The Uniform Polyhedra
A basic understanding of 3D polyhedra is important for understanding 4D polychora, because the cells of 4D polychora are 3D polyhedra. The cells of the uniform polychora are uniform polyhedra.
A polyhedron is uniform if its faces are regular polygons, and its vertices are equivalent (transitive) under its symmetry group. The uniform polyhedra include the 13 Archimedean polyhedra (15 if we count the left-handed and right-handed variants of the snub cube and snub dodecahedron separately), which have the symmetries of the Platonic solids, and two infinite families of the prisms and the antiprisms.
The above image shows a “family portrait” of the Archimedean polyhedra as well as a few examples of prisms and an antiprism. The selected examples of prisms and antiprism are those that occur as cells in the 4D uniform polychora. From left to right, back to front, the pictured polyhedra are:
(4th row) The great rhombicuboctahedron, the (small) rhombicuboctahedron, the truncated octahedron, the truncated icosahedron, the (small) rhombicosidodecahedron, and the great rhombicosidodecahedron;
As the names of the individual Archimedean polyhedra suggest, they are derived from the Platonic solids, and thus share the same symmetries. We may thus classify them according to the Platonic solids they derive their symmetry from. The prisms and antiprisms derive their symmetry from the regular polygons. The prisms are simply extrusions of 2D polygons into 3D; the antiprisms are two dual polygons joined to each other by equilateral triangles.
The snub cube and snub dodecahedron are special cases among the Archimedean polyhedra, in that they only have half the symmetries of their corresponding Platonic solids (the cube and the dodecahedron), and are chiral: they are not the same as their mirror images. For this reason, we included both variants of each of them in the family portrait. There are also two special cases among the prisms and antiprisms: the square prism is identical to the cube, and the triangular antiprism is identical to the octahedron.
The Tetrahedron Family
The tetrahedron family comprises of 7 members, including the regular tetrahedron itself. However, due to the fact that the tetrahedron is the alternated cube, three of these members coincide with members in the cube family. Furthermore, since the tetrahedron is self-dual, two of the remaining four members are identical to the other two members, except in dual orientation. Hence, there are only two distinct members in the tetrahedron family.
(Dual tetrahedron: identical to the tetrahedron.)
(Rectified tetrahedron: identical to the octahedron.)
(Truncated dual tetrahedron: identical to the truncated tetrahedron.)
Tetrahedron: bounded by 4 triangles.
(Cantellated tetrahedron: identical to the cuboctahedron.)
Truncated tetrahedron: bounded by 4 triangles and 4 hexagons.
(Omnitruncated tetrahedron: identical to the truncated octahedron.)
The Cube Family
The cube family comprises 8 members, 7 with full cubic symmetry, and the snub cube, with diminished cubic symmetry. This family includes the cube and the regular octahedron. The snub cube is chiral, and so has two enantiomorphs (mirror images).
Octahedron: bounded by 8 triangles.
Cuboctahedron: a quasi-regular polyhedron bounded by 8 triangles and 6 squares.
Truncated octahedron: bounded by 8 hexagons and 6 squares.
Cube: bounded by 6 squares.
(Small) Rhombicuboctahedron: bounded by 8 triangles and 6+12=18 squares.
Truncated cube: bounded by 8 triangles and 6 octagons.
Great rhombicuboctahedron: bounded by 8 hexagons, 6 octagons, and 12 squares.
Snub cube: bounded by 8+24=32 triangles and 6 squares.
The great rhombicuboctahedron is also known as the “truncated cuboctahedron”. However, this name is actually a misnomer, because truncating the cuboctahedron does not yield a uniform polyhedron, only a non-uniform topological equivalent to the real Archimedean polyhedron. The correct derivation of the great rhombicuboctahedron is by radially expanding the hexagonal faces of the truncated octahedron and filling in the resulting gaps with squares and octagons. It can also be derived by expanding the octagonal faces of the truncated cube or the 12 non-axial square faces (the ones that share an edge with two triangles, not the ones surrounded by other square faces) of the rhombicuboctahedron.
The Dodecahedron Family
The dodecahedron family also comprises 8 members, 7 with full icosahedral symmetry, and the snub dodecahedron, with diminished icosahedral symmetry. This family includes the regular dodecahedron and the icosahedron themselves. Just like the snub cube, the snub dodecahedron is chiral, and has two enantiomorphs (mirror images).
Icosahedron: bounded by 20 triangles.
Icosidodecahedron: bounded by 20 triangles and 12 pentagons.
Truncated icosahedron: bounded by 20 hexagons and 12 pentagons. Commonly recognized as the stitching pattern on a soccer ball, and also the shape of buckminsterfullerene.
Regular dodecahedron: bounded by 12 pentagons.
(Small) Rhombicosidodecahedron: bounded by 20 triangles, 12 pentagons, and 30 squares.
Truncated dodecahedron: bounded by 20 triangles and 12 decagons.
Great rhombicosidodecahedron: bounded by 20 hexagons, 12 decagons, and 30 squares.
Snub dodecahedron: bounded by 20+60=80 triangles and 12 pentagons.
The great rhombicosidodecahedron is also known as the “truncated icosidodecahedron”. However, just as with the “truncated cuboctahedron”, this is a misnomer, because truncating the icosidodecahedron does not yield a uniform polyhedron, only a non-uniform topological equivalent. The correct derivation of the great rhombicosidodecahedron is by radially expanding the hexagonal faces of the truncated icosahedron and filling in the gaps with squares and decagons. Other correct derivations include expanding the decagonal faces of the truncated dodecahedron, or the square faces of the (small) rhombicosidodecahedron.
The Prisms & Antiprisms
The families of the prisms and the antiprisms have infinitely many members. They derive their symmetries from 2D regular polygons. The square prism is identical to the cube, and the triangular antiprism is identical to the regular octahedron; these two members have a much greater degree of symmetry than the other members due to this coincidence.
Since it is impossible to list an infinite number of prisms and antiprisms, we only list some representative examples here.
Triangular prism: bounded by 2 triangles and 3 squares.
(Triangular antiprism: identical to the regular octahedron.)
(Square prism: identical to the cube.)
Square antiprism: bounded by 2 squares and 8 triangles.
Pentagonal prism: bounded by 2 pentagons and 5 squares.
Pentagonal antiprism: bounded by 2 pentagons and 10 triangles.
Hexagonal prism: bounded by 2 hexagons and 6 squares.
Hexagonal antiprism: bounded by 2 hexagons and 12 triangles.
Octagonal prism: bounded by 2 octagons and 8 squares.
Octagonal antiprism: bounded by 2 octagons and 16 triangles.
Decagonal prism: bounded by 2 decagons and 10 squares.
Decagonal antiprism: bounded by 2 decagons and 20 triangles.