List of uniform polychora

This list of uniform polychora is sorted based on Jonathan Bowers' Polychoron Site. Currently, 2189 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.

Name	OBSA	Picture	Coxeter diagram	Vertex figure	Cells	Army	Regiment	Company	Symmetry
Triangular duoprism	Triddip			Tetragonal disphenoid	6 trip	Triddip 9 vertices	Triddip 18 edges		A2≀S2
Triangular-square duoprism	Tisdip			Digonal disphenoid	4 trip 3 cube	Tisdip 12 vertices	Tisdip 24 edges		A2×B2
Triangular-pentagonal duoprism	Trapedip			Digonal disphenoid	5 trip 3 pip	Trapedip 15 vertices	Trapedip 30 edges		A2×H2
Triangular-pentagrammic duoprism	Tistadip			Digonal disphenoid	5 trip 3 stip	Semiuniform Trapedip	Tistadip 30 edges		A2×H2
Square-pentagonal duoprism	Squipdip			Digonal disphenoid	5 cube 4 pip	Squipdip 20 vertices	Squipdip 40 edges		B2×H2
Square-pentagrammic duoprism	Sistadip			Digonal disphenoid	5 cube 4 stip	Semiuniform Squipdip	Sistadip 40 edges		B2×H2
Pentagonal duoprism	Pedip			Tetragonal disphenoid	10 pip	Pedip 25 vertices	Pedip 50 edges		H2≀S2
Pentagonal-pentagrammic duoprism	Starpedip			Digonal disphenoid	5 pip 5 stip	Semiuniform Pedip	Starpedip 50 edges		H2×H2
Pentagrammic duoprism	Stardip			Tetragonal disphenoid	10 stip	Pedip	Stardip 50 edges		H2≀S2

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.

Index	Name	OBSA	Picture	Coxeter diagram	Vertex figure	Cells	Army	Regiment	Company	Symmetry
1	Pentachoron	Pen			Tetrahedron	5 tet	Pen 5 vertices	Pen 10 edges	Pen	A4
2	Tesseract	Tes			Tetrahedron	8 cube	Tes 16 vertices	Tes 32 edges	Tes	B4
3	Hexadecachoron	Hex			Octahedron	16 tet	Hex 8 vertices	Hex 24 edges	Hex	B4
4	Icositetrachoron	Ico			Cube	24 oct	Ico 24 vertices	Ico 96 edges	Ico	F4
5	Hecatonicosachoron	Hi			Tetrahedron	120 doe	Hi 600 vertices	Hi 1200 edges	Hi	H4
6	Hexacosichoron	Ex			Icosahedron	600 tet	Ex 120 vertices	Ex 720 edges	Ex	H4
7	Faceted hexacosichoron	Fix			Great dodecahedron	120 ike	Ex	Ex	Ex	H4
8	Great hecatonicosachoron	Gohi			Small stellated dodecahedron	120 gad	Ex	Ex	Gohi	H4
9	Grand hecatonicosachoron	Gahi			Great icosahedron	120 doe	Ex	Ex	Gohi	H4
10	Small stellated hecatonicosachoron	Sishi			Dodecahedron	120 sissid	Ex	Sishi 1200 edges	Sishi	H4
11	Great grand hecatonicosachoron	Gaghi			Great stellated dodecahedron	120 gad	Ex	Sishi	Gaghi	H4
12	Great stellated hecatonicosachoron	Gishi			Icosahedron	120 gissid	Ex	Gishi 720 edges	Gishi	H4
13	Grand stellated hecatonicosachoron	Gashi			Great dodecahedron	120 sissid	Ex	Gishi	Gishi	H4
14	Great faceted hexacosichoron	Gofix			Small stellated dodecahedron	120 gike	Ex	Gishi	Gofix	H4
15	Grand hexacosichoron	Gax			Great icosahedron	600 tet	Ex	Gishi	Gofix	H4
16	Great grand stellated hecatonicosachoron	Gogishi			Tetrahedron	120 gissid	Hi	Gogishi 1200 edges	Gogishi	H4
17	Tesseractihemioctachoron	Tho		/2	Tetrahemihexahedron	8 tet 4 oct	Hex	Hex	Hex	D4

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.

