Uniform 6-polytope

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

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}.

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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
1	A₆	[3,3,3,3,3]
2	B₆	[3,3,3,3,4]
3	D₆	[3,3,3,3^1,1]
4	E₆	[3^2,2,1]
4	E₆	[3,3^2,2]

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]	Prism family based on 5-simplex
2	B₅A₁	[4,3,3,3,2]	Prism family based on 5-cube
3a	D₅A₁	[3^2,1,1,2]	Prism family based on 5-demicube

#	Coxeter group		Notes
4	A₃I₂(p)A₁	[3,3,2,p,2]	Prism family based on tetrahedral-p-gonal duoprisms
5	B₃I₂(p)A₁	[4,3,2,p,2]	Prism family based on cubic-p-gonal duoprisms
6	H₃I₂(p)A₁	[5,3,2,p,2]	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]	Family based on 5-cell-p-gonal duoprisms.
2	B₄I₂(p)	[4,3,3,2,p]	Family based on tesseract-p-gonal duoprisms.
3	F₄I₂(p)	[3,4,3,2,p]	Family based on 24-cell-p-gonal duoprisms.
4	H₄I₂(p)	[5,3,3,2,p]	Family based on 120-cell-p-gonal duoprisms.
5	D₄I₂(p)	[3^1,1,1,2,p]	Family based on demitesseract-p-gonal duoprisms.

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

Enumerating the convex uniform 6-polytopes

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

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

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

The B₆ family

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

The D₆ family

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

The E₆ family

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		2₂₁ Icosiheptaheptacontidipeton (jak)	99	648	1080	720	216	27
116		Rectified 2₂₁ Rectified icosiheptaheptacontidipeton (rojak)	126	1350	4320	5040	2160	216
117		Truncated 2₂₁ Truncated icosiheptaheptacontidipeton (tojak)	126	1350	4320	5040	2376	432
118		Cantellated 2<sub>21</sub> Small rhombated icosiheptaheptacontidipeton (sirjak)	342	3942	15120	24480	15120	2160
119		Runcinated 2<sub>21</sub> Small demiprismated icosiheptaheptacontidipeton (shopjak)	342	4662	16200	19440	8640	1080
120		Demified icosiheptaheptacontidipeton (hejak)	342	2430	7200	7920	3240	432
121		Bitruncated 2<sub>21</sub> Bitruncated icosiheptaheptacontidipeton (botajik)						2160
122		Demirectified icosiheptaheptacontidipeton (harjak)						1080
123		Cantitruncated 2<sub>21</sub> Great rhombated icosiheptaheptacontidipeton (girjak)						4320
124		Runcitruncated 2<sub>21</sub> Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak)						4320
125		Steritruncated 2<sub>21</sub> Cellitruncated icosiheptaheptacontidipeton (catjak)						2160
126		Demitruncated icosiheptaheptacontidipeton (hotjak)						2160
127		Runcicantellated 2<sub>21</sub> Demiprismatorhombated icosiheptaheptacontidipeton (haprojak)						6480
128		Small demirhombated icosiheptaheptacontidipeton (shorjak)						4320
129		Small prismated icosiheptaheptacontidipeton (spojak)						4320
130		Tritruncated icosiheptaheptacontidipeton (titajak)						4320
131		Runcicantitruncated 2<sub>21</sub> Great demiprismated icosiheptaheptacontidipeton (ghopjak)						12960
132		Stericantitruncated 2<sub>21</sub> Celligreatorhombated icosiheptaheptacontidipeton (cograjik)						12960
133		Great demirhombated icosiheptaheptacontidipeton (ghorjak)						8640
134		Prismatotruncated icosiheptaheptacontidipeton (potjak)						12960
135		Demicellitruncated icosiheptaheptacontidipeton (hictijik)						8640
136		Prismatorhombated icosiheptaheptacontidipeton (projak)						12960
137		Great prismated icosiheptaheptacontidipeton (gapjak)						25920
138		Demicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik)						25920

