Uniform 5-polytope

Graphs of regular and uniform polytopes.

5-simplex	Rectified 5-simplex		Truncated 5-simplex
Cantellated 5-simplex	Runcinated 5-simplex		Stericated 5-simplex
5-orthoplex	Truncated 5-orthoplex		Rectified 5-orthoplex
Cantellated 5-orthoplex		Runcinated 5-orthoplex
Cantellated 5-cube	Runcinated 5-cube		Stericated 5-cube
5-cube	Truncated 5-cube		Rectified 5-cube
5-demicube		Truncated 5-demicube
Cantellated 5-demicube		Runcinated 5-demicube

In geometry, a uniform polyteron^[1]^[2] (or uniform 5-polytope) is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive and constructed from uniform polychoron facets.

The complete set of convex uniform polytera 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.

Regular 5-polytopes

Main article: List of regular polytopes#Five Dimensions

Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face. There are exactly three such regular polytopes, all convex:

{3,3,3,3} - Hexateron (5-simplex)
{4,3,3,3} - Penteract (5-hypercube)
{3,3,3,4} - Pentacross (5-orthoplex)

There are no nonconvex regular polytopes in 5 or more dimensions.

Convex uniform 5-polytopes

There are 105 known convex uniform 5-polytopes, plus a number of infinite families of duoprism prisms, and polygon-polyhedron duoprisms. All except the grand antiprism prism are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.

Reflection families

The hexateron is the regular form in the A₅ family. The penteract and pentacross are the regular forms in the B₅ family. The bifurcating graph of the D₆ family contains the pentacross, as well as a demipenteract which is an alternated penteract.

Fundamental families

#	Coxeter group
1	A₅	[3⁴]
2	B₅	[4,3³]
3	D₅	[3^2,1,1]

Uniform prisms There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes:

#	Coxeter groups
1	A₄ × A₁	[3,3,3] × [ ]
2	B₄ × A₁	[4,3,3] × [ ]
3	F₄ × A₁	[3,4,3] × [ ]
4	H₄ × A₁	[5,3,3] × [ ]
5	D₄ × A₁	[3^1,1,1] × [ ]

There is one infinite family of 5-polytopes based on prisms of the uniform duoprisms {p}×{q}×{ }:

Coxeter groups		Coxeter graph
I₂(p) × I₂(q) × A₁	[p] × [q] × [ ]

Uniform duoprisms

There are 3 categorical uniform duoprismatic families of polytopes based on Cartesian products of the uniform polyhedra and regular polygons: {q,r}×{p}:

#	Coxeter groups
1	A₃ × I₂(p)	[3,3] × [p]
2	B₃ × I₂(p)	[4,3] × [p]
3.	H₃ × I₂(p)	[5,3] × [p]

Enumerating the convex uniform 5-polytopes

Simplex family: A₅ [3⁴]
- 19 uniform 5-polytopes
Hypercube/Orthoplex family: BC₅ [4,3³]
- 31 uniform 5-polytopes
Demihypercube D₅/E₅ family: [3^2,1,1]
- 23 uniform 5-polytopes (8 unique)
Prisms and duoprisms:
- 56 uniform 5-polytope (46 unique) constructions based on prismatic families: [3,3,3]x[ ], [4,3,3]x[ ], [5,3,3]x[ ], [3^1,1,1]x[ ].
- One non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two grand antiprisms connected by polyhedral prisms.

That brings the tally to: 19+31+8+46+1=105

In addition there are:

Infinitely many uniform 5-polytope constructions based on duoprism prismatic families: [p]x[q]x[ ].
Infinitely many uniform 5-polytope constructions based on duoprismatic families: [3,3]x[p], [4,3]x[p], [5,3]x[p].

The A₅ family

There are 19 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. (16+4-1 cases)

They are named by Norman Johnson from the Wythoff construction operations upon regular 5-simplex (hexateron).

The A₅ family has symmetry of order 720 (6 factorial).

The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, all in hyperplanes with normal vector (1,1,1,1,1,1).

