List of uniform polytera

Category 1: Primaries (#1-12)
These are the 3 flag-transitive polytera and their facetings, as well as the hin (demipenteract) regiment, which has a rectified pentachoron vertex figure and contains 6 members.

Category 2: Truncates (#13-24)
These are the truncations, quasitruncations and bitruncations of the primaries, as well as their facetings. Their vertex figures are pyramids of the original edge figures (truncates) or scalenes of the original face figures.

Category 3: Rectates (#25-47)
These are the rectifications and birectifications of the primaries, with the exception of the larger regiments such as rat and nit. Rix (rectified hexateron) has 7 members, dot (dodecateron) has 5, and rin (rectified penteract) has 11. Their vertex figures are facetings of tetrahedral prisms (rix and rin) or the triangular duoprism (dot).

Category 4: Rat Regiment (#48-91)
Rat is the rectified triacontaditeron. Its vertex figure is the octahedral prism. There are 44 members.

Category 5: Nit Regiment (#92-240)
Nit is the penteractitriacontaditeron, which has a triangular-square duoprism vertex figure. It is the largest typical regiment among the nonprismatic uniform polytera, for it has 149 members.

Category 6: Sphenoverts (#241-377)
These are all of the polytera with wedge vertex figures, which includes the cantellated polytera. There are 3 regiments with 15 members each, and 1 regiment with 92.

Category 7: Birhombates (#378-447)
These are the bicantellated polytera which have duowedge vertex figures. Sibrid (small birhombidodecateron) has 23 members, while sibrant (small birhombipenteractitriacontaditeron) has 47 members.

Category 8: Fastegiumverts (#448-680)
These polytera have triangular fastegium or antifastegium vertex figures, including the runcinated polytera. There are 4 regiments with 37 members each, and 1 regiment with 85.

Category 9: Podiumverts (#681-721)
These are the expanded (stericated) polytera and all other polytera with podium vertex figures. There are 3 regiments, where 1 has 11 members and the other 2 have 15 members.

Category 10: Greater Truncates (#722-747)
This category includes the lone operative polytera other than the primaries and the truncates, with the exception of gart (great rhombated triacontaditeron) which has 2 members. Their vertex figures are irregular pentachora.

Category 11: Sphenopyriverts (#748-786)
These are the polytera with wedge pyramid vertex figures, which includes the runcitruncated polytera. There are 4 regiments with 7 members and 1 (pattit) with 11 members.

Category 12: Podipyriverts (#787-828)
These polytera have podium or antipodium pyramid vertex figures, including the steritruncated polytera. There are 6 regiments with 7 members each.

Category 13: Prismatorhombates and Kin (#829-870)
These polytera have trapezoid disphenoid or trapezoid pyramidal pyramid vertex figures. There are 14 regiments with 3 members each.

Category 14: Antisphenoverts (#871-942)
These are the stericantellated polytera, as well as polytera with related vertex figures. There are 3 regiments, where 1 has 18 members and the other 2 have 27 members.

Category 15: Siphin Regiment (#943-998)
Siphin (small prismatodemipenteract) has a tet || oct vertex figure. There are 56 members.

Category 16: Skivbadant and Gikvacadant Regiments (#999-1124)
These are the polytera with skewed duowedge vertex figures and their facetings. They have 63 members each.

Category 17: Sibacadint and Gidacadint Regiments (#1125-1154)
These are the polytera with skewed wedge pyramid vertex figures and their facetings. They have 15 members each.

Category 18: Skatbacadint Regiment (#1155-1287)
Skatbacadint (small skewtrigonary biprismatocellidis10-32teron) has a skewed antifastegium vertex figure. It is the second largest polyteron regiment, with 133 members.

Category 19: Miscellaneous (#1288-1297)
These are the atypical polytera that do not fit in any other category. This includes the two Johnson antiprisms, 6 in the hosiap regiment including itself, 2 with (F4/2×2×A1)/2 symmetry, and 3 with F4/2×A1 symmetry; and the sishia regiment which has 2 uniform members.