List of uniform polychora

This list of uniform polychora is sorted based on Jonathan Bowers' Polychoron Site. Currently, 2191 uniform polychora are known, plus two infinite families: the polygonal duoprisms and polygonal-antiprismatic prisms. At least 305 fissary polychora (plus an unknown number in the last three categories) are excluded from the main count.

Category A: Duoprisms (infinite)
These are prism products of two polygons. A subset of these are the prisms of 3-dimensional prisms. They have either tetragonal disphenoid (for n,n-duoprisms) or digonal disphenoid (m,n-duoprisms) vertex figures.

Category B: Antiduoprisms (infinite)
These are prisms of 3-dimensional antiprisms. Their vertex figures are isosceles trapezoidal pyramids or crossed isosceles trapezoidal pyramids.

Category 1: Regulars + 1 (#1-17)
These are the flag-transitive uniform polychora. There are 6 convex regulars and 10 nonconvex regulars. One quasiregular polychoron is included here for lack of a better placement.

Category 2: Truncates (#18-38)
These are the truncations and quasitruncations of the regular polychora and other non-rectates with uniform vertex figures. Their vertex figures are pyramids of the original edge figures.

Category 3: Triangular Rectates (#39-59)
These are the rectifications of regular polychora where three cells meet at an edge, and their facetings. There are 7 regiments which have 3 members in this category each (two of them have more members in other categories). Their vertex figures are triangular prisms, triangular retroprisms and triangular toroprisms, one of each in each regiment.

Category 4: Ico Regiment (#60-72)
The icositetrachoron is regular and is also the rectified hexadecachoron and the rectified demitesseract. This category contains all the members of its regiment except ico itself. There are 13 members in this category with three different types of symmetry. Their vertex figures are semi-uniform facetings of a cube. They could also be called the "square rectates."

Category 5: Pentagonal Rectates (#73-132)
These are the rectifications of the regular polychora where five cells meet at an edge. There are four regiments with 2 rectates and 13 faceted rectates each. Their vertex figures are pentagonal prisms and their facetings.

Category 6: Sphenoverts (#133-297)
This category includes the cantellated polychora, all polychora with wedge vertex figures, and their facetings. There are 24 regiments with 7 members each, although the rico regiment contains 3 members counted previously.

Category 7: Bitruncates (#298-306, F1-F2)
These are the bitruncations and biquasitruncations of the regular polychora. They have either tetragonal disphenoid (for doubled symmetry) or digonal disphenoid vertex figures. Two of them are fissary due to coinciding vertices and are generated as double covers.

Category 8: Great Rhombates (#307-329)
These polychora, which include the cantitruncated polychora, have sphenoid vertex figures.

Category 9: Omnitruncates (#330-351)
These polychora are represented by Coxeter-Dynkin diagrams with all nodes ringed. This includes the omnitruncates of regular polychora. Their vertex figures are either phyllic disphenoids (if they have doubled symmetry) or irregular tetrahedra.

Category 10: Prismatorhombates (#352-441)
These are polychora with isosceles trapezoidal pyramid or skewed rectangular pyramid vertex figures and their facetings, including runcitruncations of the regular polychora. There are 30 regiments of 3.

Category 11: Antipodiumverts (#442-481)
These are the uniform polychora with triangular antipodium (or triangular antiprism) vertex figures and their facetings. This includes the runcinations of regular polychora. There are 6 regiments, one which has 5 members and five with 7.

Category 12: Podiumverts (#482-511)
These are the uniform polychora with triangular podium vertex figures and their facetings. There are 4 regiments with 7 members each as well as 2 additional members of a previous regiment. The quasiruncinations are included in this category.

Category 13: Spic and Giddic Regiments (#512-551, F3-F6)
These are the polychora with square antiprism vertex figures and their facetings. Spic (small prismatotetracontoctachoron) is the runcination of ico, while its conjugate quippic (a member of the giddic regiment) is its conjugate and ico's quasiruncination. There are 2 regiments with 20 members each.

Category 14: Skewverts (#552-611)
These are the polychora with skewed wedge vertex figures. There are 4 regiments of 15.

Category 15: Afdec Regiment (#612-664, F7)
Afdec (antifrustary distetracontoctachoron) is a polychoron with a rectangular trapezoprism vertex figure. Its regiment has 53 members plus one fissary.

