Uniform 6-polytope

Template:Short description

Graphs of three regular and related uniform polytopes
6-simplex Template:CDD	Truncated 6-simplex Template:CDD		Rectified 6-simplex Template:CDD
Cantellated 6-simplex Template:CDD		Runcinated 6-simplex Template:CDD
Stericated 6-simplex Template:CDD		Pentellated 6-simplex Template:CDD
6-orthoplex Template:CDD	Truncated 6-orthoplex Template:CDD		Rectified 6-orthoplex Template:CDD
Cantellated 6-orthoplex Template:CDD	Runcinated 6-orthoplex Template:CDD		Stericated 6-orthoplex Template:CDD
Cantellated 6-cube Template:CDD		Runcinated 6-cube Template:CDD
Stericated 6-cube Template:CDD		Pentellated 6-cube Template:CDD
6-cube Template:CDD	Truncated 6-cube Template:CDD		Rectified 6-cube Template:CDD
6-demicube Template:CDD	Truncated 6-demicube Template:CDD		Cantellated 6-demicube Template:CDD
Runcinated 6-demicube Template:CDD		Stericated 6-demicube Template:CDD
2₂₁ Template:CDD		1₂₂ Template:CDD
Truncated 2₂₁ Template:CDD		Truncated 1₂₂ Template:CDD

In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope.

The simplest uniform polypeta are regular polytopes: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}.

History of discovery

Regular polytopes: (convex faces)
- 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions.
Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
- 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular polytera) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.^[1]
Convex uniform polytopes:
- 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
Nonregular uniform star polytopes: (similar to the nonconvex uniform polyhedra)
- Ongoing: Jonathan Bowers and other researchers search for other non-convex uniform 6-polytopes, with a current count of 41348 known uniform 6-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 5-polytopes. The list is not proven complete.^[2]^[3]

Uniform 6-polytopes by fundamental Coxeter groups

Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams.

There are four fundamental reflective symmetry groups which generate 153 unique uniform 6-polytopes.

#	Coxeter group		Coxeter-Dynkin diagram
1	A₆	[3,3,3,3,3]	Template:CDD
2	B₆	[3,3,3,3,4]	Template:CDD
3	D₆	[3,3,3,3^1,1]	Template:CDD
4	E₆	[3^2,2,1]	Template:CDD
4	E₆	[3,3^2,2]	Template:CDD

Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

Uniform prismatic families

Uniform prism

There are 6 categorical uniform prisms based on the uniform 5-polytopes.

#	Coxeter group			Notes
1	A₅A₁	[3,3,3,3,2]	Template:CDD	Prism family based on 5-simplex
2	B₅A₁	[4,3,3,3,2]	Template:CDD	Prism family based on 5-cube
3a	D₅A₁	[3^2,1,1,2]	Template:CDD	Prism family based on 5-demicube

#	Coxeter group			Notes
4	A₃I₂(p)A₁	[3,3,2,p,2]	Template:CDD	Prism family based on tetrahedral-p-gonal duoprisms
5	B₃I₂(p)A₁	[4,3,2,p,2]	Template:CDD	Prism family based on cubic-p-gonal duoprisms
6	H₃I₂(p)A₁	[5,3,2,p,2]	Template:CDD	Prism family based on dodecahedral-p-gonal duoprisms

Uniform duoprism

There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower-dimensional uniform polytopes. Five are formed as the product of a uniform 4-polytope with a regular polygon, and six are formed by the product of two uniform polyhedra:

#	Coxeter group			Notes
1	A₄I₂(p)	[3,3,3,2,p]	Template:CDD	Family based on 5-cell-p-gonal duoprisms.
2	B₄I₂(p)	[4,3,3,2,p]	Template:CDD	Family based on tesseract-p-gonal duoprisms.
3	F₄I₂(p)	[3,4,3,2,p]	Template:CDD	Family based on 24-cell-p-gonal duoprisms.
4	H₄I₂(p)	[5,3,3,2,p]	Template:CDD	Family based on 120-cell-p-gonal duoprisms.
5	D₄I₂(p)	[3^1,1,1,2,p]	Template:CDD	Family based on demitesseract-p-gonal duoprisms.

#	Coxeter group			Notes
6	A₃²	[3,3,2,3,3]	Template:CDD	Family based on tetrahedral duoprisms.
7	A₃B₃	[3,3,2,4,3]	Template:CDD	Family based on tetrahedral-cubic duoprisms.
8	A₃H₃	[3,3,2,5,3]	Template:CDD	Family based on tetrahedral-dodecahedral duoprisms.
9	B₃²	[4,3,2,4,3]	Template:CDD	Family based on cubic duoprisms.
10	B₃H₃	[4,3,2,5,3]	Template:CDD	Family based on cubic-dodecahedral duoprisms.
11	H₃²	[5,3,2,5,3]	Template:CDD	Family based on dodecahedral duoprisms.

Uniform triaprism

There is one infinite family of uniform triaprismatic families of polytopes constructed as a Cartesian products of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.

