# The Uniform Polychora

The *uniform polychora* are the 4D analogues of the
Archimedean polyhedra, prisms, and antiprisms. A
uniform polychoron has cells which are uniform
polyhedra, and its vertices are *transitive,* meaning that they are
equivalent under their symmetries.

There are 47 non-prismatic convex uniform polychora, which include the regular polychora from which they derive their symmetries. In addition to these, there are also 18 convex polyhedral prisms based on the Platonic and Archimedean solids, one of which, the cube prism, coincides with the tesseract, leaving 17 distinct polyhedral prisms. Then there are the infinite families of the antiprism prisms and duoprisms.

The 47 non-prismatic uniform polychora may be grouped into *families*
according to the regular polychora their symmetries are derived from. There are
4 families represented, one for each regular polychoron together with its
dual.

## The 5-cell family

The 5-cell family consists of the uniform truncates of the
5-cell. Since the 5-cell is self-dual, only 9 of the 15
truncates of this family are unique. Furthermore, the 5-cell's self-duality
also causes some of the members of the family (marked with an asterisk
^{*}) to have a higher degree of symmetry than the 5-cell itself.

(Dual 5-cell: identical to the 5-cell itself.)

(Rectified dual 5-cell: identical to the rectified 5-cell.)

(Truncated dual 5-cell: identical to the truncated 5-cell.)

Rectified 5-cell: a semiregular polychoron bounded by 5 tetrahedra and 5 octahedra.

(Cantellated dual 5-cell: identical to the cantellated 5-cell.)

Bitruncated 5-cell

^{*}: a cell-transitive uniform polychoron bounded by 10 truncated tetrahedra.(Cantitruncated dual 5-cell: identical to the cantitruncated 5-cell.)

5-cell: the regular member.

Runcinated 5-cell

^{*}: a pretty uniform polychoron bounded by 10 tetrahedra and 20 triangular prisms.Cantellated 5-cell: a uniform polychoron bounded by 5 octahedra, 5 cuboctahedra, and 10 triangular prisms.

(Runcitruncated dual 5-cell: identical to the runcitruncated 5-cell.)

Truncated 5-cell: a uniform polychoron bounded by 5 truncated tetrahedra and 5 tetrahedra.

Runcitruncated 5-cell: a pretty uniform polychoron bounded by 5 truncated tetrahedra, 10 hexagonal prisms, 10 triangular prisms, and 5 cuboctahedra.

Cantitruncated 5-cell: a pretty little uniform polytope bounded by 5 truncated octahedra, 5 truncated tetrahedra, and 10 triangular prisms.

Omnitruncated 5-cell

^{*}: a uniform polychoron bounded by 10 truncated octahedra and 20 hexagonal prisms.

## The tesseract/16-cell family

The tesseract/16-cell family consists of 15 members having the symmetry of the tesseract and the 16-cell. Due to the coincidence of the 24-cell with the rectified 16-cell, three of these 15 members coincide with members of the 24-cell family, thus leaving 12 unique members in this family.

16-cell: one of the two regular members.

(Rectified 16-cell: identical to the 24-cell.)

Truncated 16-cell: a uniform polychoron bounded by 16 truncated tetrahedra and 8 octahedra.

Rectified tesseract: a uniform polychoron bounded by 16 tetrahedra and 8 cuboctahedra.

(Cantellated 16-cell: identical to the rectified 24-cell.)

Bitruncated tesseract (bitruncated 16-cell): a uniform polychoron bounded by 8 truncated octahedra and 16 truncated tetrahedra.

(Cantitruncated 16-cell: identical to the truncated 24-cell; a uniform polychoron bounded by 24 truncated octahedra and 24 cubes.)

Tesseract: the other regular member of this family.

Runcinated tesseract (runcinated 16-cell): a uniform polychoron bounded by 32 cubes, 32 triangular prisms, and 16 tetrahedra.

Cantellated tesseract: a uniform polychoron bounded by 8 rhombicuboctahedra, 16 octahedra, and 32 triangular prisms.

Runcitruncated 16-cell: a pretty uniform polychoron bounded by 8 rhombicuboctahedra, 16 truncated tetrahedra, 24 cubes, and 32 hexagonal prisms.

Truncated tesseract: a uniform polychoron bounded by 8 truncated cubes and 16 tetrahedra.

Runcitruncated tesseract: a pretty uniform polychoron bounded by 8 truncated cubes, 16 cuboctahedra, 24 octagonal prisms, and 32 triangular prisms.

Cantitruncated tesseract: a pretty polychoron bounded by 8 great rhombicuboctahedra, 16 truncated tetrahedra, and 32 triangular prisms.

Omnitruncated tesseract (omnitruncated 16-cell): a uniform polychoron bounded by 8 great rhombicuboctahedra, 16 truncated octahedra, 24 octagonal prisms, and 32 hexagonal prisms.

## The 24-cell family

The 24-cell family consists of 16 members, 3 of which overlap with the
tesseract/16-cell family because of the coincidence of the
24-cell with the rectified 16-cell. Furthermore, due to
the fact that the 24-cell is self-dual, only 9 of the members of this family
are unique. The self-duality also causes some members (marked with
^{*}) of this family to have a higher degree of symmetry than the
24-cell itself.

One special member of this family, the snub 24-cell, has a diminished 24-cell symmetry. It is not derived by the usual uniform truncation processes, but by a particular partitioning of the 24-cell's edges in the Golden Ratio.

