Regular Polychora

Category 1: Regular polychora

Above are sections of four of the regular polychora from the dodecahedral perspective, the number represents the location of the sectioning realm (i.e. .30 section is 30 percent down from the top).

The regular polychora are the polychora with congruent vertices and congruent cells, where the cells are regular. Below is a list of the 16 regulars. The first 6 are convex, the others are non-convex. The regular polychora are also mentioned in H.S.M. Coxeter's book Complex Polytopes.

1. Pen - pentachoron (5-cell) - cells are 5 tetrahedra (tets), its vertex figure is also a tet, it has 5 vertices, 10 edges and 10 triangular faces.

2. Tes - tesseract - cells are 8 cubes, its vertex figure is a tet, it has 16 vertices, 32 edges, and 24 faces (squares).

3. Hex - hexadecachoron (16-cell) - cells are 16 tets, vertex figure is an octahedron (oct), it has 8 vertices, 24 edges, and 32 faces (triangles). The hex regiment also contains tho (#17).

4. Ico - icositetrachoron (24-cell) - cells are 24 octs, vertex figure is a cube, it has 24 vertices, 96 edges, and 96 triangular faces. The ico regiment contains 14 polychora (category 4)

5. Hi - hecatonicosachoron (120-cell) - cells are 120 dodecahedra (does), vertex figure is a tet, it has 600 vertices, 1200 edges, and 720 pentagonal faces.

6. Ex - hexacosichoron (600-cell) - cells are 600 tets, vertex figure is an icosahedron (ike), it has 120 vertices, 720 edges, and 1200 triangular faces. The ex regiment contains 6 polychora (ex, fix, gohi, gahi, sachi, and sisp (the small swirlprism)).

7. Fix - faceted hexacosichoron (also called icosahedral hecatonicosachoron) - cells are 120 ikes, vertex figure is a gad (great dodecahedron), it has 120 vertices, 720 edges, and 1200 triangular faces. It is in the ex regiment.

8. Gohi - great hecatonicosachoron - cells are 120 gads, vertex figure is a sissid (small stellated dodecahedron), it has 120 vertices, 720 edges, and 720 pentagonal faces. Its in the ex regiment.

9. Gahi - grand hecatonicosachoron - cells are 120 does, vertex figure is a gike (great icosahedron), it has 120 vertices, 720 edges, and 720 pentagonal faces. Its in the ex regiment.

10. Sishi - small stellated hecatonicosachoron - cells are 120 sissids, vertex figure is a doe, it has 120 vertices, 1200 edges, and 720 star (pentagram) faces. The sishi regiment contains 44 polychora (sishi, gaghi, 8 swirlprisms, and 34 others which are in category 17 (the great prismasaurus, paphacki, is one of these)).

11. Gaghi - great grand hecatonicosachoron - cells are 120 gads, vertex figure is a gissid (great stellated dodecahedron), it has 120 vertices, 1200 edges, and 720 pentagonal faces. Its in the sishi regiment.

12. Gishi - great stellated hecatonicosachoron - cells are 120 gissids, vertex figure is an ike, it has 120 vertices, 720 edges, and 720 star faces. The gishi regiment contains 6 polychora (gishi, gashi, gofix, gax, gachi, and gisp (great swirlprism)).

13. Gashi - grand stellated hecatonicosachoron - cells are 120 sissids, vertex figure is a gad, it has 120 vertices, 720 edges, and 720 star faces. Its in the gishi regiment.

14. Gofix - great faceted hexacosichoron (also called great icosahedral hecatonicosachoron) - cells are 120 gikes, vertex figure is a sissid, it has 120 vertices, 720 edges, and 1200 triangular faces. Its in the gishi regiment.

15. Gax - grand hexacosichoron - cells are 600 tets, vertex figure is a gike, it has 120 vertices, 720 edges, and 1200 triangular faces. Its in the gishi regiment.

16. Gogishi - great grand stellated hecatonicosachoron - cells are 120 gissids, vertex figure is a tet, it has 600 vertices, 1200 edges, and 720 star faces.

These next three are not regular, but they belong to the smaller regular regiments.

17. Tho - tesseractihemioctachoron - cells are 8 tets and 4 octs, vertex figure is a thah (tetrahemihexahedron). It has 8 vertices and has hemitesseractic symmetry. It also has 24 edges, and 32 triangular faces. It is in the hex regiment.

18. Sachi - small complexihecatonicosachoron - cells are 120 cids (complex icosidodecahedron - this is the exopolyhedron with the faces of ike and gad), vertex figure looks just like a gad, only the interior is different. It has 120 vertices, 720 edges, 1200 triangular faces, and 720 pentagonal faces. It looks exactly like fix, only the interior is different (I call this type of case a copycat). Its in the ex regiment.

19. Gachi - great complexihecatonicosachoron - cells are 120 gacids (great complex icosidodecahedron - this is the exopolyhedron with the faces of sissid and gike), vertex figure looks just like a gad, only the interior differs. It has 120 vertices, 720 edges, 1200 triangular faces, and 720 star faces. It is a copycat of gashi and is in the gishi regiment.