Index	Name	OBSA	Picture	Coxeter diagram	Vertex figure	Cells	Army	Regiment	Company	Symmetry
18	Truncated pentachoron	Tip			Triangular pyramid	5 tet 5 tut	Tip 20 vertices	Tip 40 edges	Tip	A4
19	Truncated tesseract	Tat			Triangular pyramid	16 tet 8 tic	Tat 64 vertices	Tat 128 edges	Tat	B4
20	Truncated hexadecachoron	Thex			Square pyramid	8 oct 16 tut	Thex 48 vertices	Thex 120 edges	Thex	B4
21	Truncated icositetrachoron	Tico			Triangular pyramid	24 cube 24 toe	Tico 192 vertices	Tico 384 edges	Tico	F4
22	Truncated hecatonicosachoron	Thi			Triangular pyramid	600 tet 120 tid	Thi 2400 vertices	Thi 4800 edges	Thi	H4
23	Truncated hexacosichoron	Tex			Pentagonal pyramid	120 ike 600 tut	Tex 1440 vertices	Tex 4320 edges	Tex	H4
24	Truncated faceted hexacosichoron	Tiffix			Pentagrammic pyramid	120 gad 120 ti	Tex	Tex	Tiffix	H4
25	Truncated great hecatonicosachoron	Tighi			Pentagonal pyramid	120 sissid 120 tigid	Semiuniform Tex	Tighi 4320 edges	Tighi	H4
26	Truncated grand hecatonicosachoron	Taghi			Pentagrammic pyramid	120 gike 120 tid	Semiuniform Tex	Tighi	Taghi	H4
27	Truncated great grand hecatonicosachoron	Tigaghi			Triangular pyramid	120 gissid 120 tigid	Sidpixhi 2400 vertices	Tigaghi 4800 edges	Tigaghi	H4
28	Truncated great faceted hexacosichoron	Tigfix			Pentagonal pyramid	120 sissid 120 tiggy	Semiuniform Tex	Tigfix 4320 edges	Tigfix	H4
29	Truncated grand hexacosichoron	Taggix			Pentagrammic pyramid	120 gike 600 tut	Semiuniform Tex	Tigfix	Taggix	H4
30	Truncated tesseractihemioctachoron	Titho		/2	Bowtie pyramid	8 thah 8 tut 4 toe	Thex	Thex	Titho	D4
31	Truncated small ditrigonary hexacosihecatonicosachoron	Tissidtixhi			Ditrigonal pyramid	120 sidtid 600 tut 120 tiggy	Semiuniform Sidpixhi	Tissidtixhi 8400 edges	Tissidtixhi	H4
33	Truncated great ditrigonary hexacosihecatonicosachoron	Tiggidtixhi			Tripod pyramid	120 gidtid 600 tut 120 ti	Semiuniform Sidpixhi	Tissidtixhi	Tiggidtixhi	H4
32	Truncated ditrigonary dishecatonicosachoron	Tidtidohi			Propeller tripod pyramid	120 ditdid 120 ti 120 tiggy	Semiuniform Sidpixhi	Tissidtixhi	Tidtidohi	H4
34	Quasitruncated tesseract	Quitit			Triangular pyramid	16 tet 8 quith	Sidpith 64 vertices	Quitit 128 edges	Quitit	B4
35	Quasitruncated small stellated hecatonicosachoron	Quit sishi			Triangular pyramid	120 doe 120 quit sissid	Tissidtixhi 2400 vertices	Quit sishi 4800 edges	Quit sishi	H4
36	Quasitruncated great stellated hecatonicosachoron	Quit gishi			Pentagonal pyramid	120 ike 120 quit gissid	Semiuniform Tex	Quit gishi 4320 edges	Quit gishi	H4
37	Quasitruncated grand stellated hecatonicosachoron	Quit gashi			Pentagrammic pyramid	120 gad 120 quit sissid	Semiuniform Tex	Quit gishi	Quit gashi	H4
38	Quasitruncated great grand stellated hecatonicosachoron	Quit gogishi			Triangular pyramid	600 tet 120 quit gissid	Semiuniform Sidpixhi	Quit gogishi 4800 edges	Quit gogishi	H4

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."

