _N_<sub>_k_</sub> denotes the number of _k_-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).

## Visualizations

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.

|5-cell||8-cell||16-cell||24-cell||120-cell||600-cell|
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|colspan=6|Wireframe orthographic projections inside Petrie polygons.||||||||||
|colspan=6|Solid orthographic projections||||||||||
|tetrahedral envelope (cell/vertex-centered)|cubic envelope (cell-centered)|octahedral envelope (vertex centered)|cuboctahedral envelope (cell-centered)|truncated rhombic triacontahedron envelope (cell-centered)|Pentakis icosidodecahedral envelope (vertex-centered)||||||
|colspan=6|Wireframe Schlegel diagrams (Perspective projection)||||||||||
|(Cell-centered)|(Cell-centered)|(Cell-centered)|(Cell-centered)|(Cell-centered)|(Vertex-centered)||||||
|colspan=6|Wireframe stereographic projections (Hyperspherical)||||||||||

## Related pages

- Regular polytope
- Platonic solid

## References

- H. S. M. Coxeter, _Introduction to Geometry, 2nd ed._, John Wiley & Sons Inc., 1969. .
- H. S. M. Coxeter, _Regular Polytopes_, 3rd. ed., Dover Publications, 1973. .
- 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

## Other websites

- 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)