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.
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:
- 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.
- 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
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.
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:
- The pentatope itself;
- The truncated pentatope (with 5 tetrahedra and 5 truncated tetrahedra);
- The rectified pentatope (5 tetrahedra and 5 octahedra);
- The bitruncated pentatope (10 truncated tetrahedra);
- The runcinated pentatope (10 tetrahedra and 20 triangular prisms).
- The 16-cell itself;
- The tesseract itself;
- The truncated 16-cell (with 8 octahedra and 16 truncated tetrahedra);
- The rectified 16-cell, which is identical to the 24-cell (24 octahedra);
- The bitruncated 16-cell, which is identical to the bitruncated tesseract (8 truncated octahedra and 16 truncated tetrahedra);
- The truncated tesseract (16 tetrahedra and 8 truncated cubes);
- The rectified tesseract (16 tetrahedra and 8 cuboctahedra);
- 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:
- The 24-cell itself;
- The truncated 24-cell (with 24 cubes and 24 truncated octahedra);
- The rectified 24-cell (24 cubes and 24 cuboctahedra);
- The bitruncated 24-cell (48 truncated cubes);
- The runcinated 24-cell (48 octahedra and 192 triangular prisms).
- The 120-cell itself;
- The 600-cell itself;
- The truncated 120-cell (with 120 truncated dodecahedra and 600 tetrahedra);
- The rectified 120-cell (120 icosidodecahedra and 600 tetrahedra);
- The bitruncated 120-cell, which is identical to the bitruncated 600-cell (120 truncated icosahedra and 600 truncated tetrahedra);
- The truncated 600-cell (600 truncated tetrahedra and 120 icosahedra);
- The rectified 600-cell (600 octahedra and 120 icosahedra);
- 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.)
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.
- Convex regular polychoron
- List of regular polytopes
- Semiregular 4-polytopes Subset of uniform polychora with regular polyhedron cells.
- 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.
- The duocylinder is a figure in 4-dimensional space related to the duoprisms. It is also not a polychoron because its bounding volumes are not polyhedral.