Index	Name	OBSA	Picture	Coxeter diagram	Vertex figure	Cells	Army	Regiment	Company	Symmetry
60	Icositetrahemiicositetrachoron	Ihi				24 thah 12 co	Ico 24 vertices	Ico 96 edges		F4/2
61	Octahemihexadecachoron	Oh		/2	Crossed square prism	8 oct 8 oho	Ico	Ico		B4
62	Small hexadecahemihexadecachoron	Shahoh			Inverted square prism	16 oct 8 oho	Ico	Ico		B4
63	Hexadecahemihexadecachoron	Huhoh				16 thah 8 cho	Ico	Ico		B4
64	Hexadecaoctahemihexadecachoron	Hohoh				8 oct 16 thah 8 cho	Ico	Ico		B4
65	Octahexadecahemihexadecachoron	Ohuhoh				8 oct 16 thah 8 co	Ico	Ico		B4
66	Great hexadecahemihexadecachoron	Ghahoh				16 thah 8 co	Ico	Ico		B4
67	Rectified tesseractihemioctachoron	Ratho		/2		8 oct 8 thah 4 co	Ico	Ico		D4
68	Octadishemioctachoron	Odho				8 thah 4 oho 4 cho	Ico	Ico		D4
69	Hexadecaoctadishemioctachoron	Hodho				16 oct 8 thah 4 oho 4 cho	Ico	Ico		D4
70	Hexadecaocta-hemihexadecintercepted hemioctachoron	Hoh honho				24 thah 8 cho 4 co	Ico	Ico		D4
71	Disocta-dishemioctintercepted hemioctachoron	Dod honho				16 thah 4 oho 4 cho 4 co	Ico	Ico		D4
72	Disocta-hemihexadecintercepted hemioctachoron	Doh honho				8 oct 8 thah 4 co 8 oho	Ico	Ico		D4

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.

Index	Name	OBSA	Picture	Coxeter diagram	Vertex figure	Cells	Army	Regiment	Company	Symmetry
298	Decachoron	Deca			Tetragonal disphenoid	10 tut	Deca 30 vertices	Deca 60 edges		A4×2
299	Tesseractihexadecachoron	Tah			Digonal disphenoid	8 toe 16 tut	Tah 96 vertices	Tah 192 edges		B4
300	Tetracontoctachoron	Cont			Tetragonal disphenoid	48 tic	Cont 288 vertices	Cont 576 edges		F4×2
301	Hexacosihecatonicosachoron	Xhi			Digonal disphenoid	120 ti 600 tut	Xhi 3600 vertices	Xhi 7200 edges		H4
302	Small hecatonicosihecatonicosachoron	Shihi			Digonal disphenoid	120 tigid 120 tid	Semiuniform Xhi	Shihi 7200 edges		H4
303	Dishecatonicosachoron	Dahi			Digonal disphenoid	120 tiggy 120 ti	Semiuniform Srahi 3600 vertices	Dahi 7200 edges		H4
304	Great hexacosihecatonicosachoron	Gixhi			Digonal disphenoid	120 tiggy 600 tut	Semiuniform Srix 3600 vertices	Gixhi 7200 edges		H4
305	Great tetracontoctachoron	Gic			Tetragonal disphenoid	48 quith	Cont	Gic 576 edges		F4×2
306	Great hecatonicosihecatonicosachoron	Ghihi			Digonal disphenoid	120 quit sissid 120 quit gissid	Semiuniform Xhi	Ghihi 7200 edges		H4
F1	Small tripesic hecatonicosachoron	Sitphi		/2	Compound of 3 tetragonal disphenoids	120 tigid	Hi 600 vertices	Sidtaxhi 3600 edges		H4
F2	Great tripesic hecatonicosachoron	Gitphi		(o5x5/3x5o)/2	Compound of 3 tetragonal disphenoids	120 quit sissid	Hi	Gadtaxady 3600 edges		H4

Category 8: Great Rhombates (#307-329)

These polychora, which include the cantitruncated polychora, have sphenoid vertex figures.

