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 or more 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:

  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.

Regular polychora

A polychoron is said to be regular if all its cells, faces, edges, and vertices, are congruent. There are exactly six convex regular polychora. These are the analogs of the five Platonic solids.

  1. pentachoron (with 5 tetrahedral cells) (also called a "4-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 non-convex regular 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

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 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.

Convex Uniform Polychora

There are forty-six Wythoffian convex non-prismatic uniform polychora. These include the six regular polychora, and have symmetry groups derived from them.

The convex uniform polychora with pentatope symmetry include:

  1. The pentatope itself;
  2. The truncated pentatope (with 5 tetrahedra and 5 truncated tetrahedra);
  3. The rectified pentatope (5 tetrahedra and 5 octahedra);
  4. The bitruncated pentatope (10 truncated tetrahedra);
  5. The runcinated pentatope (10 tetrahedra and 20 triangular prisms).

The polychora with 16-cell and tesseract symmetry include:

  1. The 16-cell itself;
  2. The tesseract itself;
  3. The truncated 16-cell (with 8 octahedra and 16 truncated tetrahedra);
  4. The rectified 16-cell, which is identical to the 24-cell (24 octahedra);
  5. The bitruncated 16-cell, which is identical to the bitruncated tesseract (8 truncated octahedra and 16 truncated tetrahedra);
  6. The truncated tesseract (16 tetrahedra and 8 truncated cubes);
  7. The rectified tesseract (16 tetrahedra and 8 cuboctahedra);
  8. The runcinated tesseract, which is identical to the runcinated 16-cell (32 cubes, 32 triangular prisms, and 16 tetrahedra).

The polychora with 24-cell symmetry include:

  1. The 24-cell itself;
  2. The truncated 24-cell (with 24 cubes and 24 truncated octahedra);
  3. The rectified 24-cell (24 cubes and 24 cuboctahedra);
  4. The bitruncated 24-cell (48 truncated cubes);
  5. The runcinated 24-cell (48 octahedra and 192 triangular prisms).

The polychora with 120-cell and 600-cell symmetry include:

  1. The 120-cell itself;
  2. The 600-cell itself;
  3. The truncated 120-cell (with 120 truncated dodecahedra and 600 tetrahedra);
  4. The rectified 120-cell (120 icosidodecahedra and 600 tetrahedra);
  5. The bitruncated 120-cell, which is identical to the bitruncated 600-cell (120 truncated icosahedra and 600 truncated tetrahedra);
  6. The truncated 600-cell (600 truncated tetrahedra and 120 icosahedra);
  7. The rectified 600-cell (600 octahedra and 120 icosahedra);
  8. The runcinated 600-cell, identical to the runcinated 120-cell (120 dodecahedra, 600 tetrahedra, 720 pentagonal prisms, and 1200 triangular prisms).

(Note: these lists are not exhaustive.)

There is a forty-seventh anomalous polychoron which is non-Wythoffian, consisting of 20 pentagonal antiprisms and 300 tetrahedra.

The remaining convex uniform polychora may be grouped into two infinite families: the duoprisms and the polyhedral prisms. The polyhedral prisms consist of pairs of any of the 3D uniform polyhedra (including antiprisms), joined to each other by suitable polygonal prisms.

See also

External links