See symmetry graphs: List of A5 polytopes

#	Base point	Johnson naming system Bowers name and (acronym) Coxeter-Dynkin	k-face element counts					Vertex figure	Facet counts by location: [3,3,3,3]
#	Base point		4	3	2	1	0	Vertex figure	[3,3,3] (6)	[3,3]×[ ] (15)	[3]×[3] (20)	[ ]×[3,3] (15)	[3,3,3] (6)
1	(0,0,0,0,0,1) or (0,1,1,1,1,1)	5-simplex hexateron (hix)	6	15	20	15	6	{3,3,3}	(5) {3,3,3}	-	-	-	-
2	(0,0,0,0,1,1) or (0,0,1,1,1,1)	Rectified 5-simplex rectified hexateron (rix)	12	45	80	60	15	t{3,3}x{ }	(4) t₁{3,3,3}	-	-	-	(2) {3,3,3}
3	(0,0,0,0,1,2) or (0,1,2,2,2,2)	Truncated 5-simplex truncated hexateron (tix)	12	60	120	90	20	Tetrah.pyr	(4) t_0,1{3,3,3}	-	-	-	(1) {3,3,3}
4	(0,0,0,1,1,1)	Birectified 5-simplex dodecateron (dot)	12	45	80	75	30	{3}x{3}	(3) t₁{3,3,3}	-	-	-	(3) t₁{3,3,3}
5	(0,0,0,1,1,2) or (0,1,1,2,2,2)	Cantellated 5-simplex small rhombated hexateron (sarx)	12	60	140	150	60	prism-wedge	(3) t_0,2{3,3,3}	-	-	(1) × { }×{3,3}	(1) t₁{3,3,3}
6	(0,0,0,1,2,2) or (0,0,1,2,2,2)	Bitruncated 5-simplex bitruncated hexateron (bittix)	27	135	290	240	60		(3) t_1,2{3,3,3}	-	-	-	(2) t_0,1{3,3,3}
7	(0,0,0,1,2,3) or (0,1,2,3,3,3)	Cantitruncated 5-simplex great rhombated hexateron (garx)	32	180	420	360	90		t_0,1,2{3,3,3}	-	-	× { }×{3,3}	t_0,1{3,3,3}
8	(0,0,1,1,1,2) or (0,1,1,1,2,2)	Runcinated 5-simplex small prismated hexateron (spix)	47	255	420	270	60		(2) t_0,3{3,3,3}	-	(3) × {3}×{3}	(3) × { }×t₁{3,3}	(1) t₁{3,3,3}
9	(0,0,1,1,2,2)	Bicantellated 5-simplex small birhombated dodecateron (sibrid)	62	180	210	120	30		(2) t_0,2{3,3,3}	-	(8) × {3}×{3}	-	(2) t_0,2{3,3,3}
10	(0,0,1,1,2,3) or (0,1,2,2,3,3)	Runcitruncated 5-simplex prismatotruncated hexateron (pattix)	27	135	290	300	120		t_0,1,3{3,3,3}	-	× {6}×{3}	× { }×t₁{3,3}	t_0,2{3,3,3}
11	(0,0,1,2,2,3) or (0,1,1,2,3,3)	Runcicantellated 5-simplex prismatorhombated hexateron (pirx)	32	180	420	450	180		t_0,1,3{3,3,3}	-	× {3}×{3}	× { }×t_0,1{3,3}	t_1,2{3,3,3}
12	(0,0,1,2,3,3)	Bicantitruncated 5-simplex great birhombated dodecateron (gibrid)	47	315	720	630	180		t_0,1,2{3,3,3}	-	× {3}×{3}	-	t_0,1,2{3,3,3}
13	(0,0,1,2,3,4) or (0,1,2,3,4,4)	Runcicantitruncated 5-simplex great prismated hexateron (gippix)	47	255	570	540	180	Irr.5-cell	t_0,1,2,3{3,3,3}	-	× {3}×{6}	× { }×t_0,1{3,3}	t_0,2{3,3,3}
14	(0,1,1,1,1,2)	Stericated 5-simplex small cellated dodecateron (scad)	62	330	570	420	120	Irr.16-cell	(1) {3,3,3}	(4) × { }×{3,3}	(6) × {3}×{3}	(4) × { }×{3,3}	(1) {3,3,3}
15	(0,1,1,1,2,3) or (0,1,2,2,2,3)	Steritruncated 5-simplex celliprismated hexateron (cappix)	62	420	900	720	180		t_0,1{3,3,3}	× { }×t_0,1{3,3}	× {3}×{6}	× { }×{3,3}	t_0,3{3,3,3}
16	(0,1,1,2,2,3)	Stericantellated 5-simplex small cellirhombated dodecateron (card)	47	315	810	900	360		t_0,2{3,3,3}	× { }×t_0,2{3,3}	× {3}×{3}	× { }×t_0,2{3,3}	t_0,2{3,3,3}
17	(0,1,1,2,3,4) or (0,1,2,3,3,4)	Stericantitruncated 5-simplex celligreatorhombated hexateron (cograx)	62	480	1140	1080	360		t_0,1,2{3,3,3}	× { }×t_0,1,2{3,3}	× {3}×{6}	× { }×t_0,2{3,3}	t_0,1,3{3,3,3}
18	(0,1,2,2,3,4)	Steriruncitruncated 5-simplex celliprismatotruncated dodecateron (captid)	62	450	1110	1080	360		t_0,1,3{3,3,3}	× { }×t_0,1{3,3}	× {6}×{6}	× { }×t_0,1,3{3,3}	t_0,1,3{3,3,3}
19	(0,1,2,3,4,5)	Omnitruncated 5-simplex great cellated dodecateron (gocad)	62	540	1560	1800	720	Irr. {3,3,3}	(1) t_0,1,2,3{3,3,3}	(1) × { }×t_0,1,2{3,3}	(1) × {6}×{6}	(1) × { }×t_0,1,2{3,3}	(1) t_0,1,2,3{3,3,3}

