The necessary condition for the polychoron to be regular (with Schläfli symbol 
) and finite is
 |
Sufficiency can be established by consideration of the six figures satisfying this condition.
There are sixteen regular polychora, six of which are convex (Wells 1986, p. 68) and ten of which are stellated (Wells 1991, p. 209). The regular convex polychora have four principal types of symmetry axes, and the projections into three-spaces orthogonal to these may be called the "canonical" projections.
Of the six regular convex polychora, five are typically regarded as being analogous to the Platonic solids: the 4-simplex (a hyper-tetrahedron), the 4-cross polytope (a hyper-octahedron), the 4-cube (a hyper-cube), the 600-cell (a hyper-icosahedron), and the 120-cell (a hyper-dodecahedron). The 24-cell, however, has no perfect analogy in higher or lower spaces. The pentatope and 24-cell are self-dual, the 16-cell is the dual of the tesseract, and the 600- and 120-cells are dual to each other.
The convex regular polychora are listed in the following table (Coxeter 1969, p. 414; Wells 1991, p. 210).
| name | Schläfli symbol | class |  |  |  |  |
| pentatope |  | simplex | 5 | 10 | 10 | 5 |
| 16-cell |  | cross polytope | 8 | 24 | 32 | 16 |
| tesseract |  | hypercube | 16 | 32 | 24 | 8 |
| 24-cell |  | 24 | 96 | 96 | 24 | |
| 120-cell |  | 600 | 1200 | 720 | 120 | |
| 600-cell |  | 120 | 720 | 1200 | 600 |
Here, 
is the number of polyhedron vertices, 
the number of polytope edges,
the number of faces, and 
the number of cells. These quantities satisfy the identity
 |
which is a version of the polyhedral formula.
|
|
|
|
The regular star polychora are known as the Schläfli-Hess polychora by Normal Johnson, and are analogs of the four regular star polyhedra, namely the Kepler-Poinsot polyhedra.
Nine of the ten regular star polychora can be obtained by faceting 
; in other words, they have the same vertices as 
. The tenth, 
, can be obtained by faceting 
. In addition, of the ten regular star polychora, several
share the same edges: 
, 
, 
, and 
; 
, 
, 
, and 
; and 
and 
. 
does not share edges with any other regular polychora.
There are therefore only four different projections (into any given plane or three-space)
of the ten regular star polychora, illustrated above.
