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 | Symmetry group | |
|---|---|---|---|---|---|---|---|---|---|---|
| 5-cell pentachoron pentatope hyperpyramid hypertetrahedron 4-simplex |
simplex (n-simplex) |
{3,3,3} | 5 | 10 | 10 triangles |
5 tetrahedra |
tetrahedra | (self-dual) | A4 | 120 |
| 8-cell octachoron tesseract hypercube 4-cube |
hypercube (n-cube) |
{4,3,3} | 16 | 32 | 24 squares |
8 cubes |
tetrahedra | 16-cell | B4 | 384 |
| 16-cell hexadecachoron hyperoctahedron 4-orthoplex |
cross-polytope (n-orthoplex) |
{3,3,4} | 8 | 24 | 32 triangles |
16 tetrahedra |
octahedra | 8-cell | B4 | 384 |
| 24-cell icositetrachoron octaplex polyoctahedron |
{3,4,3} | 24 | 96 | 96 triangles |
24 octahedra |
cubes | (self-dual) | F4 | 1152 | |
| 120-cell hecatonicosachoron dodecaplex hyperdodecahedron polydodecahedron |
dodecahedral pentagonal polytope (n-pentagonal polytope) |
{5,3,3} | 600 | 1200 | 720 pentagons |
120 dodecahedra |
tetrahedra | 600-cell | H4 | 14400 |
| 600-cell hexacosichoron tetraplex hypericosahedron polytetrahedron |
icosahedral pentagonal polytope (n-pentagonal polytope) |
{3,3,5} | 120 | 720 | 1200 triangles |
600 tetrahedra |
icosahedra | 120-cell | H4 | 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 Nk denotes the number of k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).
Read more about this topic: Convex Regular 4-polytope
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