Polychoron
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In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning "many" and choros meaning "room" or "space"), 4-polytope, or polyhedroid. The two-dimensional analogue of a polychoron is a polygon, and the three-dimensional analogue is a polyhedron.
Note that the use of the term polychoron is not entirely standard. Its use has been advocated by Norman Johnson and George Olshevsky. See the Uniform Polychora Project.
A polychoron has vertices, edges, faces, and cells. A vertex is where two or more edges meet. An edge is where two or more faces meet, and a face is where two cells meet. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron.
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.
Uniform polychora
A polychoron is said to be uniform if it is vertex-uniform (i.e. there is a symmetry taking any vertex to any other) and its cells are uniform polyhedra. A uniform polyhedron is a polyhedron that is vertex-transitive, with each face made up of regular polygons. (These include the 5 Platonic solids, 13 Archimedean solids, 4 Kepler-Poinsot solids, and 53 other nonregular, nonconvex forms).
The Uniform Polychora Project has classified the 8,186 currently known uniform polychora into 29 groups. There may be more.
There are exactly six convex regular polychora. These are the analogs of the five Platonic solids in 3 dimensions.
- pentachoron (with 5 tetrahedral cells) (also called a "4-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 non-convex regular 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 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 forty-six Wythoffian convex non-prismatic uniform polychora.
See also
- convex regular polychoron
- The 3-sphere (or glome) is another commonly discussed figure that resides in 4-dimensional space. This is not a polychoron, since it is not made up of polyhedral cells.
External links
- Polychoron on Mathworld (http://mathworld.wolfram.com/Polychoron.html)
- Four dimensional figures page (http://members.aol.com/Polycell/uniform.html)
- Multidimensional glossary (http://members.aol.com/Polycell/glossary.html) – compiled by George Olshevsky