#	Coxeter group			Notes
1	I₂(p)I₂(q)I₂(r)	[p,2,q,2,r]	Template:CDD	Family based on p,q,r-gonal triprisms

Enumerating the convex uniform 6-polytopes

Simplex family: A₆ [3⁴] - Template:CDD
- 35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular:
  1. {3⁴} - 6-simplex - Template:CDD
Hypercube/orthoplex family: B₆ [4,3⁴] - Template:CDD
- 63 uniform 6-polytopes as permutations of rings in the group diagram, including two regular forms:
  1. {4,3³} — 6-cube (hexeract) - Template:CDD
  2. {3³,4} — 6-orthoplex, (hexacross) - Template:CDD
Demihypercube D₆ family: [3^3,1,1] - Template:CDD
- 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
  1. {3,3^2,1}, 1₂₁ 6-demicube (demihexeract) - Template:CDD; also as h{4,3³}, Template:CDD
  2. {3,3,3^1,1}, 2₁₁ 6-orthoplex - Template:CDD, a half symmetry form of Template:CDD.
E₆ family: [3^3,1,1] - Template:CDD
- 39 uniform 6-polytopes as permutations of rings in the group diagram, including:
  1. {3,3,3^2,1}, 2₂₁ - Template:CDD
  2. {3,3^2,2}, 1₂₂ - Template:CDD

These fundamental families generate 153 nonprismatic convex uniform polypeta.

In addition, there are 57 uniform 6-polytope constructions based on prisms of the uniform 5-polytopes: [3,3,3,3,2], [4,3,3,3,2], [3^2,1,1,2], excluding the penteract prism as a duplicate of the hexeract.

In addition, there are infinitely many uniform 6-polytope based on:

Duoprism prism families: [3,3,2,p,2], [4,3,2,p,2], [5,3,2,p,2].
Duoprism families: [3,3,3,2,p], [4,3,3,2,p], [5,3,3,2,p].
Triaprism family: [p,2,q,2,r].

The A₆ family

Template:Further There are 32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram. All 35 are enumerated below. They are named by Norman Johnson from the Wythoff construction operations upon regular 6-simplex (heptapeton). Bowers-style acronym names are given in parentheses for cross-referencing.

The A₆ family has symmetry of order 5040 (7 factorial).

The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, all in hyperplanes with normal vector (1,1,1,1,1,1,1).

#	Coxeter-Dynkin	Johnson naming system Bowers name and (acronym)	Base point	Element counts
#	Coxeter-Dynkin	Johnson naming system Bowers name and (acronym)	Base point	5	4	3	2	1	0
1	Template:CDD	6-simplex heptapeton (hop)	(0,0,0,0,0,0,1)	7	21	35	35	21	7
2	Template:CDD	Rectified 6-simplex rectified heptapeton (ril)	(0,0,0,0,0,1,1)	14	63	140	175	105	21
3	Template:CDD	Truncated 6-simplex truncated heptapeton (til)	(0,0,0,0,0,1,2)	14	63	140	175	126	42
4	Template:CDD	Birectified 6-simplex birectified heptapeton (bril)	(0,0,0,0,1,1,1)	14	84	245	350	210	35
5	Template:CDD	Cantellated 6-simplex small rhombated heptapeton (sril)	(0,0,0,0,1,1,2)	35	210	560	805	525	105
6	Template:CDD	Bitruncated 6-simplex bitruncated heptapeton (batal)	(0,0,0,0,1,2,2)	14	84	245	385	315	105
7	Template:CDD	Cantitruncated 6-simplex great rhombated heptapeton (gril)	(0,0,0,0,1,2,3)	35	210	560	805	630	210
8	Template:CDD	Runcinated 6-simplex small prismated heptapeton (spil)	(0,0,0,1,1,1,2)	70	455	1330	1610	840	140
9	Template:CDD	Bicantellated 6-simplex small birhombated heptapeton (sabril)	(0,0,0,1,1,2,2)	70	455	1295	1610	840	140
10	Template:CDD	Runcitruncated 6-simplex prismatotruncated heptapeton (patal)	(0,0,0,1,1,2,3)	70	560	1820	2800	1890	420
11	Template:CDD	Tritruncated 6-simplex tetradecapeton (fe)	(0,0,0,1,2,2,2)	14	84	280	490	420	140
12	Template:CDD	Runcicantellated 6-simplex prismatorhombated heptapeton (pril)	(0,0,0,1,2,2,3)	70	455	1295	1960	1470	420
13	Template:CDD	Bicantitruncated 6-simplex great birhombated heptapeton (gabril)	(0,0,0,1,2,3,3)	49	329	980	1540	1260	420
14	Template:CDD	Runcicantitruncated 6-simplex great prismated heptapeton (gapil)	(0,0,0,1,2,3,4)	70	560	1820	3010	2520	840
15	Template:CDD	Stericated 6-simplex small cellated heptapeton (scal)	(0,0,1,1,1,1,2)	105	700	1470	1400	630	105
16	Template:CDD	Biruncinated 6-simplex small biprismato-tetradecapeton (sibpof)	(0,0,1,1,1,2,2)	84	714	2100	2520	1260	210
17	Template:CDD	Steritruncated 6-simplex cellitruncated heptapeton (catal)	(0,0,1,1,1,2,3)	105	945	2940	3780	2100	420
18	Template:CDD	Stericantellated 6-simplex cellirhombated heptapeton (cral)	(0,0,1,1,2,2,3)	105	1050	3465	5040	3150	630
19	Template:CDD	Biruncitruncated 6-simplex biprismatorhombated heptapeton (bapril)	(0,0,1,1,2,3,3)	84	714	2310	3570	2520	630
20	Template:CDD	Stericantitruncated 6-simplex celligreatorhombated heptapeton (cagral)	(0,0,1,1,2,3,4)	105	1155	4410	7140	5040	1260
21	Template:CDD	Steriruncinated 6-simplex celliprismated heptapeton (copal)	(0,0,1,2,2,2,3)	105	700	1995	2660	1680	420
22	Template:CDD	Steriruncitruncated 6-simplex celliprismatotruncated heptapeton (captal)	(0,0,1,2,2,3,4)	105	945	3360	5670	4410	1260
23	Template:CDD	Steriruncicantellated 6-simplex celliprismatorhombated heptapeton (copril)	(0,0,1,2,3,3,4)	105	1050	3675	5880	4410	1260
24	Template:CDD	Biruncicantitruncated 6-simplex great biprismato-tetradecapeton (gibpof)	(0,0,1,2,3,4,4)	84	714	2520	4410	3780	1260
25	Template:CDD	Steriruncicantitruncated 6-simplex great cellated heptapeton (gacal)	(0,0,1,2,3,4,5)	105	1155	4620	8610	7560	2520
26	Template:CDD	Pentellated 6-simplex small teri-tetradecapeton (staff)	(0,1,1,1,1,1,2)	126	434	630	490	210	42
27	Template:CDD	Pentitruncated 6-simplex teracellated heptapeton (tocal)	(0,1,1,1,1,2,3)	126	826	1785	1820	945	210
28	Template:CDD	Penticantellated 6-simplex teriprismated heptapeton (topal)	(0,1,1,1,2,2,3)	126	1246	3570	4340	2310	420
29	Template:CDD	Penticantitruncated 6-simplex terigreatorhombated heptapeton (togral)	(0,1,1,1,2,3,4)	126	1351	4095	5390	3360	840
30	Template:CDD	Pentiruncitruncated 6-simplex tericellirhombated heptapeton (tocral)	(0,1,1,2,2,3,4)	126	1491	5565	8610	5670	1260
31	Template:CDD	Pentiruncicantellated 6-simplex teriprismatorhombi-tetradecapeton (taporf)	(0,1,1,2,3,3,4)	126	1596	5250	7560	5040	1260
32	Template:CDD	Pentiruncicantitruncated 6-simplex terigreatoprismated heptapeton (tagopal)	(0,1,1,2,3,4,5)	126	1701	6825	11550	8820	2520
33	Template:CDD	Pentisteritruncated 6-simplex tericellitrunki-tetradecapeton (tactaf)	(0,1,2,2,2,3,4)	126	1176	3780	5250	3360	840
34	Template:CDD	Pentistericantitruncated 6-simplex tericelligreatorhombated heptapeton (tacogral)	(0,1,2,2,3,4,5)	126	1596	6510	11340	8820	2520
35	Template:CDD	Omnitruncated 6-simplex great teri-tetradecapeton (gotaf)	(0,1,2,3,4,5,6)	126	1806	8400	16800	15120	5040

