V - E + F - C = 0

where

V = number of vertices
E = number of edges
F = number of faces
C = number of (3-dimensional) cells



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

V = 120
E = 720
F = 1200
C = 600
and we note that 120 - 720 + 1200 - 600 = 0.

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.