# Polychoron

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:

- 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.

## Contents

## 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.

- 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

## 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:

- 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 polychora with 16-cell and tesseract symmetry include:

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

- 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.)

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

- 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.

## External links

- Polychoron on Mathworld
- Four dimensional figures page
- Multidimensional glossary – compiled by George Olshevsky