The B₆ family

Template:Further There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

The B₆ family has symmetry of order 46080 (6 factorial x 2⁶).

They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube and 6-orthoplex. Bowers names and acronym names are given for cross-referencing.

#	Coxeter-Dynkin diagram	Schläfli symbol	Names	Element counts
#	Coxeter-Dynkin diagram	Schläfli symbol	Names	5	4	3	2	1	0
36	Template:CDD	t₀{3,3,3,3,4}	6-orthoplex Hexacontatetrapeton (gee)	64	192	240	160	60	12
37	Template:CDD	t₁{3,3,3,3,4}	Rectified 6-orthoplex Rectified hexacontatetrapeton (rag)	76	576	1200	1120	480	60
38	Template:CDD	t₂{3,3,3,3,4}	Birectified 6-orthoplex Birectified hexacontatetrapeton (brag)	76	636	2160	2880	1440	160
39	Template:CDD	t₂{4,3,3,3,3}	Birectified 6-cube Birectified hexeract (brox)	76	636	2080	3200	1920	240
40	Template:CDD	t₁{4,3,3,3,3}	Rectified 6-cube Rectified hexeract (rax)	76	444	1120	1520	960	192
41	Template:CDD	t₀{4,3,3,3,3}	6-cube Hexeract (ax)	12	60	160	240	192	64
42	Template:CDD	t_0,1{3,3,3,3,4}	Truncated 6-orthoplex Truncated hexacontatetrapeton (tag)	76	576	1200	1120	540	120
43	Template:CDD	t_0,2{3,3,3,3,4}	Cantellated 6-orthoplex Small rhombated hexacontatetrapeton (srog)	136	1656	5040	6400	3360	480
44	Template:CDD	t_1,2{3,3,3,3,4}	Bitruncated 6-orthoplex Bitruncated hexacontatetrapeton (botag)					1920	480
45	Template:CDD	t_0,3{3,3,3,3,4}	Runcinated 6-orthoplex Small prismated hexacontatetrapeton (spog)					7200	960
46	Template:CDD	t_1,3{3,3,3,3,4}	Bicantellated 6-orthoplex Small birhombated hexacontatetrapeton (siborg)					8640	1440
47	Template:CDD	t_2,3{4,3,3,3,3}	Tritruncated 6-cube Hexeractihexacontitetrapeton (xog)					3360	960
48	Template:CDD	t_0,4{3,3,3,3,4}	Stericated 6-orthoplex Small cellated hexacontatetrapeton (scag)					5760	960
49	Template:CDD	t_1,4{4,3,3,3,3}	Biruncinated 6-cube Small biprismato-hexeractihexacontitetrapeton (sobpoxog)					11520	1920
50	Template:CDD	t_1,3{4,3,3,3,3}	Bicantellated 6-cube Small birhombated hexeract (saborx)					9600	1920
51	Template:CDD	t_1,2{4,3,3,3,3}	Bitruncated 6-cube Bitruncated hexeract (botox)					2880	960
52	Template:CDD	t_0,5{4,3,3,3,3}	Pentellated 6-cube Small teri-hexeractihexacontitetrapeton (stoxog)					1920	384
53	Template:CDD	t_0,4{4,3,3,3,3}	Stericated 6-cube Small cellated hexeract (scox)					5760	960
54	Template:CDD	t_0,3{4,3,3,3,3}	Runcinated 6-cube Small prismated hexeract (spox)					7680	1280
55	Template:CDD	t_0,2{4,3,3,3,3}	Cantellated 6-cube Small rhombated hexeract (srox)					4800	960
56	Template:CDD	t_0,1{4,3,3,3,3}	Truncated 6-cube Truncated hexeract (tox)	76	444	1120	1520	1152	384
57	Template:CDD	t_0,1,2{3,3,3,3,4}	Cantitruncated 6-orthoplex Great rhombated hexacontatetrapeton (grog)					3840	960
58	Template:CDD	t_0,1,3{3,3,3,3,4}	Runcitruncated 6-orthoplex Prismatotruncated hexacontatetrapeton (potag)					15840	2880
59	Template:CDD	t_0,2,3{3,3,3,3,4}	Runcicantellated 6-orthoplex Prismatorhombated hexacontatetrapeton (prog)					11520	2880
60	Template:CDD	t_1,2,3{3,3,3,3,4}	Bicantitruncated 6-orthoplex Great birhombated hexacontatetrapeton (gaborg)					10080	2880
61	Template:CDD	t_0,1,4{3,3,3,3,4}	Steritruncated 6-orthoplex Cellitruncated hexacontatetrapeton (catog)					19200	3840
62	Template:CDD	t_0,2,4{3,3,3,3,4}	Stericantellated 6-orthoplex Cellirhombated hexacontatetrapeton (crag)					28800	5760
63	Template:CDD	t_1,2,4{3,3,3,3,4}	Biruncitruncated 6-orthoplex Biprismatotruncated hexacontatetrapeton (boprax)					23040	5760
64	Template:CDD	t_0,3,4{3,3,3,3,4}	Steriruncinated 6-orthoplex Celliprismated hexacontatetrapeton (copog)					15360	3840
65	Template:CDD	t_1,2,4{4,3,3,3,3}	Biruncitruncated 6-cube Biprismatotruncated hexeract (boprag)					23040	5760
66	Template:CDD	t_1,2,3{4,3,3,3,3}	Bicantitruncated 6-cube Great birhombated hexeract (gaborx)					11520	3840