The B₅ family

The B₅ family has symmetry of order 3840 (5!×2⁵).

This family has 2⁵−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter-Dynkin diagram.

For simplicity it is divided into two subgroups, each with 12 forms, and 7 "middle" forms which equally belong in both.

The penteractic family of polytera are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform polyteron. All coordinates correspond with uniform polytera of edge length 2.

See symmetry graph: List of B5 polytopes

#	Base point	Name Coxeter-Dynkin	Element counts					Vertex figure	Facet counts by location: [4,3,3,3]
#	Base point	Name Coxeter-Dynkin	4	3	2	1	0	Vertex figure	[4,3,3] (10)	[4,3]×[ ] (40)	[4]×[3] (80)	[ ]×[3,3] (80)	[3,3,3] (32)
1	(0,0,0,0,1)√2	5-orthoplex (Quadrirectified 5-cube)	10	40	80	80	32	{3,3,4}	{3,3,3}	-	-	-	-
2	(0,0,0,1,1)√2	Rectified 5-orthoplex (Trirectified 5-cube)	42	200	400	320	80	{ }×{3,4}	{3,3,4}	-	-	-	t₁{3,3,3}
3	(0,0,0,1,2)√2	Truncated 5-orthoplex (Quadritruncated 5-cube)	42	200	400	400	160	(Octah.pyr)	t_0,1{3,3,3}	{3,3,3}	-	-	-
4	(0,0,1,1,1)√2	Birectified 5-cube (Birectified 5-orthoplex)	42	280	640	480	80	{4}×{3}	t₁{3,3,4}	-	-	-	t₁{3,3,3}
5	(0,0,1,1,2)√2	Cantellated 5-orthoplex (Tricantellated 5-cube)	122	680	1520	1280	320	Prism-wedge	t₁{3,3,4}	{ }×{3,4}	-	-	t_0,2{3,3,3}
6	(0,0,1,2,2)√2	Bitruncated 5-orthoplex (tritruncated 5-cube)	42	280	720	800	320		t_0,1{3,3,4}	-	-	-	t_1,2{3,3,3}
7	(0,0,1,2,3)√2	Cantitruncated 5-orthoplex (tricantitruncated 5-orthoplex)	122	680	1520	1600	640		t_0,2{3,3,4}	{ }×t₁{3,4}	{6}×{4}	-	t_0,1,3{3,3,3}
8	(0,1,1,1,1)√2	Rectified 5-cube	42	240	400	240	40	{3,3}x{ }	t₁{4,3,3}	-	-	-	{3,3,3}
9	(0,1,1,1,2)√2	Runcinated 5-orthoplex	202	1240	2160	1440	320		t₁{4,3,3}	-	{3}×{4}		t_0,3{3,3,3}
10	(0,1,1,2,2)√2	Bicantellated 5-cube (Bicantellated 5-orthoplex)	122	840	2160	1920	480		t_0,2{4,3,3}	-	{4}×{3}	-	t_0,2{3,3,3}
11	(0,1,1,2,3)√2	Runcitruncated 5-orthoplex	202	1560	3760	3360	960		t_0,2{3,3,4}	{ }×t₁{3,4}	{6}×{4}	-	t_0,1,3{3,3,3}
12	(0,1,2,2,2)√2	Bitruncated 5-cube	42	280	720	720	240		t_1,2{4,3,3}	-	-	-	t_0,1{3,3,3}
13	(0,1,2,2,3)√2	Runcicantellated 5-orthoplex	202	1240	2960	2880	960		{ }×t_0,1{3,4}	t_1,2{3,3,4}	{3}×{4}	-	t_0,1,3{3,3,3}
14	(0,1,2,3,3)√2	Bicantitruncated 5-cube (Bicantitruncated 5-orthoplex)	122	840	2160	2400	960		t_0,2{4,3,3}	-	{4}×{3}	-	t_0,2{3,3,3}
