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.

Index	Name	OBSA	Picture	Coxeter diagram	Vertex figure	Tera	Army	Regiment	Company	Symmetry
1	Hexateron	Hix			Pentachoron	6 pen	Hix 6 vertices	Hix 15 edges	Hix	A5
2	Penteract	Pent			Pentachoron	10 tes	Pent 32 vertices	Pent 80 edges	Pent	B5
3	Triacontaditeron	Tac			Hexadecachoron	32 pen	Tac 10 vertices	Tac 40 edges	Tac	B5
4	Hexadecahemidecateron	Hehad		/2	Tesseractihemioctachoron	16 pen 5 hex	Tac	Tac	Tac	D5
5	Pentachoric hemiantiprism	Phap				2+10 pen 5 tho	Tac	Tac	Tac	A4x2xA1+
6	Spinopentachoric hemiantiprism	Nophap				20 pen 5 tho	Tac	Tac	Tac	A4x2xA1+
7	Demipenteract	Hin			Rectified pentachoron	10 hex 16 pen	Hin 16 vertices	Hin 80 edges	Hin	D5
8	Dishexadecateron	Dah		/2	Facetorectified pentachoron	16 pen 16 rap	Hin	Hin	Hin	D5
9	Hexadecapenteract	Han			Prismatointercepted pentachoron	10 hex 16 rap	Hin	Hin	Hin	D5
10	Retrodishexadecateron	Radah				16 pen 16 firp	Hin	Hin	Radah	D5
11	Retropenteractihexadecateron	Rinah				10 hex 16 firp	Hin	Hin	Radah	D5
12	Hexadecateron	Hit			Noble faceting of rectified pentachoron	16 pinnip	Hin	Hin	Radah	D5

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.

Index	Name	OBSA	Picture	Coxeter diagram	Vertex figure	Tera	Army	Regiment	Company	Symmetry
13	Truncated hexateron	Tix			Tetrahedral pyramid	6 pen 6 tip	Tix 30 vertices	Tix 75 edges	Tix	A5
14	Truncated penteract	Tan			Tetrahedral pyramid	10 tat 32 pen	Tan 160 vertices	Tan 400 edges	Tan	B5
15	Truncated triacontaditeron	Tot			Octahedral pyramid	32 tip 10 hex	Tot 80 vertices	Tot 280 edges	Tot	B5
16	Truncated hexadecahemidecateron	Thehid		/2	Tetrahemihexahedral pyramid	16 tip 5 thex 10 tho	Tot	Tot	Tot	D5
17	Quasitruncated penteract	Quittin			Tetrahedral pyramid	10 quitit 32 pen	Scant 160 vertices	Quittin 400 edges	Quittin	B5
18	Truncated demipenteract	Thin			Triangular prismatic pyramid	10 thex 16 tip 16 rap	Thin 160 vertices	Thin 560 edges	Thin	D5
19	Truncated dishexadecateron	Tadah		/2	Triangular retroprismatic pyramid	16 tip 16 grip 16 firp	Thin	Thin	Tadah	D5
20	Truncated hexadecapenteract	Tohan			Triangular toroprismatic pyramid	10 thex 16 grip 16 pinnip	Thin	Thin	Tohan	D5
21	Bitruncated hexateron	Bittix			Triangular scalene	6 deca 6 tip	Bittix 60 vertices	Bittix 150 edges	Bittix	A5
22	Bitruncated penteract	Bittin			Triangular scalene	10 tah 32 tip	Bittin 320 vertices	Bittin 800 edges	Bittin	B5
23	Bitruncated triacontaditeron	Bittit			Square scalene	32 deca 10 thex	Bittit 240 vertices	Bittit 720 edges	Bittit	B5
24	Bitruncated hexadecahemidecateron	Bathehad		/2	Bowtie scalene	16 deca 10 titho 5 tah	Bittit	Bittit	Bathehad	D5

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.

Index	Name	OBSA	Picture	Coxeter diagram	Vertex figure	Tera	Army	Regiment	Company	Symmetry
1288	Small ditetrahedronary hexacosihecatonicosachoric antiprism	Sidtaxhiap			Tetrahedron atop truncated tetrahedron	1200 pen 120 sidtidap 2 sidtaxhi	Hipe 1200 vertices	Sidtaxhiap 9600 edges	Sidtaxhiap	H4×A1
1289	Grand ditetrahedronary hexacosihecatonicosachoric antiprism	Gadtaxhiap		oo3oo3xo5ox3/2*b&#x	Tetrahedron atop quasitruncated tetrahedron	1200 pen 120 gidtidap 2 gadtaxhi	Hipe 1200 vertices	Gadtaxhiap 9600 edges	Gadtaxhiap	H4×A1
1290	Hollow stellated icositetrachoric antiprism	Hosiap				48 hex as 24 haddet 96 pen as 48 sopy	Icope 48 vertices	Hosiap 336 edges	Hosiap	F4×A1
1291	Inverted hollow stellated icositetrachoric antiprism	Ihsiap				48 hex as 24 haddet 48 pen 48 rap	Icope 48 vertices	Hosiap 336 edges	Hosiap	(F4/2×2×A1)/2
1292	Hollow stellated icositetrachoric retroantiprism	Hesirap				48 hex as 24 haddet 48 pen 48 firp	Icope 48 vertices	Hosiap 336 edges	Hesirap	(F4/2×2×A1)/2
	Retrohollow stellated icositetrachoric antiprism	Rahsiap				48 pen 48 tepe 48 firp	Icope 48 vertices	Hosiap 336 edges	Rahsiap	F4/2×A1
	Spinohollow stellated icositetrachoric hemiantiprism	Nahsihap				48 tho 48 rap 48 pen	Icope 48 vertices	Hosiap 336 edges	Nahsihap	F4/2×A1
	Hollow stellated icositetrachoric hemiantiprism	Hesihap				96 pen 48 tepe 48 pinnip 48 tho	Icope 48 vertices	Hosiap 336 edges	Hesihap	F4/2×A1
1293	Dodecahedronary prismatohexacosichoric alterprism	Dopixa				120 spid as 2 dopix 1200 firp 2400 tepe	Exip 240 vertices	Sishia 3840 edges		H4×A1
1294	Invertidodecahedronary hecatonicosachoric alterprism	Idhia				2 idhi 120 ditdidap 720 pappip 1440 trapedip	Exip 240 vertices	Sishia 3840 edges		H4×A1