(Dual 24-cell: identical to the 24-cell.)

(Rectified dual 24-cell: identical to the rectified 24-cell.)

(Truncated dual 24-cell: identical to the truncated 24-cell.)

Rectified 24-cell: a pretty uniform polychoron bounded by 24 cubes and 24 cuboctahedra.

(Cantellated dual 24-cell: identical to the cantellated 24-cell.)

Bitruncated 24-cell

^{*}: a beautiful uniform polychoron bounded by 48 truncated cubes.(Cantitruncated dual 24-cell: identical to the cantitruncated 24-cell.)

24-cell: the regular member of this family.

Runcinated 24-cell

^{*}: a uniform polychoron bounded by 48 octahedra and 192 triangular prisms.Cantellated 24-cell: a pretty uniform polychoron bounded by 24 rhombicuboctahedra, 24 cuboctahedra, and 96 triangular prisms.

(Runcitruncated dual 24-cell: identical to the runcitruncated 24-cell.)

Truncated 24-cell: a uniform polychoron bounded by 24 cubes and 24 truncated octahedra.

Runcitruncated 24-cell: a beautiful uniform polychoron bounded by 24 truncated octahedra, 24 rhombicuboctahedra, 96 triangular prisms, and 96 hexagonal prisms.

Cantitruncated 24-cell: a pretty polychoron bounded by 24 great rhombicuboctahedra, 24 truncated cubes, and 96 triangular prisms.

Omnitruncated 24-cell

^{*}: a beautiful uniform polychoron bounded by 48 great rhombicuboctahedra and 192 hexagonal prisms.Snub 24-cell: a semi-regular polychoron bounded by 24 icosahedra and 120 tetrahedra. It has a diminished 24-cell symmetry, and can be used as an intermediate to derive the 600-cell from the 24-cell.

## The 120-cell/600-cell family

The 120-cell/600-cell family comprises 15 uniform truncates. All 15 are
distinct. In addition, we list the grand antiprism here,
even though strictly speaking it does not belong in this family, but it
*can* be constructed from the 600-cell by removing two rings of 10
vertices each.

600-cell: one of the regular members of this family.

Rectified 600-cell: a beautiful quasiregular polychoron bounded by 120 regular icosahedra and 600 octahedra.

Truncated 600-cell: a beautiful uniform polychoron bounded by 120 regular icosahedra and 600 truncated tetrahedra.

Rectified 120-cell: a beautiful uniform polychoron bounded by 120 icosidodecahedra and 600 tetrahedra.

Cantellated 600-cell: a beautiful uniform polychoron bounded by 120 icosidodecahedra, 600 cuboctahedra, and 720 pentagonal prisms.

Bitruncated 120-cell (bitruncated 600-cell): a beautiful uniform polychoron bounded by 120 truncated icosahedra and 600 truncated tetrahedra.

Cantitruncated 600-cell: a beautiful uniform polychoron with 120 truncated icosahedra, 720 pentagonal prisms, and 600 truncated octahedra.

120-cell: the other regular member of this family.

Runcinated 120-cell (runcinated 600-cell): a beautiful uniform polychoron bounded by 120 regular dodecahedra, 600 tetrahedra, 720 pentagonal prisms, and 1200 triangular prisms, for a whopping total of 2640 cells.

Cantellated 120-cell: a beautiful uniform polychoron with 120 rhombicosidodecahedra, 600 octahedra, and 1200 triangular prisms.

Runcitruncated 600-cell: a beautiful uniform polychoron bounded by 120 rhombicosidodecahedra, 600 truncated tetrahedra, 720 pentagonal prisms, and 1200 hexagonal prisms.

Truncated 120-cell: a cute uniform polychoron bounded by 120 truncated dodecahedra and 600 tetrahedra.

Runcitruncated 120-cell: a cute uniform polychoron with 120 truncated dodecahedra, 720 decagonal prisms, 1200 triangular prisms, and 600 cuboctahedra.

Cantitruncated 120-cell: a beautiful uniform polychoron with 120 great rhombicosidodecahedra, 1200 hexagonal prisms, and 600 truncated tetrahedra.

Omnitruncated 120-cell (omnitruncated 600-cell): the grand-daddy of them all, the largest convex uniform polychoron, having 120 great rhombicosidodecahedra, 720 decagonal prisms, 1200 hexagonal prisms, and 600 truncated octahedra.

Grand antiprism: an unusual polychoron bounded by 20 pentagonal antiprisms and 300 tetrahedra. It is the closest uniform 4D equivalent to the 3D family of antiprisms.

## Prismatic Uniform Polychora

In addition to the uniform polychora derived from the symmetries of regular
polychora, there are also the 18 *polyhedral prisms*, which are simply
extrusions of the 3D Platonic solids and Archimedean polyhedra. One of them,
the cubic prism, is simply the tesseract, which is already
included in the tesseract family, so leaving 17 distinct polyhedral prisms.
Then there are the prisms of the 3D polygonal antiprisms, which form an
infinite family.

Furthermore, there is a second family of prismatic (prism-like) polychora,
the **duoprisms**. This is the family of the
*Cartesian products* of two polygons. They consists of two mutually
perpendicular rings of 3D prisms. These include the prisms of the 3D prisms,
which are equivalent to the Cartesian product of a polygon and a square.