15	(0,1,2,3,4)√2	Runcicantitruncated 5-orthoplex	202	1560	4240	4800	1920		t_0,1,2{3,3,4}	{ }×t_0,1{3,4}	{6}×{4}	-	t_0,1,2,3{3,3,3}
16	(1,1,1,1,1)	5-cube	32	80	80	40	10	{3,3,3}	{4,3,3}	-	-	-	-
17	(1,1,1,1,1) + (0,0,0,0,1)√2	Stericated 5-cube (Stericated 5-orthoplex)	242	800	1040	640	160	Tetr.antiprm	{4,3,3}	{4,3}×{ }	{4}×{3}	{ }×{3,3}	{3,3,3}
18	(1,1,1,1,1) + (0,0,0,1,1)√2	Runcinated 5-cube	162	1200	2160	1440	320		t_0,3{4,3,3}	-	{4}×{3}	{ }×t₁{3,3}	{3,3,3}
19	(1,1,1,1,1) + (0,0,0,1,2)√2	Steritruncated 5-orthoplex	242	1600	2960	2240	640		t_0,3{3,3,4}	{ }×{4,3}	-	-	t_0,1{3,3,3}
20	(1,1,1,1,1) + (0,0,1,1,1)√2	Cantellated 5-cube	82	640	1520	1200	240	Prism-wedge	t_0,2{4,3,3}	-	-	{ }×{3,3}	t₁{3,3,3}
21	(1,1,1,1,1) + (0,0,1,1,2)√2	Stericantellated 5-cube (Stericantellated 5-orthoplex)	242	2080	4720	3840	960		t_0,2{4,3,3}	t_0,2{4,3}×{ }	{4}×{3}	{ }×t_0,2{3,3}	t_0,2{3,3,3}
22	(1,1,1,1,1) + (0,0,1,2,2)√2	Runcicantellated 5-cube	162	1200	2960	2880	960		t_0,1,3{4,3,3}	-	{4}×{3}	{ }×t_0,1{3,3}	t_1,2{3,3,3}
23	(1,1,1,1,1) + (0,0,1,2,3)√2	Stericantitruncated 5-orthoplex	242	2400	6000	5760	1920		{ }×t_0,2{3,4}	t_0,1,3{3,3,4}	{6}×{4}	{ }×t_0,1{3,3}	t_0,1,2{3,3,3}
24	(1,1,1,1,1) + (0,1,1,1,1)√2	Truncated 5-cube	42	240	400	280	80	Tetrah.pyr	t_0,1{4,3,3}	-	-	-	{3,3,3}
25	(1,1,1,1,1) + (0,1,1,1,2)√2	Steritruncated 5-cube	242	1520	2880	2240	640		t_0,1{4,3,3}	t_0,1{4,3}×{ }	{8}×{3}	{ }×{3,3}	t_0,3{3,3,3}
26	(1,1,1,1,1) + (0,1,1,2,2)√2	Runcitruncated 5-cube	162	1440	3680	3360	960		t_0,1,3{4,3,3}	{ }×t_0,1{4,3}	{6}×{8}	{ }×t_0,1{3,3}	t_0,1,3{3,3,3}]]
27	(1,1,1,1,1) + (0,1,1,2,3)√2	Steriruncitruncated 5-cube (Steriruncitruncated 5-orthoplex)	242	2160	5760	5760	1920		t_0,1,3{4,3,3}	t_0,1{4,3}×{ }	{8}×{6}	{ }×t_0,1{3,3}	t_0,1,3{3,3,3}
28	(1,1,1,1,1) + (0,1,2,2,2)√2	cantitruncated 5-cube	82	640	1520	1440	480		t_0,1,2{4,3,3}	-	-	{ }×{3,3}	t_0,1{3,3,3}
29	(1,1,1,1,1) + (0,1,2,2,3)√2	Stericantitruncated 5-cube	242	2320	5920	5760	1920		t_0,1,2{4,3,3}	t_0,1,2{4,3}×{ }	{8}×{3}	{ }×t_0,2{3,3}	t_0,1,3{3,3,3}
30	(1,1,1,1,1) + (0,1,2,3,3)√2	Runcicantitruncated 5-cube	162	1440	4160	4800	1920		t_0,1,2,3{4,3,3}	-	{8}×{3}	{ }×t_0,1{3,3}	t_0,1,2{3,3,3}
31	(1,1,1,1,1) + (0,1,2,3,4)√2	Omnitruncated 5-cube (omnitruncated 5-orthoplex)	242	2640	8160	9600	3840	Irr. {3,3,3}	t_0,1,2{4,3}×{ }	t_0,1,2{4,3}×{ }	{8}×{6}	{ }×t_0,1,2{3,3}	t_0,1,2,3{3,3,3}

The D₅ family

The D₅ family has symmetry of order 1920 (5! x 2⁴).