Index	Name	OBSA	Picture	Coxeter diagram	Vertex figure	Cells	Army	Regiment	Company	Symmetry
307	Great rhombated pentachoron	Grip			Sphenoid	5 toe 5 tut 10 trip	Grip 60 vertices	Grip 120 edges		A4
308	Great rhombated tesseract	Grit			Sphenoid	8 girco 16 tut 32 trip	Grit 192 vertices	Grit 384 edges		B4
309	Great rhombated icositetrachoron	Grico			Sphenoid	24 girco 24 tic 96 trip	Grico 576 vertices	Grico 1152 edges		F4
310	Great rhombated hecatonicosachoron	Grahi			Sphenoid	120 grid 600 tut 1200 trip	Grahi 7200 vertices	Grahi 14400 edges		H4
311	Great rhombated hexacosichoron	Grix			Sphenoid	600 toe 120 ti 720 pip	Grix 7200 vertices	Grix 14400 edges		H4
312	Great rhombated faceted hexacosichoron	Girfix			Sphenoid	720 stip 120 tigid 120 grid	Semiuniform Grix	Girfix 14400 edges		H4
313	Great rhombated grand hecatonicosachoron	Graghi			Sphenoid	720 stip 120 tiggy 120 grid	Semiuniform Prahi 7200 vertices	Graghi 14400 edges		H4
314	Great rhombated grand hexacosichoron	Graggix			Sphenoid	720 stip 600 toe 120 tiggy	Semiuniform Prahi	Graggix 14400 edges		H4
315	Great quasirhombated tesseract	Gaqrit			Sphenoid	32 trip 16 tut 8 quitco	Semiuniform Grit	Gaqrit 384 edges		B4
316	Great quasirhombated icositetrachoron	Gaqri			Sphenoid	96 trip 24 quith 24 quitco	Semiuniform Prico 576 vertices	Gaqri 1152 edges		F4
317	Great quasirhombated great grand hecatonicosachoron	Gaqrigaghi			Sphenoid	1200 trip 120 quit gissid 120 quitdid	Semiuniform Prahi	Gaqrigaghi 14400 edges		H4
318	Great quasirhombated great faceted hexacosichoron	Gaqrigafix			Sphenoid	720 pip 120 quit sissid 120 gaquatid	Semiuniform Grix	Gaqrigafix 14400 edges		H4
319	Great quasirhombated small stellated hecatonicosachoron	Gaqrisashi			Sphenoid	1200 trip 120 tid 120 quitdid	Semiuniform Grahi	Gaqrisashi 14400 edges		H4
320	Great quasirhombated great grand stellated hecatonicosachoron	Gaqrigagishi			Sphenoid	1200 trip 600 tut 120 gaquatid	Semiuniform Grix	Gaqrigagishi 14400 edges		H4
321	Great quasirhombated great stellated hecatonicosachoron	Gaqrigashi			Sphenoid	720 pip 120 ti 120 gaquatid	Semiuniform Prix	Gaqrigashi 14400 edges		H4
322	Tesseractihexadecatruncated tesseractihexadecachoron	Thatoth			Sphenoid	16 tut 8 tic 8 cotco	Semiuniform Grit	Thatoth 384 edges		B4
323	Tesseractihexadecaquasitruncated tesseractihexadecachoron	Thaquitoth			Sphenoid	16 tut 8 quith 8 cotco	Semiuniform Prit 192 vertices	Thaquitoth 384 edges		B4
324	Icositetraquasitruncated tetracontoctachoron	Iquatoc			Sphenoid	24 tic 24 quith 24 cotco	Semiuniform Prico	Iquatoc 1152 edges		F4
325	Hecatonicosihexacositruncated hecatonicosihexacosichoron	Hixthix			Sphenoid	600 tut 120 tid 120 idtid	Semiuniform Grahi	Hixthix 14400 edges		H4
326	Dishecatonicosatruncated dishecatonicosachoron	Dahitady			Sphenoid	120 tigid 120 tiggy 120 idtid	Semiuniform Prix 7200 vertices	Dahitady 14400 edges		H4
327	Hecatonicosihecatonicosaquasitruncated dishecatonicosachoron	Hihiquatady			Sphenoid	120 tid 120 quit gissid 120 idtid	Semiuniform Prahi	Hihiquatady 14400 edges		H4
328	Dishecatonicosaquasitruncated dishecatonicosachoron	Dahiquatady			Sphenoid	120 quit sissid 120 ti 120 idtid	Semiuniform Prahi	Dahiquatady 14400 edges		H4
329	Hecatonicosihexacosiquasitruncated hecatonicosihexacosichoron	Hixquathix			Sphenoid	600 tut 120 quit gissid 120 idtid	Semiuniform Prahi	Hixquathix 14400 edges		H4

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.