The use of the term "polychoron" for such figures has been advocated by George Olshevsky[?], and is also supported by Norman W. Johnson[?].
The twodimensional analogue of a polychoron is a polygon, and the threedimensional 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 threedimensional 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 fourdimensional figure bounded by cells with the requirements that:
All uniform polychora are vertextransitive (i.e. all vertices are equivalent), and are made up of uniform cells. A uniform cell is a cell that is vertextransitive, 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 CoxeterDynkin 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 nonconvex polychora:
There are fortysix Wythoffian convex nonprismatic uniform polychora.
Another commonly discussed figure that resides in 4dimensional space is the 3sphere, 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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