This family has 23 Wythoffian uniform polyhedra, from 3x8-1 permutations of the D₅ Coxeter-Dynkin diagram with one or more rings. 15 (2x8-1) are repeated from the B₅ family and 8 are unique to this family.

See symmetry graphs: List of D5 polytopes

#	Coxeter-Dynkin diagram Schläfli symbol symbols Johnson and Bowers names	Element counts					Vertex figure	Facets by location: [3^1,2,1]
#		4	3	2	1	0	Vertex figure	[3,3,3] (16)	[3^1,1,1] (10)	[3,3]×[ ] (40)	[ ]×[3]×[ ] (80)	[3,3,3] (16)
51	(1₂₁) 5-demicube Hemipenteract (hin)	26	120	160	80	16	t₁{3,3,3}	{3,3,3}	t₀(1₁₁)	-	-	-
52	t_0,1(1₂₁) Truncated 5-demicube Truncated hemipenteract (thin)	42	280	640	560	160
53	t_0,2(1₂₁) Cantellated 5-demicube Small rhombated hemipenteract (sirhin)	42	360	880	720	160
54	t_0,3(1₂₁) Runcinated 5-demicube Small prismated hemipenteract (siphin)	82	480	720	400	80
55	t_0,1,2(1₂₁) Cantitruncated 5-demicube Great rhombated hemipenteract (girhin)	42	360	1040	1200	480
56	t_0,1,3(1₂₁) Runcitruncated 5-demicube Prismatotruncated hemipenteract (pithin)	82	720	1840	1680	480
57	t_0,2,3(1₂₁) Runcicantellated 5-demicube Prismatorhombated hemipenteract (pirhin)	82	560	1280	1120	320
58	t_0,1,2,3(1₂₁) Runcicantitruncated 5-demicube Great prismated hemipenteract (giphin)	82	720	2080	2400	960

Uniform prismatic forms

There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes:

A₄ × A₁

This prismatic family has 9 forms:

The A₁ x A₄ family has symmetry of order 240 (2*5!).

#	Coxeter-Dynkin and Schläfli symbols Name	Element counts
#	Coxeter-Dynkin and Schläfli symbols Name	Facets	Cells	Faces	Edges	Vertices
59	{3,3,3}x{ } 5-cell prism	7	20	30	25	10
60	t₁{3,3,3}x{ } Rectified 5-cell prism	12	50	90	70	20
61	t_0,1{3,3,3}x{ } Truncated 5-cell prism	12	50	100	100	40
62	t_0,2{3,3,3}x{ } Cantellated 5-cell prism	22	120	250	210	60
63	t_0,3{3,3,3}x{ } Runcinated 5-cell prism	32	130	200	140	40
64	t_1,2{3,3,3}x{ } Bitruncated 5-cell prism	12	60	140	150	60
65	t_0,1,2{3,3,3}x{ } Cantitruncated 5-cell prism	22	120	280	300	120
66	t_0,1,3{3,3,3}x{ } Runcitruncated 5-cell prism	32	180	390	360	120
67	t_0,1,2,3{3,3,3}x{ } Omnitruncated 5-cell prism	32	210	540	600	240

B₄ × A₁

This prismatic family has 16 forms. (Three are shared with [3,4,3]×[ ] family)

The A₁ x B₄ family has symmetry of order 768 (2*2^4*4!).