Category 16: Affixthi Regiment (#665-763, F8)
Affixthi (antifrustary 600tris120choron) is a polychoron with a rectangular frustum vertex figure. Its regiment has 99 members plus one fissary.

Category 17: Sishi Regiment (#764-777, F9-F10)
The small stellated hecatonicosachoron (short name sishi), a regular polychoron with a dodecahedron vertex figure, has several uniform facetings. The following are those that are fully symmetric but not regular, numbering 14 plus 2 fissaries.

Category 18: Ditetrahedrals (#778-888, F11-F22)
This category contains three regiments with 37 members and at least 4 fissaries each (the fissaries of category 7 could also be placed here). Their vertex figures are facetings of truncated tetrahedra or a rhombitetratetrahedron. The 4D equivalents of sidtid, ditdid and gidtid are in this category.

Category 19: Prisms (#889-962)
These are the prisms of the uniform polyhedra, except for the cube whose prism is the tesseract, and the nonregular prisms and antiprisms whose prisms are in categories A and B. There are 74 of these. Their vertex figures are the pyramids of those of the polyhedra.

Category 20: Miscellaneous (#963-984, 1846-1855, 2189-2191, 60 fissaries)
These are the atypical polychora that do not fit in any other large category. They include uniform swirlprisms, eight sidpith or gittith-related polychora, normal snubs and 4D (duo)antiprism(oid)s.

Category 21: Padohi Regiment (#985-1065, 78 fissaries)
Padohi (prismatodis120choron) is the runcinated small stellated hecatonicosachoron. Its vertex figure is a pentagonal antipodium. There are 81 members.

Category 22: Gidipthi Regiment (#1066-1146, 78 fissaries)
Gidipthi (great dipentary tris120choron) is a polychoron with a pentagonal podium vertex figure. There are 81 members in its regiment, including quipdohi (the conjugate of padohi).

Category 23: Rissidtixhi Regiment (#1147-1303, F28-F30)
Rissidtixhi is the rectification of sidtixhi from category 17. Its vertex figure is a ditrigonal prism. There are 157 members.

Category 24: Stut Phiddix Regiment (#1304-1382, 30 fissaries)
There are 79 uniform polychora in this regiment. The vertex figure is a triangular cupola.

Category 25: Getit Xethi Regiment (#1383-1461, 30 fissaries)
There are 79 uniform polychora in this regiment, including getit phiddix (the conjugate of stut phiddix). The vertex figure is a triangular cupola.

Category 26: Sabbadipady Regiment (Blends) (#1462-1473, F24-F27)
The regiment of the polychoron sabbadipady, created by blending siddapady and quit sishi. It has 12 members. If gohi is cantitruncated, the result is an exotic polychoroid that is part of this regiment.

Category 27: Sidtaps and Gidtaps (#1474-1491)
These are blended snubs; there are 2 regiments of 9. Sadsadox is the blend of 10 rox, while gadsadox is the blend of 10 raggix. They are called the "baby monster snubs." Their vertex figures are facetings of blends of two pentagonal prisms or pentagrammic prisms. Sadsadox and gadsadox were originally called sidtapsadix and gidtapsadix, hence the names.

Category 28: Sadros Daskydox Regiment (Idcossids) (#1492-1668)
This regiment is based on the compound of 10 padohi. It has many facetings that blend, causing them to be true uniform polychora, numbering 177. The name "idcossid" derives from the first member discovered, which is exotic and no longer counted. Under the old system where it was counted, there were 2,749 idcossids. Together with the next category, it forms the "monster snubs."

Category 29: Gadros Daskydox Regiment (Dircospids) (#1669-1845)
This regiment is based on the compound of 10 gidipthi. It has many facetings that blend, causing them to be true uniform polychora, numbering 177. The name "dircospid" derives from the first member discovered, which is exotic and no longer counted. Together with the previous category, it forms the "monster snubs." Together with the previous regiment which is its conjugate regiment, it was once the largest known polychoron regiment.

Category 30: Disdi Regiment (Idtessids) (#1856-2188)
This regiment was first discovered in 2006 with the scaliform member disdi, but in 2021, 333 uniform members were discovered, making it the current largest known regiment of uniform polychora. They have either pyritotesseractic or chiral demitesseractic symmetry. "Idtessid" was never an actual OBSA, it is just a pun on "idcossid" and "tesseractic".