67	Template:CDD	t_0,1,5{3,3,3,3,4}	Pentitruncated 6-orthoplex Teritruncated hexacontatetrapeton (tacox)					8640	1920
68	Template:CDD	t_0,2,5{3,3,3,3,4}	Penticantellated 6-orthoplex Terirhombated hexacontatetrapeton (tapox)					21120	3840
69	Template:CDD	t_0,3,4{4,3,3,3,3}	Steriruncinated 6-cube Celliprismated hexeract (copox)					15360	3840
70	Template:CDD	t_0,2,5{4,3,3,3,3}	Penticantellated 6-cube Terirhombated hexeract (topag)					21120	3840
71	Template:CDD	t_0,2,4{4,3,3,3,3}	Stericantellated 6-cube Cellirhombated hexeract (crax)					28800	5760
72	Template:CDD	t_0,2,3{4,3,3,3,3}	Runcicantellated 6-cube Prismatorhombated hexeract (prox)					13440	3840
73	Template:CDD	t_0,1,5{4,3,3,3,3}	Pentitruncated 6-cube Teritruncated hexeract (tacog)					8640	1920
74	Template:CDD	t_0,1,4{4,3,3,3,3}	Steritruncated 6-cube Cellitruncated hexeract (catax)					19200	3840
75	Template:CDD	t_0,1,3{4,3,3,3,3}	Runcitruncated 6-cube Prismatotruncated hexeract (potax)					17280	3840
76	Template:CDD	t_0,1,2{4,3,3,3,3}	Cantitruncated 6-cube Great rhombated hexeract (grox)					5760	1920
77	Template:CDD	t_0,1,2,3{3,3,3,3,4}	Runcicantitruncated 6-orthoplex Great prismated hexacontatetrapeton (gopog)					20160	5760
78	Template:CDD	t_0,1,2,4{3,3,3,3,4}	Stericantitruncated 6-orthoplex Celligreatorhombated hexacontatetrapeton (cagorg)					46080	11520
79	Template:CDD	t_0,1,3,4{3,3,3,3,4}	Steriruncitruncated 6-orthoplex Celliprismatotruncated hexacontatetrapeton (captog)					40320	11520
80	Template:CDD	t_0,2,3,4{3,3,3,3,4}	Steriruncicantellated 6-orthoplex Celliprismatorhombated hexacontatetrapeton (coprag)					40320	11520
81	Template:CDD	t_1,2,3,4{4,3,3,3,3}	Biruncicantitruncated 6-cube Great biprismato-hexeractihexacontitetrapeton (gobpoxog)					34560	11520
82	Template:CDD	t_0,1,2,5{3,3,3,3,4}	Penticantitruncated 6-orthoplex Terigreatorhombated hexacontatetrapeton (togrig)					30720	7680
83	Template:CDD	t_0,1,3,5{3,3,3,3,4}	Pentiruncitruncated 6-orthoplex Teriprismatotruncated hexacontatetrapeton (tocrax)					51840	11520
84	Template:CDD	t_0,2,3,5{4,3,3,3,3}	Pentiruncicantellated 6-cube Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)					46080	11520
85	Template:CDD	t_0,2,3,4{4,3,3,3,3}	Steriruncicantellated 6-cube Celliprismatorhombated hexeract (coprix)					40320	11520
86	Template:CDD	t_0,1,4,5{4,3,3,3,3}	Pentisteritruncated 6-cube Tericelli-hexeractihexacontitetrapeton (tactaxog)					30720	7680
87	Template:CDD	t_0,1,3,5{4,3,3,3,3}	Pentiruncitruncated 6-cube Teriprismatotruncated hexeract (tocrag)					51840	11520
88	Template:CDD	t_0,1,3,4{4,3,3,3,3}	Steriruncitruncated 6-cube Celliprismatotruncated hexeract (captix)					40320	11520
89	Template:CDD	t_0,1,2,5{4,3,3,3,3}	Penticantitruncated 6-cube Terigreatorhombated hexeract (togrix)					30720	7680
90	Template:CDD	t_0,1,2,4{4,3,3,3,3}	Stericantitruncated 6-cube Celligreatorhombated hexeract (cagorx)					46080	11520
91	Template:CDD	t_0,1,2,3{4,3,3,3,3}	Runcicantitruncated 6-cube Great prismated hexeract (gippox)					23040	7680
92	Template:CDD	t_0,1,2,3,4{3,3,3,3,4}	Steriruncicantitruncated 6-orthoplex Great cellated hexacontatetrapeton (gocog)					69120	23040
93	Template:CDD	t_0,1,2,3,5{3,3,3,3,4}	Pentiruncicantitruncated 6-orthoplex Terigreatoprismated hexacontatetrapeton (tagpog)					80640	23040
94	Template:CDD	t_0,1,2,4,5{3,3,3,3,4}	Pentistericantitruncated 6-orthoplex Tericelligreatorhombated hexacontatetrapeton (tecagorg)					80640	23040
95	Template:CDD	t_0,1,2,4,5{4,3,3,3,3}	Pentistericantitruncated 6-cube Tericelligreatorhombated hexeract (tocagrax)					80640	23040
96	Template:CDD	t_0,1,2,3,5{4,3,3,3,3}	Pentiruncicantitruncated 6-cube Terigreatoprismated hexeract (tagpox)					80640	23040
97	Template:CDD	t_0,1,2,3,4{4,3,3,3,3}	Steriruncicantitruncated 6-cube Great cellated hexeract (gocax)					69120	23040
98	Template:CDD	t_0,1,2,3,4,5{4,3,3,3,3}	Omnitruncated 6-cube Great teri-hexeractihexacontitetrapeton (gotaxog)					138240	46080