#	Coxeter-Dynkin and Schläfli symbols Name	Element counts
#	Coxeter-Dynkin and Schläfli symbols Name	Facets	Cells	Faces	Edges	Vertices
68	{4,3,3}x{ } Tesseractic prism	10	40	80	80	32
69	t₁{4,3,3}x{ } Rectified tesseractic prism	26	136	272	224	64
70	t_0,1{4,3,3}x{ } Truncated tesseractic prism	26	136	304	320	128
71	t_0,2{4,3,3}x{ } Cantellated tesseractic prism	58	360	784	672	192
72	t_0,3{4,3,3}x{ } Runcinated tesseractic prism	82	368	608	448	128
73	t_1,2{4,3,3}x{ } Bitruncated tesseractic prism	26	168	432	480	192
74	t_0,1,2{4,3,3}x{ } Cantitruncated tesseractic prism	58	360	880	960	384
75	t_0,1,3{4,3,3}x{ } Runcitruncated tesseractic prism	82	528	1216	1152	384
76	t_0,1,2,3{4,3,3}x{ } Omnitruncated tesseractic prism	82	624	1696	1920	768
77	{3,3,4}x{ } 16-cell prism	18	64	88	56	16
78	t₁{3,3,4}x{ } Rectified 16-cell prism (Same as 24-cell prism)	26	144	288	216	48
79	t_0,1{3,3,4}x{ } Truncated 16-cell prism	26	144	312	288	96
80	t_0,2{3,3,4}x{ } Cantellated 16-cell prism (Same as rectified 24-cell prism)	50	336	768	672	192
81	t_0,1,2{3,3,4}x{ } Cantitruncated 16-cell prism (Same as truncated 24-cell prism)	50	336	864	960	384
82	t_0,1,3{3,3,4}x{ } Runcitruncated 16-cell prism	82	528	1216	1152	384

F₄ × A₁

This prismatic family has 10 forms.

The A₁ x F₄ family has symmetry of order 2304 (2*1152).

#	Coxeter-Dynkin and Schläfli symbols Name	Element counts
#	Coxeter-Dynkin and Schläfli symbols Name	Facets	Cells	Faces	Edges	Vertices
[79]	{3,4,3}x{ } 24-cell prism	26	144	288	216	48
[80]	t₁{3,4,3}x{ } rectified 24-cell prism	50	336	768	672	192
[81]	t_0,1{3,4,3}x{ } truncated 24-cell prism	50	336	864	960	384
84	t_0,2{3,4,3}x{ } cantellated 24-cell prism	146	1008	2304	2016	576
85	t_0,3{3,4,3}x{ } runcinated 24-cell prism	242	1152	1920	1296	288
86	t_1,2{3,4,3}x{ } bitruncated 24-cell prism	50	432	1248	1440	576
87	t_0,1,2{3,4,3}x{ } cantitruncated 24-cell prism	146	1008	2592	2880	1152
88	t_0,1,3{3,4,3}x{ } runcitruncated 24-cell prism	242	1584	3648	3456	1152
89	t_0,1,2,3{3,4,3}x{ } omnitruncated 24-cell prism	242	1872	5088	5760	2304
[83]	h_0,1{3,4,3}x{ } snub 24-cell prism	146	768	1392	960	192

H₄ × A₁

This prismatic family has 15 forms:

The A₁ x H₄ family has symmetry of order 28800 (2*14400).

#	Coxeter-Dynkin and Schläfli symbols Name	Element counts
#	Coxeter-Dynkin and Schläfli symbols Name	Facets	Cells	Faces	Edges	Vertices
90	{5,3,3}x{ } 120-cell prism	122	960	2640	3000	1200
91	t₁{5,3,3}x{ } Rectified 120-cell prism	722	4560	9840	8400	2400
92	t_0,1{5,3,3}x{ } Truncated 120-cell prism	722	4560	11040	12000	4800
93	t_0,2{5,3,3}x{ } Cantellated 120-cell prism	1922	12960	29040	25200	7200
94	t_0,3{5,3,3}x{ } Runcinated 120-cell prism	2642	12720	22080	16800	4800
95	t_1,2{5,3,3}x{ } Bitruncated 120-cell prism	722	5760	15840	18000	7200
96	t_0,1,2{5,3,3}x{ } Cantitruncated 120-cell prism	1922	12960	32640	36000	14400
97	t_0,1,3{5,3,3}x{ } Runcitruncated 120-cell prism	2642	18720	44880	43200	14400
98	t_0,1,2,3{5,3,3}x{ } Omnitruncated 120-cell prism	2642	22320	62880	72000	28800
99	{3,3,5}x{ } 600-cell prism	602	2400	3120	1560	240
100	t₁{3,3,5}x{ } Rectified 600-cell prism	722	5040	10800	7920	1440
101	t_0,1{3,3,5}x{ } Truncated 600-cell prism	722	5040	11520	10080	2880
102	t_0,2{3,3,5}x{ } Cantellated 600-cell prism	1442	11520	28080	25200	7200
103	t_0,1,2{3,3,5}x{ } Cantitruncated 600-cell prism	1442	11520	31680	36000	14400
104	t_0,1,3{3,3,5}x{ } Runcitruncated 600-cell prism	2642	18720	44880	43200	14400

Grand antiprism prism

The grand antiprism prism is the only known convex nonwythoffian uniform polyteron. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (300 tetrahedrons, 20 pentagonal antiprisms, 700 triangular prisms, 20 pentagonal prisms), 322 hypercells (2 grand antiprisms , 20 pentagonal antiprism prisms , and 300 tetrahedral prisms ).

