Redirected from Polychora
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:-
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:-
There are ten regular non-convex polychora:-
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
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