The D₆ family

Template:Further The D₆ family has symmetry of order 23040 (6 factorial x 2⁵).

This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D₆ Coxeter-Dynkin diagram. Of these, 31 (2×16−1) are repeated from the B₆ family and 16 are unique to this family. The 16 unique forms are enumerated below. Bowers-style acronym names are given for cross-referencing.

#	Coxeter diagram	Names	Base point (Alternately signed)	Element counts						Circumrad
#	Coxeter diagram	Names	Base point (Alternately signed)	5	4	3	2	1	0	Circumrad
99	Template:CDD = Template:CDD	6-demicube Hemihexeract (hax)	(1,1,1,1,1,1)	44	252	640	640	240	32	0.8660254
100	Template:CDD = Template:CDD	Cantic 6-cube Truncated hemihexeract (thax)	(1,1,3,3,3,3)	76	636	2080	3200	2160	480	2.1794493
101	Template:CDD = Template:CDD	Runcic 6-cube Small rhombated hemihexeract (sirhax)	(1,1,1,3,3,3)					3840	640	1.9364916
102	Template:CDD = Template:CDD	Steric 6-cube Small prismated hemihexeract (sophax)	(1,1,1,1,3,3)					3360	480	1.6583123
103	Template:CDD = Template:CDD	Pentic 6-cube Small cellated demihexeract (sochax)	(1,1,1,1,1,3)					1440	192	1.3228756
104	Template:CDD = Template:CDD	Runcicantic 6-cube Great rhombated hemihexeract (girhax)	(1,1,3,5,5,5)					5760	1920	3.2787192
105	Template:CDD = Template:CDD	Stericantic 6-cube Prismatotruncated hemihexeract (pithax)	(1,1,3,3,5,5)					12960	2880	2.95804
106	Template:CDD = Template:CDD	Steriruncic 6-cube Prismatorhombated hemihexeract (prohax)	(1,1,1,3,5,5)					7680	1920	2.7838821
107	Template:CDD = Template:CDD	Penticantic 6-cube Cellitruncated hemihexeract (cathix)	(1,1,3,3,3,5)					9600	1920	2.5980761
108	Template:CDD = Template:CDD	Pentiruncic 6-cube Cellirhombated hemihexeract (crohax)	(1,1,1,3,3,5)					10560	1920	2.3979158
109	Template:CDD = Template:CDD	Pentisteric 6-cube Celliprismated hemihexeract (cophix)	(1,1,1,1,3,5)					5280	960	2.1794496
110	Template:CDD = Template:CDD	Steriruncicantic 6-cube Great prismated hemihexeract (gophax)	(1,1,3,5,7,7)					17280	5760	4.0926762
111	Template:CDD = Template:CDD	Pentiruncicantic 6-cube Celligreatorhombated hemihexeract (cagrohax)	(1,1,3,5,5,7)					20160	5760	3.7080991
112	Template:CDD = Template:CDD	Pentistericantic 6-cube Celliprismatotruncated hemihexeract (capthix)	(1,1,3,3,5,7)					23040	5760	3.4278274
113	Template:CDD = Template:CDD	Pentisteriruncic 6-cube Celliprismatorhombated hemihexeract (caprohax)	(1,1,1,3,5,7)					15360	3840	3.2787192
114	Template:CDD = Template:CDD	Pentisteriruncicantic 6-cube Great cellated hemihexeract (gochax)	(1,1,3,5,7,9)					34560	11520	4.5552168

The E₆ family

Template:Further There are 39 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Bowers-style acronym names are given for cross-referencing. The E₆ family has symmetry of order 51,840.

