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:-
- 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.
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:-
- 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)
There are ten regular non-convex polychora:-
- 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
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