where
The above picture shows a 2-dimensional projection of the regular polyhedron in 4-dimensional Euclidean space with 600 tetrahedral cells, sometimes called a hypericosahedron. It has
It turns out that there are 6 regular hyperpolyhedra in 4-dimensional Euclidean space
as shown in the following table:
| Figure | Vertices | Edges | Faces | Cells | Euler's formula |
|---|---|---|---|---|---|
| 4-simplex (hypertetrahedron) | 5 | 10 | 10 triangles | 5 tetrahedra | 5 - 10 + 10 - 5 = 0 |
| hypercube (tesseract) | 16 | 32 | 24 squares | 8 cubes | 8 - 16 + 32 - 24 = 0 |
| hyperoctahedron | 8 | 24 | 32 triangles | 16 tetrahedra | 24 - 32 +16 - 8 = 0 |
| 24-cell | 24 | 96 | 96 triangles | 24 octahedra | 24 - 96 + 96 - 24 = 0 |
| hyperdodecahedron | 600 | 1200 | 720 pentagons | 120 dodecahedra | 600 - 1200 + 720 - 120 = 0 |
| hypericosahedron | 120 | 720 | 1200 triangles | 600 icosahedra | 120 - 720 + 1200 - 600 = 0 |
Surprisingly in Euclidean spaces of dimension n>4 there are only 3 different
kinds of regular hyperpolyhedra: the simplex (hypertetrahedron), the hypercube and the
hyperoctahedron. The table below shows the number of i-dimensional faces in
each of these is shown in the table below.