#	Coxeter diagram	Names	Element counts
#	Coxeter diagram	Names	5-faces	4-faces	Cells	Faces	Edges	Vertices
115	Template:CDD	2₂₁ Icosiheptaheptacontidipeton (jak)	99	648	1080	720	216	27
116	Template:CDD	Rectified 2₂₁ Rectified icosiheptaheptacontidipeton (rojak)	126	1350	4320	5040	2160	216
117	Template:CDD	Truncated 2₂₁ Truncated icosiheptaheptacontidipeton (tojak)	126	1350	4320	5040	2376	432
118	Template:CDD	Cantellated 2₂₁ Small rhombated icosiheptaheptacontidipeton (sirjak)	342	3942	15120	24480	15120	2160
119	Template:CDD	Runcinated 2₂₁ Small demiprismated icosiheptaheptacontidipeton (shopjak)	342	4662	16200	19440	8640	1080
120	Template:CDD	Demified icosiheptaheptacontidipeton (hejak)	342	2430	7200	7920	3240	432
121	Template:CDD	Bitruncated 2₂₁ Bitruncated icosiheptaheptacontidipeton (botajik)						2160
122	Template:CDD	Demirectified icosiheptaheptacontidipeton (harjak)						1080
123	Template:CDD	Cantitruncated 2₂₁ Great rhombated icosiheptaheptacontidipeton (girjak)						4320
124	Template:CDD	Runcitruncated 2₂₁ Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak)						4320
125	Template:CDD	Steritruncated 2₂₁ Cellitruncated icosiheptaheptacontidipeton (catjak)						2160
126	Template:CDD	Demitruncated icosiheptaheptacontidipeton (hotjak)						2160
127	Template:CDD	Runcicantellated 2₂₁ Demiprismatorhombated icosiheptaheptacontidipeton (haprojak)						6480
128	Template:CDD	Small demirhombated icosiheptaheptacontidipeton (shorjak)						4320
129	Template:CDD	Small prismated icosiheptaheptacontidipeton (spojak)						4320
130	Template:CDD	Tritruncated icosiheptaheptacontidipeton (titajak)						4320
131	Template:CDD	Runcicantitruncated 2₂₁ Great demiprismated icosiheptaheptacontidipeton (ghopjak)						12960
132	Template:CDD	Stericantitruncated 2₂₁ Celligreatorhombated icosiheptaheptacontidipeton (cograjik)						12960
133	Template:CDD	Great demirhombated icosiheptaheptacontidipeton (ghorjak)						8640
134	Template:CDD	Prismatotruncated icosiheptaheptacontidipeton (potjak)						12960
135	Template:CDD	Demicellitruncated icosiheptaheptacontidipeton (hictijik)						8640
136	Template:CDD	Prismatorhombated icosiheptaheptacontidipeton (projak)						12960
137	Template:CDD	Great prismated icosiheptaheptacontidipeton (gapjak)						25920
138	Template:CDD	Demicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik)						25920

#	Coxeter diagram	Names	Element counts
#	Coxeter diagram	Names	5-faces	4-faces	Cells	Faces	Edges	Vertices
139	Template:CDD = Template:CDD	1₂₂ Pentacontatetrapeton (mo)	54	702	2160	2160	720	72
140	Template:CDD = Template:CDD	Rectified 1₂₂ Rectified pentacontatetrapeton (ram)	126	1566	6480	10800	6480	720
141	Template:CDD = Template:CDD	Birectified 1₂₂ Birectified pentacontatetrapeton (barm)	126	2286	10800	19440	12960	2160
142	Template:CDD = Template:CDD	Trirectified 1₂₂ Trirectified pentacontatetrapeton (trim)	558	4608	8640	6480	2160	270
143	Template:CDD = Template:CDD	Truncated 1₂₂ Truncated pentacontatetrapeton (tim)					13680	1440
144	Template:CDD = Template:CDD	Bitruncated 1₂₂ Bitruncated pentacontatetrapeton (bitem)						6480
145	Template:CDD = Template:CDD	Tritruncated 1₂₂ Tritruncated pentacontatetrapeton (titam)						8640
146	Template:CDD = Template:CDD	Cantellated 1₂₂ Small rhombated pentacontatetrapeton (sram)						6480
147	Template:CDD = Template:CDD	Cantitruncated 1₂₂ Great rhombated pentacontatetrapeton (gram)						12960
148	Template:CDD = Template:CDD	Runcinated 1₂₂ Small prismated pentacontatetrapeton (spam)						2160
149	Template:CDD = Template:CDD	Bicantellated 1₂₂ Small birhombated pentacontatetrapeton (sabrim)						6480
150	Template:CDD = Template:CDD	Bicantitruncated 1₂₂ Great birhombated pentacontatetrapeton (gabrim)						12960
151	Template:CDD = Template:CDD	Runcitruncated 1₂₂ Prismatotruncated pentacontatetrapeton (patom)						12960
152	Template:CDD = Template:CDD	Runcicantellated 1₂₂ Prismatorhombated pentacontatetrapeton (prom)						25920
153	Template:CDD = Template:CDD	Omnitruncated 1₂₂ Great prismated pentacontatetrapeton (gopam)						51840

Triaprisms

Uniform triaprisms, {p}×{q}×{r}, form an infinite class for all integers p,q,r>2. {4}×{4}×{4} makes a lower symmetry form of the 6-cube.