#	Name	Element counts
#	Name	Facets	Cells	Faces	Edges	Vertices
105	grand antiprism prism Gappip	322	1360	1940	1100	200

Notes on the Wythoff construction for the uniform 5-polytopes

Construction of the reflective 5-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 5-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 5-polytopes.

The last operation, the snub, and more generally the alternation, are the operation that can create nonreflective forms. These are drawn with "hollow rings" at the nodes.

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}	Any regular 5-polytope
Rectified	t₁{p,q,r,s}	The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual.
Birectified	t₂{p,q,r,s}	Birectification reduces cells to their duals.
Truncated	t_0,1{p,q,r,s}	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 5-polytope. The 5-polytope has its original faces doubled in sides, and contains the faces of the dual.
Cantellated	t_0,2{p,q,r,s}	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Runcinated	t_0,3{p,q,r,s}	Runcination reduces cells and creates new cells at the vertices and edges.
Stericated	t_0,4{p,q,r,s}	Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for polyterons.)
Omnitruncated	t_0,1,2,3,4{p,q,r,s}	All four operators, truncation, cantellation, runcination, and sterication are applied.

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.^[3]

Fundamental groups
#	Coxeter group
1	${\tilde{A}}_4$	[3^[5]]	[(3,3,3,3,3)]
2	${\tilde{B}}_4$	[4,3,3^1,1]
3	${\tilde{C}}_4$	[4,3,3,4]	h[4,3,3,4]
4	${\tilde{D}}_4$	[3^1,1,1,1]	q[4,3,3,4]
5	${\tilde{F}}_4$	[3,4,3,3]

Prismatic groups
#	Coxeter group
1	${\tilde{C}}_3$ x ${\tilde{I}}_1$	[4,3,4]x[∞]
2	${\tilde{B}}_3$ x ${\tilde{I}}_1$	[4,3^1,1]x[∞]
3	${\tilde{A}}_3$ x ${\tilde{I}}_1$	[3^[4]]x[∞]
4	${\tilde{C}}_2$ x ${\tilde{I}}_1$ x ${\tilde{I}}_1$	[4,4]x[∞]x[∞]
5	${\tilde{H}}_2$ x ${\tilde{I}}_1$ x ${\tilde{I}}_1$	[6,3]x[∞]x[∞]
6	${\tilde{A}}_2$ x ${\tilde{I}}_1$ x ${\tilde{I}}_1$	[3^[3]]x[∞]x[∞]
7	${\tilde{I}}_1$ x ${\tilde{I}}_1$ x ${\tilde{I}}_1$ x ${\tilde{I}}_1$	[∞]x[∞]x[∞]x[∞]
8	${\tilde{A}}_2$ x ${\tilde{A}}_2$	[3^[3]]x[3^[3]]
9	${\tilde{A}}_2$ x ${\tilde{B}}_2$	[3^[3]]x[4,4]
10	${\tilde{A}}_2$ x ${\tilde{G}}_2$	[3^[3]]x[6,3]
11	${\tilde{B}}_2$ x ${\tilde{B}}_2$	[4,4]x[4,4]
12	${\tilde{B}}_2$ x ${\tilde{G}}_2$	[4,4]x[6,3]
13	${\tilde{G}}_2$ x ${\tilde{G}}_2$	[6,3]x[6,3]

There are three regular honeycombs of Euclidean 4-space:

tesseractic honeycomb, with symbols {4,3,3,4}, = . There are 19 uniform honeycombs in this family.
24-cell honeycomb, with symbols {3,4,3,3}, . There are 31 reflective uniform honeycombs in this family, and one alternated form.
- Snub 24-cell honeycomb, with symbols h_0,1{3,4,3,3}, constructed by four snub 24-cell, one 16-cell, and five 5-cells at each vertex.
4-demicube honeycomb, with symbols {3,3,4,3},

Other families that generate uniform honeycombs:

There are 23 uniform honeycombs, 4 unique in the demitesseractic honeycomb family. With symbols h{4,3²,4} it is geometrically identical to the hexadecachoric honeycomb, =
There are 7 uniform honeycombs from the , family, all unique, including:
There are 9 uniform honeycombs in the ${\tilde{D}}_4$ : [3^1,1,1,1] family, all repeated in other families, including the demitesseractic honeycomb.

Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.

