The use of the term "polychoron" for such figures has been advocated by George Olshevsky, and is also supported by Norman W. Johnson.

The two-dimensional analogue of a polychoron is a polygon, and the three-dimensional analogue is a polyhedron.

A polychoron has vertices, edges, faces, and cells. A vertex is where one or more edges meet. An edge is where one or more faces meet, and a face is where one or more cells meet. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron.

Jonathan Bowers has classified the 8,186 currently known uniform polychora into 29 groups. There may be more.

A polychoron is a closed four-dimensional figure bounded by cells with the requirements that:-

- Each face must join exactly two cells.
- Adjacent cells are not in the same three-dimensional space.
- The figure is not a compound of other figures which meet the requirements.

There is a technique called the Coxeter-Dynkin system for performing Wythoff's construction for producing uniform polytopes. This method allows the polychora to be effectively enumerated.

There are six regular convex polychora:-

- pentachoron (with 5 tetrahedral cells) (also called a "simplex")
- tesseract (with 8 cubic cells) (also called a "hypercube")
- hexadecachoron (with 16 tetrahedral cells)
- icositetrachoron (with 24 octahedral cells)
- hecatonicosachoron (with 120 dodecahedral cells)
- hexacosichoron (with 600 tetrahedral cells)

- faceted hexacosichoron (also called icosahedral hecatonicosachoron)
- great hecatonicosachoron
- grand hecatonicosachoron
- small stellated hecatonicosachoron
- great grand hecatonicosachoron
- great stellated hecatonicosachoron
- grand stellated hecatonicosachoron
- great faceted hexacosichoron
- grand hexacosichoron
- great grand stellated hecatonicosachoron

Another commonly discussed figure that resides in 4-dimensional space is the 3-sphere, for which the term *glome* has been proposed. This is not a polychoron, since it is not made up of polyhedral cells.

See also: hypersphere, tesseract, simplex