The extended f-vector is (p,p,1)*(q,q,1)*(r,r,1)=(pqr,3pqr,3pqr+pq+pr+qr,3p(p+1),3p,1).

Coxeter diagram	Names	Element counts
Coxeter diagram	Names	5-faces	4-faces	Cells	Faces	Edges	Vertices
Template:CDD	{p}×{q}×{r} ^[4]	p+q+r	pq+pr+qr+p+q+r	pqr+2(pq+pr+qr)	3pqr+pq+pr+qr	3pqr	pqr
Template:CDD	{p}×{p}×{p}	3p	3p(p+1)	p²(p+6)	3p²(p+1)	3p³	p³
Template:CDD	{3}×{3}×{3} (trittip)	9	36	81	99	81	27
Template:CDD	{4}×{4}×{4} = 6-cube	12	60	160	240	192	64

Non-Wythoffian 6-polytopes

In 6 dimensions and above, there are an infinite amount of non-Wythoffian convex uniform polytopes: the Cartesian product of the grand antiprism in 4 dimensions and any regular polygon in 2 dimensions. It is not yet proven whether or not there are more.

Regular and uniform honeycombs

Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

There are four fundamental affine Coxeter groups and 27 prismatic groups that generate regular and uniform tessellations in 5-space:

#	Coxeter group		Coxeter diagram	Forms
1	${\tilde{A}}_{5}$	[3^[6]]	Template:CDD	12
2	${\tilde{C}}_{5}$	[4,3³,4]	Template:CDD	35
3	${\tilde{B}}_{5}$	[4,3,3^1,1] [4,3³,4,1⁺]	Template:CDD Template:CDD	47 (16 new)
4	${\tilde{D}}_{5}$	[3^1,1,3,3^1,1] [1⁺,4,3³,4,1⁺]	Template:CDD Template:CDD	20 (3 new)

Regular and uniform honeycombs include:

A~5 There are 12 unique uniform honeycombs, including:
C~5 There are 35 uniform honeycombs, including:
- Regular hypercube honeycomb of Euclidean 5-space, the 5-cube honeycomb, with symbols {4,3³,4}, Template:CDD = Template:CDD
B~5 There are 47 uniform honeycombs, 16 new, including:
- The uniform alternated hypercube honeycomb, 5-demicubic honeycomb, with symbols h{4,3³,4}, Template:CDD = Template:CDD = Template:CDD
${\tilde{D}}_{5}$ , [3^1,1,3,3^1,1]: There are 20 unique ringed permutations, and 3 new ones. Coxeter calls the first one a quarter 5-cubic honeycomb, with symbols q{4,3³,4}, Template:CDD = Template:CDD. The other two new ones are Template:CDD = Template:CDD, Template:CDD = Template:CDD.

Prismatic groups
#	Coxeter group		Coxeter-Dynkin diagram
1	${\tilde{A}}_{4}$ x ${\tilde{I}}_{1}$	[3^[5],2,∞]	Template:CDD
2	${\tilde{B}}_{4}$ x ${\tilde{I}}_{1}$	[4,3,3^1,1,2,∞]	Template:CDD
3	${\tilde{C}}_{4}$ x ${\tilde{I}}_{1}$	[4,3,3,4,2,∞]	Template:CDD
4	${\tilde{D}}_{4}$ x ${\tilde{I}}_{1}$	[3^1,1,1,1,2,∞]	Template:CDD
5	${\tilde{F}}_{4}$ x ${\tilde{I}}_{1}$	[3,4,3,3,2,∞]	Template:CDD
6	${\tilde{C}}_{3}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,3,4,2,∞,2,∞]	Template:CDD
7	${\tilde{B}}_{3}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,3^1,1,2,∞,2,∞]	Template:CDD
8	${\tilde{A}}_{3}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[3^[4],2,∞,2,∞]	Template:CDD
9	${\tilde{C}}_{2}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,4,2,∞,2,∞,2,∞]	Template:CDD
10	${\tilde{H}}_{2}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[6,3,2,∞,2,∞,2,∞]	Template:CDD
11	${\tilde{A}}_{2}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[3^[3],2,∞,2,∞,2,∞]	Template:CDD
12	${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[∞,2,∞,2,∞,2,∞,2,∞]	Template:CDD
13	${\tilde{A}}_{2}$ x ${\tilde{A}}_{2}$ x ${\tilde{I}}_{1}$	[3^[3],2,3^[3],2,∞]	Template:CDD
14	${\tilde{A}}_{2}$ x ${\tilde{B}}_{2}$ x ${\tilde{I}}_{1}$	[3^[3],2,4,4,2,∞]	Template:CDD
15	${\tilde{A}}_{2}$ x ${\tilde{G}}_{2}$ x ${\tilde{I}}_{1}$	[3^[3],2,6,3,2,∞]	Template:CDD
16	${\tilde{B}}_{2}$ x ${\tilde{B}}_{2}$ x ${\tilde{I}}_{1}$	[4,4,2,4,4,2,∞]	Template:CDD
17	${\tilde{B}}_{2}$ x ${\tilde{G}}_{2}$ x ${\tilde{I}}_{1}$	[4,4,2,6,3,2,∞]	Template:CDD
18	${\tilde{G}}_{2}$ x ${\tilde{G}}_{2}$ x ${\tilde{I}}_{1}$	[6,3,2,6,3,2,∞]	Template:CDD
19	${\tilde{A}}_{3}$ x ${\tilde{A}}_{2}$	[3^[4],2,3^[3]]	Template:CDD
20	${\tilde{B}}_{3}$ x ${\tilde{A}}_{2}$	[4,3^1,1,2,3^[3]]	Template:CDD
21	${\tilde{C}}_{3}$ x ${\tilde{A}}_{2}$	[4,3,4,2,3^[3]]	Template:CDD
22	${\tilde{A}}_{3}$ x ${\tilde{B}}_{2}$	[3^[4],2,4,4]	Template:CDD
23	${\tilde{B}}_{3}$ x ${\tilde{B}}_{2}$	[4,3^1,1,2,4,4]	Template:CDD
24	${\tilde{C}}_{3}$ x ${\tilde{B}}_{2}$	[4,3,4,2,4,4]	Template:CDD
25	${\tilde{A}}_{3}$ x ${\tilde{G}}_{2}$	[3^[4],2,6,3]	Template:CDD
26	${\tilde{B}}_{3}$ x ${\tilde{G}}_{2}$	[4,3^1,1,2,6,3]	Template:CDD
27	${\tilde{C}}_{3}$ x ${\tilde{G}}_{2}$	[4,3,4,2,6,3]	Template:CDD

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 6, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 12 paracompact hyperbolic Coxeter groups of rank 6, each generating uniform honeycombs in 5-space as permutations of rings of the Coxeter diagrams.

Hyperbolic paracompact groups
${\bar{P}}_{5}$ = [3,3^[5]]: Template:CDD ${\hat{A U}}_{5}$ = [(3,3,3,3,3,4)]: Template:CDD ${\hat{A R}}_{5}$ = [(3,3,4,3,3,4)]: Template:CDD	${\bar{S}}_{5}$ = [4,3,3^2,1]: Template:CDD ${\bar{O}}_{5}$ = [3,4,3^1,1]: Template:CDD ${\bar{N}}_{5}$ = [3,(3,4)^1,1]: Template:CDD	${\bar{U}}_{5}$ = [3,3,3,4,3]: Template:CDD ${\bar{X}}_{5}$ = [3,3,4,3,3]: Template:CDD ${\bar{R}}_{5}$ = [3,4,3,3,4]: Template:CDD	${\bar{Q}}_{5}$ = [3^2,1,1,1]: Template:CDD ${\bar{M}}_{5}$ = [4,3,3^1,1,1]: Template:CDD ${\bar{L}}_{5}$ = [3^1,1,1,1,1]: Template:CDD

Notes on the Wythoff construction for the uniform 6-polytopes

Construction of the reflective 6-dimensional uniform polytopes are done through a Wythoff construction process, and represented through a Coxeter-Dynkin diagram, where each node represents a mirror. Nodes are ringed to imply which mirrors are active. The full set of uniform polytopes generated are based on the unique permutations of ringed nodes. Uniform 6-polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may have two ways of naming them.

Here's the primary operators available for constructing and naming the uniform 6-polytopes.

The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.

Operation	Extended Schläfli symbol	Coxeter- Dynkin diagram	Description
Parent	t₀{p,q,r,s,t}	Template:CDD	Any regular 6-polytope
Rectified	t₁{p,q,r,s,t}	Template:CDD	The edges are fully truncated into single points. The 6-polytope now has the combined faces of the parent and dual.
Birectified	t₂{p,q,r,s,t}	Template:CDD	Birectification reduces cells to their duals.
Truncated	t_0,1{p,q,r,s,t}	Template:CDD	Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 6-polytope. The 6-polytope has its original faces doubled in sides, and contains the faces of the dual.
Bitruncated	t_1,2{p,q,r,s,t}	Template:CDD	Bitrunction transforms cells to their dual truncation.
Tritruncated	t_2,3{p,q,r,s,t}	Template:CDD	Tritruncation transforms 4-faces to their dual truncation.
Cantellated	t_0,2{p,q,r,s,t}	Template:CDD	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.
Bicantellated	t_1,3{p,q,r,s,t}	Template:CDD	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.
Runcinated	t_0,3{p,q,r,s,t}	Template:CDD	Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated	t_1,4{p,q,r,s,t}	Template:CDD	Runcination reduces cells and creates new cells at the vertices and edges.
Stericated	t_0,4{p,q,r,s,t}	Template:CDD	Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
Pentellated	t_0,5{p,q,r,s,t}	Template:CDD	Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps. (expansion operation for polypeta)
Omnitruncated	t_0,1,2,3,4,5{p,q,r,s,t}	Template:CDD	All five operators, truncation, cantellation, runcination, sterication, and pentellation are applied.

Notes

Template:Reflist

References

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
H.S.M. Coxeter:
- H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Template:Isbn
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Template:KlitzingPolytopes
Template:KlitzingPolytopes

External links

Polytope names
Polytopes of Various Dimensions, Jonathan Bowers
Multi-dimensional Glossary
Template:PolyCell

Template:Polytopes

Template:Navbar-collapsible
Space	Family	${\tilde{A}}_{n - 1}$	${\tilde{C}}_{n - 1}$	${\tilde{B}}_{n - 1}$	${\tilde{D}}_{n - 1}$	${\tilde{G}}_{2}$ / ${\tilde{F}}_{4}$ / ${\tilde{E}}_{n - 1}$
E²	Uniform tiling	0_[3]	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	0_[4]	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	0_[5]	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	0_[6]	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	0_[7]	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	0_[8]	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	0_[9]	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	0_[10]	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	0_[11]	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

↑ T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
↑ Uniform Polypeta, Jonathan Bowers
↑ Uniform polytope
↑ Template:Cite web

[1] T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900

[2] Uniform Polypeta, Jonathan Bowers

[3] Uniform polytope

[4] Template:Cite web

[1]

[2]

[3]

[4]

Uniform 6-polytope

Contents

History of discovery

Uniform 6-polytopes by fundamental Coxeter groups

Uniform prismatic families