#	Coxeter diagram	Names	Element counts
#	Coxeter diagram	Names	5-faces	4-faces	Cells	Faces	Edges	Vertices
139	=	1₂₂ Pentacontatetrapeton (mo)	54	702	2160	2160	720	72
140	=	Rectified 1₂₂ Rectified pentacontatetrapeton (ram)	126	1566	6480	10800	6480	720
141	=	Birectified 1₂₂ Birectified pentacontatetrapeton (barm)	126	2286	10800	19440	12960	2160
142	=	Trirectified 1<sub>22</sub> Trirectified pentacontatetrapeton (trim)	558	4608	8640	6480	2160	270
143	=	Truncated 1₂₂ Truncated pentacontatetrapeton (tim)					13680	1440
144	=	Bitruncated 1<sub>22</sub> Bitruncated pentacontatetrapeton (bitem)						6480
145	=	Tritruncated 1<sub>22</sub> Tritruncated pentacontatetrapeton (titam)						8640
146	=	Cantellated 1<sub>22</sub> Small rhombated pentacontatetrapeton (sram)						6480
147	=	Cantitruncated 1<sub>22</sub> Great rhombated pentacontatetrapeton (gram)						12960
148	=	Runcinated 1<sub>22</sub> Small prismated pentacontatetrapeton (spam)						2160
149	=	Bicantellated 1<sub>22</sub> Small birhombated pentacontatetrapeton (sabrim)						6480
150	=	Bicantitruncated 1<sub>22</sub> Great birhombated pentacontatetrapeton (gabrim)						12960
151	=	Runcitruncated 1<sub>22</sub> Prismatotruncated pentacontatetrapeton (patom)						12960
152	=	Runcicantellated 1<sub>22</sub> Prismatorhombated pentacontatetrapeton (prom)						25920
153	=	Omnitruncated 1<sub>22</sub> 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
	{p}×{q}×{r} ^[4]	p+q+r	pq+pr+qr+p+q+r	pqr+2(pq+pr+qr)	3pqr+pq+pr+qr	3pqr	pqr
	{p}×{p}×{p}	3p	3p(p+1)	p²(p+6)	3p²(p+1)	3p³	p³
	{3}×{3}×{3} (trittip)	9	36	81	99	81	27
	{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		Forms
1	${\tilde {A}}_{5}$	[3^[6]]	12
2	${\tilde {C}}_{5}$	[4,3³,4]	35
3	${\tilde {B}}_{5}$	[4,3,3^1,1] [4,3³,4,1⁺]	47 (16 new)
4	${\tilde {D}}_{5}$	[3^1,1,3,3^1,1] [1⁺,4,3³,4,1⁺]	20 (3 new)

Regular and uniform honeycombs include:

${\tilde {A}}_{5}$ ${\tilde {A}}_{5}$ There are 12 unique uniform honeycombs, including:
${\tilde {C}}_{5}$ ${{\tilde {C}}}_{5}$ There are 35 uniform honeycombs, including:
- Regular hypercube honeycomb of Euclidean 5-space, the 5-cube honeycomb, with symbols {4,3³,4}, =
${\tilde {B}}_{5}$ ${{\tilde {B}}}_{5}$ There are 47 uniform honeycombs, 16 new, including:
- The uniform alternated hypercube honeycomb, 5-demicubic honeycomb, with symbols h{4,3³,4}, = =
${\tilde {D}}_{5}$ , [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}, = . The other two new ones are = , = .

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

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]]: ${\widehat {AU}}_{5}$ = [(3,3,3,3,3,4)]: ${\widehat {AR}}_{5}$ = [(3,3,4,3,3,4)]:	${\bar {S}}_{5}$ = [4,3,3^2,1]: ${\bar {O}}_{5}$ = [3,4,3^1,1]: ${\bar {N}}_{5}$ = [3,(3,4)^1,1]:	${\bar {U}}_{5}$ = [3,3,3,4,3]: ${\bar {X}}_{5}$ = [3,3,4,3,3]: ${\bar {R}}_{5}$ = [3,4,3,3,4]:	${\bar {Q}}_{5}$ = [3^2,1,1,1]: ${\bar {M}}_{5}$ = [4,3,3^1,1,1]: ${\bar {L}}_{5}$ = [3^1,1,1,1,1]:

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	Description
Parent	t₀{p,q,r,s,t}	Any regular 6-polytope
Rectified	t₁{p,q,r,s,t}	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}	Birectification reduces cells to their duals.
Truncated	t_0,1{p,q,r,s,t}	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}	Bitrunction transforms cells to their dual truncation.
Tritruncated	t_2,3{p,q,r,s,t}	Tritruncation transforms 4-faces to their dual truncation.
Cantellated	t_0,2{p,q,r,s,t}	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}	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}	Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated	t_1,4{p,q,r,s,t}	Runcination reduces cells and creates new cells at the vertices and edges.
Stericated	t_0,4{p,q,r,s,t}	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}	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}	All five operators, truncation, cantellation, runcination, sterication, and pentellation are applied.

Notes

^ 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
^ "N,m,k-tip".

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, ISBN 978-0-471-01003-6
- (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
Klitzing, Richard. "6D uniform polytopes (polypeta)".
Klitzing, Richard. "Uniform polytopes truncation operators".

External links

Polytope names
Polytopes of Various Dimensions, Jonathan Bowers
Multi-dimensional Glossary
Glossary for hyperspace, George Olshevsky.

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
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	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	{3^[11]}	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

This page was last edited on 17 November 2023, at 06:24