Polychoron

A polychoron (plural: polychora) (from Greek poly meaning "many" and choros meaning "room" or "space") is a four-dimensional polytope, also known as a 4-polytope, or polyhedroid.

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:-

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

All uniform polychora are vertex-transitive (i.e. all vertices are equivalent), and are made up of uniform cells. A uniform cell is a cell that is vertex-transitive, with each face made up of regular polygons. A regiment is a group of polytopes with the same set of vertices and edges.

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:-

1. pentachoron (with 5 tetrahedral cells) (also called a "simplex")
2. tesseract (with 8 cubic cells) (also called a "hypercube")
3. hexadecachoron (with 16 tetrahedral cells)
4. icositetrachoron (with 24 octahedral cells)
5. hecatonicosachoron (with 120 dodecahedral cells)
6. hexacosichoron (with 600 tetrahedral cells)

There are ten regular non-convex polychora:-

1. faceted hexacosichoron (also called icosahedral hecatonicosachoron)
2. great hecatonicosachoron
3. grand hecatonicosachoron
4. small stellated hecatonicosachoron
5. great grand hecatonicosachoron
6. great stellated hecatonicosachoron
7. grand stellated hecatonicosachoron
8. great faceted hexacosichoron
9. grand hexacosichoron
10. great grand stellated hecatonicosachoron

There are forty-six Wythoffian convex non-prismatic uniform polychora.

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