Regular tessellations of hyperbolic 4-space

There are five kinds of convex regular honeycombs and four kinds of star-honeycombs in H⁴ space:^[4]

Honeycomb name	Schläfli Symbol {p,q,r,s}	Facet type {p,q,r}	Cell type {p,q}	Face type {p}	Face figure {s}	Edge figure {r,s}	Vertex figure {q,r,s}	Dual
Order-5 pentachoric	{3,3,3,5}	{3,3,3}	{3,3}	{3}	{5}	{3,5}	{3,3,5}	{5,3,3,3}
Order-3 hecatonicosachoric	{5,3,3,3}	{5,3,3}	{5,3}	{5}	{3}	{3,3}	{3,3,3}	{3,3,3,5}
Order-5 tesseractic	{4,3,3,5}	{4,3,3}	{4,3}	{4}	{5}	{3,5}	{3,3,5}	{5,3,3,4}
Order-4 hecatonicosachoric	{5,3,3,4}	{5,3,3}	{5,3}	{5}	{4}	{3,4}	{3,3,4}	{4,3,3,5}
Order-5 hecatonicosachoric	{5,3,3,5}	{5,3,3}	{5,3}	{5}	{5}	{3,5}	{3,3,5}	Self-dual

There are four regular star-honeycombs in H⁴ space:

Honeycomb name	Schläfli Symbol {p,q,r,s}	Facet type {p,q,r}	Cell type {p,q}	Face type {p}	Face figure {s}	Edge figure {r,s}	Vertex figure {q,r,s}	Dual
Order-3 stellated hecatonicosachoric	{5/2,5,3,3}	{5/2,5,3}	{5/2,5}	{5}	{5}	{3,3}	{5,3,3}	{3,3,5,5/2}
Order-5/2 hexacosichoric	{3,3,5,5/2}	{3,3,5}	{3,3}	{3}	{5/2}	{5,5/2}	{3,5,5/2}	{5/2,5,3,3}
Order-5 icosahedral hecatonicosachoric	{3,5,5/2,5}	{3,5,5/2}	{3,5}	{3}	{5}	{5/2,5}	{5,5/2,5}	{5,5/2,5,3}
Order-3 great hecatonicosachoric	{5,5/2,5,3}	{5,5/2,5}	{5,5/2}	{5}	{3}	{5,3}	{5/2,5,3}	{3,5,5/2,5}

Regular and uniform hyperbolic honeycombs

There are 5 compact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams. There are also 9 noncompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Noncompact groups generate honeycombs with inifinite facets or vertex figures.

Compact hyperbolic groups
${\widehat{AF}}_4$ = [(3,3,3,3,4)]:	${\bar{DH}}_4$ = [5,3,3^1,1]:	${\bar{H}}_4$ = [3,3,3,5]: ${\bar{BH}}_4$ = [4,3,3,5]: ${\bar{K}}_4$ = [5,3,3,5]:

Noncompact hyperbolic groups
${\bar{P}}_4$ = [3,3^[4]]: ${\bar{BP}}_4$ = [4,3^[4]]: ${\bar{FR}}_4$ = [(3,3,4,3,4)]: ${\bar{DP}}_4$ = [3^{[3]x[ ]}]:	${\bar{N}}_4$ = [4,/3\,3,4]: ${\bar{O}}_4$ = [3,4,3^1,1]: ${\bar{S}}_4$ = [4,3^2,1]: ${\bar{M}}_4$ = [4,3^1,1,1]:	${\bar{R}}_4$ = [3,4,3,4]:

Notes

^ A proposed name polyteron (plural: polytera) has been advocated, from the Greek root poly- meaning "many", a shortened tetra- meaning "four", and suffix -on. "Four" refers to the dimension of the 5-polytope facets.
^ http://www.steelpillow.com/polyhedra/ditela.html
^ George Olshevsky (2006), Uniform Panoploid Tetracombs, manuscript. Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs.
^ Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213

References

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 (3 regular and one semiregular 4-polytope)
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, Regular Polytopes, 3rd Edition, Dover New York, 1973 (p. 297 Fundamental regions for irreducible groups generated by reflections, Spherical and Euclidean)
- H.S.M. Coxeter, The Beauty of Geometry: Twelve Essays (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (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] (p. 287 5D Euclidean groups)
- (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
Richard Klitzing, 5D, uniform polytopes (polytera)
James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990) (Page 141, 6.9 List of hyperbolic Coxeter groups, figure 2) [2]

External links

Polytope names, Guy Inchbald
Polytopes of Various Dimensions, Jonathan Bowers
Glossary for hyperspace: Polytope, George Olshevsky
Multi-dimensional Glossary, Garrett Jones
polytope names, Wendy Krieger


Family	A_n	BC_n	D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes