Uniform 5-polytope

Template:Short description

Graphs of regular and uniform 5-polytopes.
File:5-simplex t0.svg 5-simplex Template:CDD	File:5-simplex t1.svg Rectified 5-simplex Template:CDD		File:5-simplex t01.svg Truncated 5-simplex Template:CDD
File:5-simplex t02.svg Cantellated 5-simplex Template:CDD	File:5-simplex t03.svg Runcinated 5-simplex Template:CDD		File:5-simplex t04.svg Stericated 5-simplex Template:CDD
File:5-cube t4.svg 5-orthoplex Template:CDD	File:5-cube t34.svg Truncated 5-orthoplex Template:CDD		File:5-cube t3.svg Rectified 5-orthoplex Template:CDD
File:5-cube t24.svg Cantellated 5-orthoplex Template:CDD		File:5-cube t14.svg Runcinated 5-orthoplex Template:CDD
File:5-cube t02.svg Cantellated 5-cube Template:CDD	File:5-cube t03.svg Runcinated 5-cube Template:CDD		File:5-cube t04.svg Stericated 5-cube Template:CDD
File:5-cube t0.svg 5-cube Template:CDD	File:5-cube t01.svg Truncated 5-cube Template:CDD		File:5-cube t1.svg Rectified 5-cube Template:CDD
File:5-demicube t0 D5.svg 5-demicube Template:CDD		File:5-demicube t01 D5.svg Truncated 5-demicube Template:CDD
File:5-demicube t02 D5.svg Cantellated 5-demicube Template:CDD		File:5-demicube t03 D5.svg Runcinated 5-demicube Template:CDD

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

The complete set of convex uniform 5-polytopes has not been determined, but many 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 diagrams.

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 4-polytopes) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.^[1]
Convex uniform polytopes:
- 1940-1988: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes I, II, and III.
- 1966: Norman W. Johnson completed his Ph.D. Dissertation under Coxeter, The Theory of Uniform Polytopes and Honeycombs, University of Toronto
Non-convex uniform polytopes:
- 1966: Johnson describes two non-convex uniform antiprisms in 5-space in his dissertation.^[2]
- 2000-2024: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes,^[3] with a current count of 1348 known uniform 5-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 4-polytopes. The list is not proven complete.^[4]^[5]

Regular 5-polytopes

Script error: No such module "Labelled list hatnote". Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} 4-polytope facets around each face. There are exactly three such regular polytopes, all convex:

{3,3,3,3} - 5-simplex
{4,3,3,3} - 5-cube
{3,3,3,4} - 5-orthoplex

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

Convex uniform 5-polytopes

Unsolved problem in mathematics

What is the complete set of convex uniform 5-polytopes?^[6]

Symmetry of uniform 5-polytopes in four dimensions

The 5-simplex is the regular form in the A₅ family. The 5-cube and 5-orthoplex are the regular forms in the B₅ family. The bifurcating graph of the D₅ family contains the 5-orthoplex, as well as a 5-demicube which is an alternated 5-cube.

Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,b,a], have an extended symmetry, [[a,b,b,a]], like [3,3,3,3], doubling the symmetry order. Uniform polytopes in these group with symmetric rings contain this extended symmetry.

If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 5-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.

File:Coxeter diagram finite rank5 correspondence.png

Coxeter 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.

Fundamental families^[7]

Group symbol	Order	Coxeter graph		Bracket notation	Commutator subgroup	Coxeter number (h)	Reflections m=5/2 h^[8]
A₅	720	Template:CDD	Template:CDD	[3,3,3,3]	[3,3,3,3]⁺	6		15 Template:CDD
D₅	1920	Template:CDD	Template:CDD	[3,3,3^1,1]	[3,3,3^1,1]⁺	8		20 Template:CDD
B₅	3840	Template:CDD	Template:CDD	[4,3,3,3]	[3,3,3^1,1]⁺	10	5 Template:CDD	20 Template:CDD

Uniform prisms

There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. There is one infinite family of 5-polytopes based on prisms of the uniform duoprisms {p}×{q}×{ }.

Coxeter group	Order	Coxeter diagram		Coxeter notation	Commutator subgroup	Reflections
A₄A₁	120	Template:CDD	Template:CDD	[3,3,3,2] = [3,3,3]×[ ]	[3,3,3]⁺			10 Template:CDD		1 Template:CDD
D₄A₁	384	Template:CDD	Template:CDD	[3^1,1,1,2] = [3^1,1,1]×[ ]	[3^1,1,1]⁺			12 Template:CDD		1 Template:CDD
B₄A₁	768	Template:CDD	Template:CDD	[4,3,3,2] = [4,3,3]×[ ]	[3^1,1,1]⁺		4 Template:CDD	12 Template:CDD		1 Template:CDD
F₄A₁	2304	Template:CDD	Template:CDD	[3,4,3,2] = [3,4,3]×[ ]	[3⁺,4,3⁺]		12 Template:CDD	12 Template:CDD		1 Template:CDD
H₄A₁	28800	Template:CDD	Template:CDD	[5,3,3,2] = [3,4,3]×[ ]	[5,3,3]⁺			60 Template:CDD		1 Template:CDD
Duoprismatic prisms (use 2p and 2q for evens)
I₂(p)I₂(q)A₁	8pq	Template:CDD	Template:CDD	[p,2,q,2] = [p]×[q]×[ ]	[p⁺,2,q⁺]		p Template:CDD	q Template:CDD		1 Template:CDD
I₂(2p)I₂(q)A₁	16pq	Template:CDD	Template:CDD	[2p,2,q,2] = [2p]×[q]×[ ]		p Template:CDD	p Template:CDD	q Template:CDD		1 Template:CDD
I₂(2p)I₂(2q)A₁	32pq	Template:CDD	Template:CDD	[2p,2,2q,2] = [2p]×[2q]×[ ]		p Template:CDD	p Template:CDD	q Template:CDD	q Template:CDD	1 Template:CDD

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 group	Order	Coxeter diagram		Coxeter notation	Commutator subgroup	Reflections
Prismatic groups (use 2p for even)
A₃I₂(p)	48p	Template:CDD	Template:CDD	[3,3,2,p] = [3,3]×[p]	[(3,3)⁺,2,p⁺]		6 Template:CDD	p Template:CDD
A₃I₂(2p)	96p	Template:CDD	Template:CDD	[3,3,2,2p] = [3,3]×[2p]			6 Template:CDD	p Template:CDD	p Template:CDD
B₃I₂(p)	96p	Template:CDD	Template:CDD	[4,3,2,p] = [4,3]×[p]		3 Template:CDD	6Template:CDD	p Template:CDD
B₃I₂(2p)	192p	Template:CDD	Template:CDD	[4,3,2,2p] = [4,3]×[2p]		3 Template:CDD	6 Template:CDD	p Template:CDD	p Template:CDD
H₃I₂(p)	240p	Template:CDD	Template:CDD	[5,3,2,p] = [5,3]×[p]	[(5,3)⁺,2,p⁺]		15 Template:CDD	p Template:CDD
H₃I₂(2p)	480p	Template:CDD	Template:CDD	[5,3,2,2p] = [5,3]×[2p]	[(5,3)⁺,2,p⁺]		15 Template:CDD	p Template:CDD	p Template:CDD

Enumerating the convex uniform 5-polytopes

Simplex family: A₅ [3⁴]
- 19 uniform 5-polytopes
Hypercube/Orthoplex family: B₅ [4,3³]
- 31 uniform 5-polytopes
Demihypercube D₅/E₅ family: [3^2,1,1]
- 23 uniform 5-polytopes (8 unique)
Polychoral prisms:
- 56 uniform 5-polytope (45 unique) constructions based on prismatic families: [3,3,3]×[ ], [4,3,3]×[ ], [5,3,3]×[ ], [3^1,1,1]×[ ].
- 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+45+1=104

In addition there are:

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

The A₅ family

Template:See

There are 19 forms based on all permutations of the Coxeter 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). 7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440.

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).

#	Base point	Johnson naming system Bowers name and (acronym) Coxeter diagram	k-face element counts					Vertex figure	Facet counts by location: [3,3,3,3]
#	Base point		4	3	2	1	0	Vertex figure	Template:CDD [3,3,3] (6)	Template:CDD [3,3,2] (15)	Template:CDD [3,2,3] (20)	Template:CDD [2,3,3] (15)	Template:CDD [3,3,3] (6)	Alt
1	(0,0,0,0,0,1) or (0,1,1,1,1,1)	5-simplex hexateron (hix) Template:CDD	6	15	20	15	6	File:5-simplex verf.png {3,3,3}	File:Schlegel wireframe 5-cell.png {3,3,3}	-	-	-	-
2	(0,0,0,0,1,1) or (0,0,1,1,1,1)	Rectified 5-simplex rectified hexateron (rix) Template:CDD	12	45	80	60	15	File:Rectified 5-simplex verf.png t{3,3}×{ }	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}	-	-	-	File:Schlegel wireframe 5-cell.png {3,3,3}
3	(0,0,0,0,1,2) or (0,1,2,2,2,2)	Truncated 5-simplex truncated hexateron (tix) Template:CDD	12	45	80	75	30	File:Truncated 5-simplex verf.png Tetrah.pyr	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}	-	-	-	File:Schlegel wireframe 5-cell.png {3,3,3}
4	(0,0,0,1,1,2) or (0,1,1,2,2,2)	Cantellated 5-simplex small rhombated hexateron (sarx) Template:CDD	27	135	290	240	60	File:Cantellated hexateron verf.png prism-wedge	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}	-	-	File:Tetrahedral prism.png { }×{3,3}	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
5	(0,0,0,1,2,2) or (0,0,1,2,2,2)	Bitruncated 5-simplex bitruncated hexateron (bittix) Template:CDD	12	60	140	150	60	File:Bitruncated 5-simplex verf.png	File:Schlegel half-solid bitruncated 5-cell.png 2t{3,3,3}	-	-	-	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}
6	(0,0,0,1,2,3) or (0,1,2,3,3,3)	Cantitruncated 5-simplex great rhombated hexateron (garx) Template:CDD	27	135	290	300	120	File:Canitruncated 5-simplex verf.png	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}	-	-	File:Tetrahedral prism.png { }×{3,3}	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}
7	(0,0,1,1,1,2) or (0,1,1,1,2,2)	Runcinated 5-simplex small prismated hexateron (spix) Template:CDD	47	255	420	270	60	File:Runcinated 5-simplex verf.png	File:Schlegel half-solid runcinated 5-cell.png t_0,3{3,3,3}	-	File:3-3 duoprism.png {3}×{3}	File:Octahedral prism.png { }×r{3,3}	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
8	(0,0,1,1,2,3) or (0,1,2,2,3,3)	Runcitruncated 5-simplex prismatotruncated hexateron (pattix) Template:CDD	47	315	720	630	180	File:Runcitruncated 5-simplex verf.png	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}	-	File:3-6 duoprism.png {6}×{3}	File:Octahedral prism.png { }×r{3,3}	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
9	(0,0,1,2,2,3) or (0,1,1,2,3,3)	Runcicantellated 5-simplex prismatorhombated hexateron (pirx) Template:CDD	47	255	570	540	180	File:Runcicantellated 5-simplex verf.png	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}	-	File:3-3 duoprism.png {3}×{3}	File:Truncated tetrahedral prism.png { }×t{3,3}	File:Schlegel half-solid bitruncated 5-cell.png 2t{3,3,3}
10	(0,0,1,2,3,4) or (0,1,2,3,4,4)	Runcicantitruncated 5-simplex great prismated hexateron (gippix) Template:CDD	47	315	810	900	360	File:Runcicantitruncated 5-simplex verf.png Irr.5-cell	File:Schlegel half-solid omnitruncated 5-cell.png t_0,1,2,3{3,3,3}	-	File:3-6 duoprism.png {3}×{6}	File:Truncated tetrahedral prism.png { }×t{3,3}	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}
11	(0,1,1,1,2,3) or (0,1,2,2,2,3)	Steritruncated 5-simplex celliprismated hexateron (cappix) Template:CDD	62	330	570	420	120	File:Steritruncated 5-simplex verf.png	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}	File:Truncated tetrahedral prism.png { }×t{3,3}	File:3-6 duoprism.png {3}×{6}	File:Tetrahedral prism.png { }×{3,3}	File:Schlegel half-solid runcinated 5-cell.png t_0,3{3,3,3}
12	(0,1,1,2,3,4) or (0,1,2,3,3,4)	Stericantitruncated 5-simplex celligreatorhombated hexateron (cograx) Template:CDD	62	480	1140	1080	360	File:Stericanitruncated 5-simplex verf.png	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}	File:Truncated octahedral prism.png { }×tr{3,3}	File:3-6 duoprism.png {3}×{6}	File:Cuboctahedral prism.png { }×rr{3,3}	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
13	(0,0,0,1,1,1)	Birectified 5-simplex dodecateron (dot) Template:CDD	12	60	120	90	20	File:Birectified hexateron verf.png {3}×{3}	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}	-	-	-	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
14	(0,0,1,1,2,2)	Bicantellated 5-simplex small birhombated dodecateron (sibrid) Template:CDD	32	180	420	360	90	File:Bicantellated 5-simplex verf.png	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}	-	File:3-3 duoprism.png {3}×{3}	-	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
15	(0,0,1,2,3,3)	Bicantitruncated 5-simplex great birhombated dodecateron (gibrid) Template:CDD	32	180	420	450	180	File:Bicanitruncated 5-simplex verf.png	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}	-	File:3-3 duoprism.png {3}×{3}	-	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}
16	(0,1,1,1,1,2)	Stericated 5-simplex small cellated dodecateron (scad) Template:CDD	62	180	210	120	30	File:Stericated hexateron verf.png Irr.16-cell	File:Schlegel wireframe 5-cell.png {3,3,3}	File:Tetrahedral prism.png { }×{3,3}	File:3-3 duoprism.png {3}×{3}	File:Tetrahedral prism.png { }×{3,3}	File:Schlegel wireframe 5-cell.png {3,3,3}
17	(0,1,1,2,2,3)	Stericantellated 5-simplex small cellirhombated dodecateron (card) Template:CDD	62	420	900	720	180	File:Stericantellated 5-simplex verf.png	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}	File:Cuboctahedral prism.png { }×rr{3,3}	File:3-3 duoprism.png {3}×{3}	File:Cuboctahedral prism.png { }×rr{3,3}	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
18	(0,1,2,2,3,4)	Steriruncitruncated 5-simplex celliprismatotruncated dodecateron (captid) Template:CDD	62	450	1110	1080	360	File:Steriruncitruncated 5-simplex verf.png	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}	File:Tetrahedral prism.png { }×t{3,3}	File:6-6 duoprism.png {6}×{6}	File:Tetrahedral prism.png { }×t{3,3}	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
19	(0,1,2,3,4,5)	Omnitruncated 5-simplex great cellated dodecateron (gocad) Template:CDD	62	540	1560	1800	720	File:Omnitruncated 5-simplex verf.png Irr. {3,3,3}	File:Schlegel half-solid omnitruncated 5-cell.png t_0,1,2,3{3,3,3}	File:Truncated octahedral prism.png { }×tr{3,3}	File:6-6 duoprism.png {6}×{6}	File:Truncated octahedral prism.png { }×tr{3,3}	File:Schlegel half-solid omnitruncated 5-cell.png t_0,1,2,3{3,3,3}
Nonuniform		Omnisnub 5-simplex snub dodecateron (snod) snub hexateron (snix) Template:CDD	422	2340	4080	2520	360		ht_0,1,2,3{3,3,3}	ht_0,1,2,3{3,3,2}	ht_0,1,2,3{3,2,3}	ht_0,1,2,3{3,3,2}	ht_0,1,2,3{3,3,3}	(360) File:Schlegel wireframe 5-cell.png Irr. {3,3,3}

The B₅ family

Template:See 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 diagram. Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D₅ family as Template:CDD... = Template:CDD..... (There are more alternations that are not listed because they produce only repetitions, as Template:CDD... = Template:CDD.... and Template:CDD... = Template:CDD.... These would give a complete duplication of the uniform 5-polytopes numbered 20 through 34 with symmetry broken in half.)

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

The 5-cube family of 5-polytopes 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 5-polytope. All coordinates correspond with uniform 5-polytopes of edge length 2.

#	Base point	Name Coxeter diagram	Element counts					Vertex figure	Facet counts by location: [4,3,3,3]
#	Base point	Name Coxeter diagram	4	3	2	1	0	Vertex figure	Template:CDD [4,3,3] (10)	Template:CDD [4,3,2] (40)	Template:CDD [4,2,3] (80)	Template:CDD [2,3,3] (80)	Template:CDD [3,3,3] (32)	Alt
20	(0,0,0,0,1)√2	5-orthoplex triacontaditeron (tac) Template:CDD	32	80	80	40	10	File:Pentacross verf.png {3,3,4}	-	-	-	-	File:Schlegel wireframe 5-cell.png {3,3,3}
21	(0,0,0,1,1)√2	Rectified 5-orthoplex rectified triacontaditeron (rat) Template:CDD	42	240	400	240	40	File:Rectified pentacross verf.png { }×{3,4}	File:Schlegel wireframe 16-cell.png {3,3,4}	-	-	-	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
22	(0,0,0,1,2)√2	Truncated 5-orthoplex truncated triacontaditeron (tot) Template:CDD	42	240	400	280	80	File:Truncated pentacross.png (Octah.pyr)	File:Schlegel wireframe 16-cell.png {3,3,4}	-	-	-	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}
23	(0,0,1,1,1)√2	Birectified 5-cube penteractitriacontaditeron (nit) (Birectified 5-orthoplex) Template:CDD	42	280	640	480	80	File:Birectified penteract verf.png {4}×{3}	File:Schlegel half-solid rectified 16-cell.png r{3,3,4}	-	-	-	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
24	(0,0,1,1,2)√2	Cantellated 5-orthoplex small rhombated triacontaditeron (sart) Template:CDD	82	640	1520	1200	240	File:Cantellated pentacross verf.png Prism-wedge	File:Schlegel half-solid rectified 16-cell.png r{3,3,4}	File:Octahedral prism.png { }×{3,4}	-	-	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
25	(0,0,1,2,2)√2	Bitruncated 5-orthoplex bitruncated triacontaditeron (bittit) Template:CDD	42	280	720	720	240	File:Bitruncated pentacross verf.png	File:Schlegel half-solid truncated 16-cell.png t{3,3,4}	-	-	-	File:Schlegel half-solid bitruncated 5-cell.png 2t{3,3,3}
26	(0,0,1,2,3)√2	Cantitruncated 5-orthoplex great rhombated triacontaditeron (gart) Template:CDD	82	640	1520	1440	480	File:Canitruncated 5-orthoplex verf.png	File:Schlegel half-solid truncated 16-cell.png t{3,3,4}	File:Octahedral prism.png { }×{3,4}	-	-	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
27	(0,1,1,1,1)√2	Rectified 5-cube rectified penteract (rin) Template:CDD	42	200	400	320	80	File:Rectified 5-cube verf.png {3,3}×{ }	File:Schlegel half-solid rectified 8-cell.png r{4,3,3}	-	-	-	File:Schlegel wireframe 5-cell.png {3,3,3}
28	(0,1,1,1,2)√2	Runcinated 5-orthoplex small prismated triacontaditeron (spat) Template:CDD	162	1200	2160	1440	320	File:Runcinated pentacross verf.png	File:Schlegel half-solid rectified 8-cell.png r{4,3,3}	File:Cuboctahedral prism.png { }×r{3,4}	File:3-4 duoprism.png {3}×{4}		File:Schlegel half-solid runcinated 5-cell.png t_0,3{3,3,3}
29	(0,1,1,2,2)√2	Bicantellated 5-cube small birhombated penteractitriacontaditeron (sibrant) (Bicantellated 5-orthoplex) Template:CDD	122	840	2160	1920	480	File:Bicantellated penteract verf.png	File:Schlegel half-solid cantellated 16-cell.png rr{3,3,4}	-	File:3-4 duoprism.png {4}×{3}	-	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
30	(0,1,1,2,3)√2	Runcitruncated 5-orthoplex prismatotruncated triacontaditeron (pattit) Template:CDD	162	1440	3680	3360	960	File:Runcitruncated 5-orthoplex verf.png	File:Schlegel half-solid cantellated 16-cell.png rr{3,3,4}	File:Cuboctahedral prism.png { }×r{3,4}	File:6-4 duoprism.png {6}×{4}	-	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
31	(0,1,2,2,2)√2	Bitruncated 5-cube bitruncated penteract (bittin) Template:CDD	42	280	720	800	320	File:Bitruncated penteract verf.png	File:Schlegel half-solid bitruncated 8-cell.png 2t{4,3,3}	-	-	-	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}
32	(0,1,2,2,3)√2	Runcicantellated 5-orthoplex prismatorhombated triacontaditeron (pirt) Template:CDD	162	1200	2960	2880	960	File:Runcicantellated 5-orthoplex verf.png	File:Schlegel half-solid bitruncated 8-cell.png 2t{4,3,3}	File:Truncated octahedral prism.png { }×t{3,4}	File:3-4 duoprism.png {3}×{4}	-	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
33	(0,1,2,3,3)√2	Bicantitruncated 5-cube great birhombated triacontaditeron (gibrant) (Bicantitruncated 5-orthoplex) Template:CDD	122	840	2160	2400	960	File:Bicantellated penteract verf.png	File:Schlegel half-solid cantitruncated 16-cell.png tr{3,3,4}	-	File:3-4 duoprism.png {4}×{3}	-	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
34	(0,1,2,3,4)√2	Runcicantitruncated 5-orthoplex great prismated triacontaditeron (gippit) Template:CDD	162	1440	4160	4800	1920	File:Runcicantitruncated 5-orthoplex verf.png	File:Schlegel half-solid cantitruncated 16-cell.png tr{3,3,4}	File:Truncated octahedral prism.png { }×t{3,4}	File:6-4 duoprism.png {6}×{4}	-	File:Schlegel half-solid omnitruncated 5-cell.png t_0,1,2,3{3,3,3}
35	(1,1,1,1,1)	5-cube penteract (pent) Template:CDD	10	40	80	80	32	File:5-cube verf.svg {3,3,3}	File:Schlegel wireframe 8-cell.png {4,3,3}	-	-	-	-
36	(1,1,1,1,1) + (0,0,0,0,1)√2	Stericated 5-cube small cellated penteractitriacontaditeron (scant) (Stericated 5-orthoplex) Template:CDD	242	800	1040	640	160	File:Stericated penteract verf.png Tetr.antiprm	File:Schlegel wireframe 8-cell.png {4,3,3}	File:Schlegel wireframe 8-cell.png {4,3}×{ }	File:3-4 duoprism.png {4}×{3}	File:Tetrahedral prism.png { }×{3,3}	File:Schlegel wireframe 5-cell.png {3,3,3}
37	(1,1,1,1,1) + (0,0,0,1,1)√2	Runcinated 5-cube small prismated penteract (span) Template:CDD	202	1240	2160	1440	320	File:Runcinated penteract verf.png	File:Schlegel half-solid runcinated 8-cell.png t_0,3{4,3,3}	-	File:3-4 duoprism.png {4}×{3}	File:Octahedral prism.png { }×r{3,3}	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
38	(1,1,1,1,1) + (0,0,0,1,2)√2	Steritruncated 5-orthoplex celliprismated triacontaditeron (cappin) Template:CDD	242	1520	2880	2240	640	File:Steritruncated 5-orthoplex verf.png	File:Schlegel half-solid runcinated 8-cell.png t_0,3{4,3,3}	File:Schlegel wireframe 8-cell.png {4,3}×{ }	File:6-4 duoprism.png {6}×{4}	File:Truncated tetrahedral prism.png { }×t{3,3}	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}
39	(1,1,1,1,1) + (0,0,1,1,1)√2	Cantellated 5-cube small rhombated penteract (sirn) Template:CDD	122	680	1520	1280	320	File:Cantellated 5-cube vertf.png Prism-wedge	File:Schlegel half-solid cantellated 8-cell.png rr{4,3,3}	-	-	File:Tetrahedral prism.png { }×{3,3}	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
40	(1,1,1,1,1) + (0,0,1,1,2)√2	Stericantellated 5-cube cellirhombated penteractitriacontaditeron (carnit) (Stericantellated 5-orthoplex) Template:CDD	242	2080	4720	3840	960	File:Stericantellated 5-orthoplex verf.png	File:Schlegel half-solid cantellated 8-cell.png rr{4,3,3}	File:Rhombicuboctahedral prism.png rr{4,3}×{ }	File:3-4 duoprism.png {4}×{3}	File:Cuboctahedral prism.png { }×rr{3,3}	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
41	(1,1,1,1,1) + (0,0,1,2,2)√2	Runcicantellated 5-cube prismatorhombated penteract (prin) Template:CDD	202	1240	2960	2880	960	File:Runcicantellated 5-cube verf.png	File:Runcitruncated 16-cell.png t_0,2,3{4,3,3}	-	File:3-4 duoprism.png {4}×{3}	File:Truncated tetrahedral prism.png { }×t{3,3}	File:Schlegel half-solid bitruncated 5-cell.png 2t{3,3,3}
42	(1,1,1,1,1) + (0,0,1,2,3)√2	Stericantitruncated 5-orthoplex celligreatorhombated triacontaditeron (cogart) Template:CDD	242	2320	5920	5760	1920	File:Stericanitruncated 5-orthoplex verf.png	File:Runcitruncated 16-cell.png t_0,2,3{4,3,3}	File:Rhombicuboctahedral prism.png rr{4,3}×{ }	File:6-4 duoprism.png {6}×{4}	File:Truncated octahedral prism.png { }×tr{3,3}	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}
43	(1,1,1,1,1) + (0,1,1,1,1)√2	Truncated 5-cube truncated penteract (tan) Template:CDD	42	200	400	400	160	File:Truncated 5-cube verf.png Tetrah.pyr	File:Schlegel half-solid truncated tesseract.png t{4,3,3}	-	-	-	File:Schlegel wireframe 5-cell.png {3,3,3}
44	(1,1,1,1,1) + (0,1,1,1,2)√2	Steritruncated 5-cube celliprismated triacontaditeron (capt) Template:CDD	242	1600	2960	2240	640	File:Steritruncated 5-cube verf.png	File:Schlegel half-solid truncated tesseract.png t{4,3,3}	File:Truncated cubic prism.png t{4,3}×{ }	File:8-3 duoprism.png {8}×{3}	File:Tetrahedral prism.png { }×{3,3}	File:Schlegel half-solid runcinated 5-cell.png t_0,3{3,3,3}
45	(1,1,1,1,1) + (0,1,1,2,2)√2	Runcitruncated 5-cube prismatotruncated penteract (pattin) Template:CDD	202	1560	3760	3360	960	File:Runcitruncated 5-cube verf.png	File:Schlegel half-solid runcitruncated 8-cell.png t_0,1,3{4,3,3}	-	File:8-3 duoprism.png {8}×{3}	File:Octahedral prism.png { }×r{3,3}	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
46	(1,1,1,1,1) + (0,1,1,2,3)√2	Steriruncitruncated 5-cube celliprismatotruncated penteractitriacontaditeron (captint) (Steriruncitruncated 5-orthoplex) Template:CDD	242	2160	5760	5760	1920	File:Steriruncitruncated 5-cube verf.png	File:Schlegel half-solid runcitruncated 8-cell.png t_0,1,3{4,3,3}	File:Truncated cubic prism.png t{4,3}×{ }	File:8-6 duoprism.png {8}×{6}	File:Truncated tetrahedral prism.png { }×t{3,3}	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
47	(1,1,1,1,1) + (0,1,2,2,2)√2	Cantitruncated 5-cube great rhombated penteract (girn) Template:CDD	122	680	1520	1600	640	File:Canitruncated 5-cube verf.png	File:Schlegel half-solid cantitruncated 8-cell.png tr{4,3,3}	-	-	File:Tetrahedral prism.png { }×{3,3}	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}
48	(1,1,1,1,1) + (0,1,2,2,3)√2	Stericantitruncated 5-cube celligreatorhombated penteract (cogrin) Template:CDD	242	2400	6000	5760	1920	File:Stericanitruncated 5-cube verf.png	File:Schlegel half-solid cantitruncated 8-cell.png tr{4,3,3}	File:Truncated cuboctahedral prism.png tr{4,3}×{ }	File:8-3 duoprism.png {8}×{3}	File:Cuboctahedral prism.png { }×rr{3,3}	File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
49	(1,1,1,1,1) + (0,1,2,3,3)√2	Runcicantitruncated 5-cube great prismated penteract (gippin) Template:CDD	202	1560	4240	4800	1920	File:Runcicantitruncated 5-cube verf.png	File:Schlegel half-solid omnitruncated 8-cell.png t_0,1,2,3{4,3,3}	-	File:8-3 duoprism.png {8}×{3}	File:Truncated tetrahedral prism.png { }×t{3,3}	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}
50	(1,1,1,1,1) + (0,1,2,3,4)√2	Omnitruncated 5-cube great cellated penteractitriacontaditeron (gacnet) (omnitruncated 5-orthoplex) Template:CDD	242	2640	8160	9600	3840	File:Omnitruncated 5-cube verf.png Irr. {3,3,3}	File:Schlegel half-solid omnitruncated 8-cell.png tr{4,3}×{ }	File:Truncated cuboctahedral prism.png tr{4,3}×{ }	File:8-6 duoprism.png {8}×{6}	File:Truncated octahedral prism.png { }×tr{3,3}	File:Schlegel half-solid omnitruncated 5-cell.png t_0,1,2,3{3,3,3}
51		5-demicube hemipenteract (hin) Template:CDD = Template:CDD	26	120	160	80	16	File:Demipenteract verf.png r{3,3,3}	File:Schlegel wireframe 16-cell.png h{4,3,3}	-	-	-	-	(16) File:Schlegel wireframe 5-cell.png {3,3,3}
52		Cantic 5-cube Truncated hemipenteract (thin) Template:CDD = Template:CDD	42	280	640	560	160	File:Truncated 5-demicube verf.png	File:Schlegel half-solid truncated 16-cell.png h₂{4,3,3}	-	-	-	(16) File:Schlegel half-solid rectified 5-cell.png r{3,3,3}	(16) File:Schlegel half-solid truncated pentachoron.png t{3,3,3}
53		Runcic 5-cube Small rhombated hemipenteract (sirhin) Template:CDD = Template:CDD	42	360	880	720	160		File:Schlegel half-solid rectified 8-cell.png h₃{4,3,3}	-	-	-	(16) File:Schlegel half-solid rectified 5-cell.png r{3,3,3}	(16) File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
54		Steric 5-cube Small prismated hemipenteract (siphin) Template:CDD = Template:CDD	82	480	720	400	80		File:Schlegel wireframe 16-cell.png h{4,3,3}	File:Tetrahedral prism.png h{4,3}×{}	-	-	(16) File:Schlegel wireframe 5-cell.png {3,3,3}	(16) File:Schlegel half-solid runcinated 5-cell.png t_0,3{3,3,3}
55		Runcicantic 5-cube Great rhombated hemipenteract (girhin) Template:CDD = Template:CDD	42	360	1040	1200	480		File:Schlegel half-solid bitruncated 8-cell.png h_2,3{4,3,3}	-	-	-	(16) File:Schlegel half-solid bitruncated 5-cell.png 2t{3,3,3}	(16) File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}
56		Stericantic 5-cube Prismatotruncated hemipenteract (pithin) Template:CDD = Template:CDD	82	720	1840	1680	480		File:Schlegel half-solid truncated 16-cell.png h₂{4,3,3}	File:Truncated tetrahedral prism.png h₂{4,3}×{}	-	-	(16) File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}	(16) File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
57		Steriruncic 5-cube Prismatorhombated hemipenteract (pirhin) Template:CDD = Template:CDD	82	560	1280	1120	320		File:Schlegel half-solid rectified 8-cell.png h₃{4,3,3}	File:Tetrahedral prism.png h{4,3}×{}	-	-	(16) File:Schlegel half-solid truncated pentachoron.png t{3,3,3}	(16) File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}
58		Steriruncicantic 5-cube Great prismated hemipenteract (giphin) Template:CDD = Template:CDD	82	720	2080	2400	960		File:Schlegel half-solid bitruncated 8-cell.png h_2,3{4,3,3}	File:Truncated tetrahedral prism.png h₂{4,3}×{}	-	-	(16) File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}	(16) File:Schlegel half-solid omnitruncated 5-cell.png t_0,1,2,3{3,3,3}
Nonuniform		Alternated runcicantitruncated 5-orthoplex Snub prismatotriacontaditeron (snippit) Snub hemipenteract (snahin) Template:CDD = Template:CDD	1122	6240	10880	6720	960		File:Schlegel half-solid alternated cantitruncated 16-cell.png sr{3,3,4}	sr{2,3,4}	sr{3,2,4}	-	ht_0,1,2,3{3,3,3}	(960) File:Schlegel wireframe 5-cell.png Irr. {3,3,3}
Nonuniform		Edge-snub 5-orthoplex Pyritosnub penteract (pysnan) Template:CDD	1202	7920	15360	10560	1920		sr₃{3,3,4}	sr₃{2,3,4}	sr₃{3,2,4}	File:Icosahedral prism.png s{3,3}×{ }	ht_0,1,2,3{3,3,3}	(960) File:Tetrahedral prism.png Irr. {3,3}×{ }
Nonuniform		Snub 5-cube Snub penteract (snan) Template:CDD	2162	12240	21600	13440	960		ht_0,1,2,3{3,3,4}	ht_0,1,2,3{2,3,4}	ht_0,1,2,3{3,2,4}	ht_0,1,2,3{3,3,2}	ht_0,1,2,3{3,3,3}	(1920) File:Schlegel wireframe 5-cell.png Irr. {3,3,3}

The D₅ family

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

This family has 23 Wythoffian uniform polytopes, from 3×8-1 permutations of the D₅ Coxeter diagram with one or more rings. 15 (2×8-1) are repeated from the B₅ family and 8 are unique to this family, though even those 8 duplicate the alternations from the B₅ family.

In the 15 repeats, both of the nodes terminating the length-1 branches are ringed, so the two kinds of Template:CDD element are identical and the symmetry doubles: the relations are Template:CDD... = Template:CDD.... and Template:CDD... = Template:CDD..., creating a complete duplication of the uniform 5-polytopes 20 through 34 above. The 8 new forms have one such node ringed and one not, with the relation Template:CDD... = Template:CDD... duplicating uniform 5-polytopes 51 through 58 above.

#	Coxeter diagram Schläfli symbol symbols Johnson and Bowers names	Element counts					Vertex figure	Facets by location: File:CD B5 nodes.png [3^1,2,1]
#		4	3	2	1	0	Vertex figure	Template:CDD [3,3,3] (16)	Template:CDD [3^1,1,1] (10)	Template:CDD [3,3]×[ ] (40)	Template:CDD [ ]×[3]×[ ] (80)	Template:CDD [3,3,3] (16)	Alt
[51]	Template:CDD = Template:CDD h{4,3,3,3}, 5-demicube Hemipenteract (hin)	26	120	160	80	16	File:Demipenteract verf.png r{3,3,3}	File:Schlegel wireframe 5-cell.png {3,3,3}	File:Schlegel wireframe 16-cell.png h{4,3,3}	-	-	-
[52]	Template:CDD = Template:CDD h₂{4,3,3,3}, cantic 5-cube Truncated hemipenteract (thin)	42	280	640	560	160	File:Truncated 5-demicube verf.png	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}	File:Schlegel half-solid truncated 16-cell.png h₂{4,3,3}	-	-	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
[53]	Template:CDD = Template:CDD h₃{4,3,3,3}, runcic 5-cube Small rhombated hemipenteract (sirhin)	42	360	880	720	160		File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}	File:Schlegel half-solid rectified 8-cell.png h₃{4,3,3}	-	-	File:Schlegel half-solid rectified 5-cell.png r{3,3,3}
[54]	Template:CDD = Template:CDD h₄{4,3,3,3}, steric 5-cube Small prismated hemipenteract (siphin)	82	480	720	400	80		File:Schlegel half-solid runcinated 5-cell.png t_0,3{3,3,3}	File:Schlegel wireframe 16-cell.png h{4,3,3}	File:Tetrahedral prism.png h{4,3}×{}	-	File:Schlegel wireframe 5-cell.png {3,3,3}
[55]	Template:CDD = Template:CDD h_2,3{4,3,3,3}, runcicantic 5-cube Great rhombated hemipenteract (girhin)	42	360	1040	1200	480		File:Schlegel half-solid bitruncated 5-cell.png 2t{3,3,3}	File:Schlegel half-solid bitruncated 8-cell.png h_2,3{4,3,3}	-	-	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}
[56]	Template:CDD = Template:CDD h_2,4{4,3,3,3}, stericantic 5-cube Prismatotruncated hemipenteract (pithin)	82	720	1840	1680	480		File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}	File:Schlegel half-solid truncated 16-cell.png h₂{4,3,3}	File:Truncated tetrahedral prism.png h₂{4,3}×{}	-	File:Schlegel half-solid cantellated 5-cell.png rr{3,3,3}
[57]	Template:CDD = Template:CDD h_3,4{4,3,3,3}, steriruncic 5-cube Prismatorhombated hemipenteract (pirhin)	82	560	1280	1120	320		File:Schlegel half-solid runcitruncated 5-cell.png t_0,1,3{3,3,3}	File:Schlegel half-solid rectified 8-cell.png h₃{4,3,3}	File:Tetrahedral prism.png h{4,3}×{}	-	File:Schlegel half-solid truncated pentachoron.png t{3,3,3}
[58]	Template:CDD = Template:CDD h_2,3,4{4,3,3,3}, steriruncicantic 5-cube Great prismated hemipenteract (giphin)	82	720	2080	2400	960		File:Schlegel half-solid omnitruncated 5-cell.png t_0,1,2,3{3,3,3}	File:Schlegel half-solid bitruncated 8-cell.png h_2,3{4,3,3}	File:Truncated tetrahedral prism.png h₂{4,3}×{}	-	File:Schlegel half-solid cantitruncated 5-cell.png tr{3,3,3}
Nonuniform	Template:CDD = Template:CDD ht_0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplex Snub hemipenteract (snahin)	1122	6240	10880	6720	960		ht_0,1,2,3{3,3,3}	File:Schlegel half-solid alternated cantitruncated 16-cell.png sr{3,3,4}	sr{2,3,4}	sr{3,2,4}	ht_0,1,2,3{3,3,3}	(960) File:Schlegel wireframe 5-cell.png Irr. {3,3,3}

Uniform prismatic forms

There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. For simplicity, most alternations are not shown.

A₄ × A₁

This prismatic family has 9 forms:

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

#	Coxeter diagram and Schläfli symbols Name	Element counts
#	Coxeter diagram and Schläfli symbols Name	Facets	Cells	Faces	Edges	Vertices
59	Template:CDD = {3,3,3}×{ } 5-cell prism (penp)	7	20	30	25	10
60	Template:CDD = r{3,3,3}×{ } Rectified 5-cell prism (rappip)	12	50	90	70	20
61	Template:CDD = t{3,3,3}×{ } Truncated 5-cell prism (tippip)	12	50	100	100	40
62	Template:CDD = rr{3,3,3}×{ } Cantellated 5-cell prism (srippip)	22	120	250	210	60
63	Template:CDD = t_0,3{3,3,3}×{ } Runcinated 5-cell prism (spiddip)	32	130	200	140	40
64	Template:CDD = 2t{3,3,3}×{ } Bitruncated 5-cell prism (decap)	12	60	140	150	60
65	Template:CDD = tr{3,3,3}×{ } Cantitruncated 5-cell prism (grippip)	22	120	280	300	120
66	Template:CDD = t_0,1,3{3,3,3}×{ } Runcitruncated 5-cell prism (prippip)	32	180	390	360	120
67	Template:CDD = t_0,1,2,3{3,3,3}×{ } Omnitruncated 5-cell prism (gippiddip)	32	210	540	600	240

B₄ × A₁

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

The A₁×B₄ family has symmetry of order 768 (2⁵4!).

The last three snubs can be realised with equal-length edges, but turn out nonuniform anyway because some of their 4-faces are not uniform 4-polytopes.

#	Coxeter diagram and Schläfli symbols Name	Element counts
#	Coxeter diagram and Schläfli symbols Name	Facets	Cells	Faces	Edges	Vertices
[16]	Template:CDD = {4,3,3}×{ } Tesseractic prism (pent) (Same as 5-cube)	10	40	80	80	32
68	Template:CDD = r{4,3,3}×{ } Rectified tesseractic prism (rittip)	26	136	272	224	64
69	Template:CDD = t{4,3,3}×{ } Truncated tesseractic prism (tattip)	26	136	304	320	128
70	Template:CDD = rr{4,3,3}×{ } Cantellated tesseractic prism (srittip)	58	360	784	672	192
71	Template:CDD = t_0,3{4,3,3}×{ } Runcinated tesseractic prism (sidpithip)	82	368	608	448	128
72	Template:CDD = 2t{4,3,3}×{ } Bitruncated tesseractic prism (tahp)	26	168	432	480	192
73	Template:CDD = tr{4,3,3}×{ } Cantitruncated tesseractic prism (grittip)	58	360	880	960	384
74	Template:CDD = t_0,1,3{4,3,3}×{ } Runcitruncated tesseractic prism (prohp)	82	528	1216	1152	384
75	Template:CDD = t_0,1,2,3{4,3,3}×{ } Omnitruncated tesseractic prism (gidpithip)	82	624	1696	1920	768
76	Template:CDD = {3,3,4}×{ } 16-cell prism (hexip)	18	64	88	56	16
77	Template:CDD = r{3,3,4}×{ } Rectified 16-cell prism (icope) (Same as 24-cell prism)	26	144	288	216	48
78	Template:CDD = t{3,3,4}×{ } Truncated 16-cell prism (thexip)	26	144	312	288	96
79	Template:CDD = rr{3,3,4}×{ } Cantellated 16-cell prism (ricope) (Same as rectified 24-cell prism)	50	336	768	672	192
80	Template:CDD = tr{3,3,4}×{ } Cantitruncated 16-cell prism (ticope) (Same as truncated 24-cell prism)	50	336	864	960	384
81	Template:CDD = t_0,1,3{3,3,4}×{ } Runcitruncated 16-cell prism (prittip)	82	528	1216	1152	384
82	Template:CDD = sr{3,3,4}×{ } snub 24-cell prism (sadip)	146	768	1392	960	192
Nonuniform	Template:CDD rectified tesseractic alterprism (rita)	50	288	464	288	64
Nonuniform	Template:CDD truncated 16-cell alterprism (thexa)	26	168	384	336	96
Nonuniform	Template:CDD bitruncated tesseractic alterprism (taha)	50	288	624	576	192

F₄ × A₁

This prismatic family has 10 forms.

The A₁ x F₄ family has symmetry of order 2304 (2*1152). Three polytopes 85, 86 and 89 (green background) have double symmetry [[3,4,3],2], order 4608. The last one, snub 24-cell prism, (blue background) has [3⁺,4,3,2] symmetry, order 1152.

#	Coxeter diagram and Schläfli symbols Name	Element counts
#	Coxeter diagram and Schläfli symbols Name	Facets	Cells	Faces	Edges	Vertices
[77]	Template:CDD = {3,4,3}×{ } 24-cell prism (icope)	26	144	288	216	48
[79]	Template:CDD = r{3,4,3}×{ } rectified 24-cell prism (ricope)	50	336	768	672	192
[80]	Template:CDD = t{3,4,3}×{ } truncated 24-cell prism (ticope)	50	336	864	960	384
83	Template:CDD = rr{3,4,3}×{ } cantellated 24-cell prism (sricope)	146	1008	2304	2016	576
84	Template:CDD = t_0,3{3,4,3}×{ } runcinated 24-cell prism (spiccup)	242	1152	1920	1296	288
85	Template:CDD = 2t{3,4,3}×{ } bitruncated 24-cell prism (contip)	50	432	1248	1440	576
86	Template:CDD = tr{3,4,3}×{ } cantitruncated 24-cell prism (gricope)	146	1008	2592	2880	1152
87	Template:CDD = t_0,1,3{3,4,3}×{ } runcitruncated 24-cell prism (pricope)	242	1584	3648	3456	1152
88	Template:CDD = t_0,1,2,3{3,4,3}×{ } omnitruncated 24-cell prism (gippiccup)	242	1872	5088	5760	2304
[82]	Template:CDD = s{3,4,3}×{ } snub 24-cell prism (sadip)	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 diagram and Schläfli symbols Name	Element counts
#	Coxeter diagram and Schläfli symbols Name	Facets	Cells	Faces	Edges	Vertices
89	Template:CDD = {5,3,3}×{ } 120-cell prism (hipe)	122	960	2640	3000	1200
90	Template:CDD = r{5,3,3}×{ } Rectified 120-cell prism (rahipe)	722	4560	9840	8400	2400
91	Template:CDD = t{5,3,3}×{ } Truncated 120-cell prism (thipe)	722	4560	11040	12000	4800
92	Template:CDD = rr{5,3,3}×{ } Cantellated 120-cell prism (srahip)	1922	12960	29040	25200	7200
93	Template:CDD = t_0,3{5,3,3}×{ } Runcinated 120-cell prism (sidpixhip)	2642	12720	22080	16800	4800
94	Template:CDD = 2t{5,3,3}×{ } Bitruncated 120-cell prism (xhip)	722	5760	15840	18000	7200
95	Template:CDD = tr{5,3,3}×{ } Cantitruncated 120-cell prism (grahip)	1922	12960	32640	36000	14400
96	Template:CDD = t_0,1,3{5,3,3}×{ } Runcitruncated 120-cell prism (prixip)	2642	18720	44880	43200	14400
97	Template:CDD = t_0,1,2,3{5,3,3}×{ } Omnitruncated 120-cell prism (gidpixhip)	2642	22320	62880	72000	28800
98	Template:CDD = {3,3,5}×{ } 600-cell prism (exip)	602	2400	3120	1560	240
99	Template:CDD = r{3,3,5}×{ } Rectified 600-cell prism (roxip)	722	5040	10800	7920	1440
100	Template:CDD = t{3,3,5}×{ } Truncated 600-cell prism (texip)	722	5040	11520	10080	2880
101	Template:CDD = rr{3,3,5}×{ } Cantellated 600-cell prism (srixip)	1442	11520	28080	25200	7200
102	Template:CDD = tr{3,3,5}×{ } Cantitruncated 600-cell prism (grixip)	1442	11520	31680	36000	14400
103	Template:CDD = t_0,1,3{3,3,5}×{ } Runcitruncated 600-cell prism (prahip)	2642	18720	44880	43200	14400

Duoprism prisms

Uniform duoprism prisms, {p}×{q}×{ }, form an infinite class for all integers p,q>2. {4}×{4}×{ } makes a lower symmetry form of the 5-cube.

The extended f-vector of {p}×{q}×{ } is computed as (p,p,1)*(q,q,1)*(2,1) = (2pq,5pq,4pq+2p+2q,3pq+3p+3q,p+q+2,1).

Coxeter diagram	Names	Element counts
Coxeter diagram	Names	4-faces	Cells	Faces	Edges	Vertices
Template:CDD	{p}×{q}×{ }^[9]	p+q+2	3pq+3p+3q	4pq+2p+2q	5pq	2pq
Template:CDD	{p}²×{ }	2(p+1)	3p(p+1)	4p(p+1)	5p²	2p²
Template:CDD	{3}²×{ }	8	36	48	45	18
Template:CDD	{4}²×{ } = 5-cube	10	40	80	80	32

Grand antiprism prism

The grand antiprism prism is the only known convex non-Wythoffian uniform 5-polytope. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (600 tetrahedra, 40 pentagonal antiprisms, 700 triangular prisms, 20 pentagonal prisms), and 322 hypercells (2 grand antiprisms File:Grand antiprism.png, 20 pentagonal antiprism prisms File:Pentagonal antiprismatic prism.png, and 300 tetrahedral prisms File:Tetrahedral prism.png).

#	Name	Element counts
#	Name	Facets	Cells	Faces	Edges	Vertices
104	grand antiprism prism (gappip)^[10]	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 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 are the primary operators available for constructing and naming the uniform 5-polytopes.

The last operation, the snub, and more generally the alternation, are the operations 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		Coxeter diagram	Description
Parent	t₀{p,q,r,s}	{p,q,r,s}	Template:CDD	Any regular 5-polytope
Rectified	t₁{p,q,r,s}	r{p,q,r,s}	Template:CDD	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}	2r{p,q,r,s}	Template:CDD	Birectification reduces faces to points, cells to their duals.
Trirectified	t₃{p,q,r,s}	3r{p,q,r,s}	Template:CDD	Trirectification reduces cells to points. (Dual rectification)
Quadrirectified	t₄{p,q,r,s}	4r{p,q,r,s}	Template:CDD	Quadrirectification reduces 4-faces to points. (Dual)
Truncated	t_0,1{p,q,r,s}	t{p,q,r,s}	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 5-polytope. The 5-polytope has its original faces doubled in sides, and contains the faces of the dual. File:Cube truncation sequence.svg
Cantellated	t_0,2{p,q,r,s}	rr{p,q,r,s}	Template:CDD	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. File:Cube cantellation sequence.svg
Runcinated	t_0,3{p,q,r,s}		Template:CDD	Runcination reduces cells and creates new cells at the vertices and edges.
Stericated	t_0,4{p,q,r,s}	2r2r{p,q,r,s}	Template:CDD	Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for 5-polytopes.)
Omnitruncated	t_0,1,2,3,4{p,q,r,s}		Template:CDD	All four operators, truncation, cantellation, runcination, and sterication are applied.
Half	h{2p,3,q,r}		Template:CDD	Alternation, same as Template:CDD
Cantic	h₂{2p,3,q,r}		Template:CDD	Same as Template:CDD
Runcic	h₃{2p,3,q,r}		Template:CDD	Same as Template:CDD
Runcicantic	h_2,3{2p,3,q,r}		Template:CDD	Same as Template:CDD
Steric	h₄{2p,3,q,r}		Template:CDD	Same as Template:CDD
Steriruncic	h_3,4{2p,3,q,r}		Template:CDD	Same as Template:CDD
Stericantic	h_2,4{2p,3,q,r}		Template:CDD	Same as Template:CDD
Steriruncicantic	h_2,3,4{2p,3,q,r}		Template:CDD	Same as Template:CDD
Snub	s{p,2q,r,s}		Template:CDD	Alternated truncation
Snub rectified	sr{p,q,2r,s}		Template:CDD	Alternated truncated rectification
	ht_0,1,2,3{p,q,r,s}		Template:CDD	Alternated runcicantitruncation
Full snub	ht_0,1,2,3,4{p,q,r,s}		Template:CDD	Alternated omnitruncation

Regular and uniform honeycombs

File:Coxeter diagram affine rank5 correspondence.png

Coxeter 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 five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.^[11]^[12]

Fundamental groups
#	Coxeter group			Coxeter diagram	Forms
1	${\tilde{A}}_{4}$	[3^[5]]	[(3,3,3,3,3)]	Template:CDD	7
2	${\tilde{C}}_{4}$	[4,3,3,4]		Template:CDD	19
3	${\tilde{B}}_{4}$	[4,3,3^1,1]	[4,3,3,4,1⁺]	Template:CDD = Template:CDD	23 (8 new)
4	${\tilde{D}}_{4}$	[3^1,1,1,1]	[1⁺,4,3,3,4,1⁺]	Template:CDD = Template:CDD	9 (0 new)
5	${\tilde{F}}_{4}$	[3,4,3,3]		Template:CDD	31 (21 new)

There are three regular honeycombs of Euclidean 4-space:

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

Other families that generate uniform honeycombs:

There are 23 uniquely ringed forms, 8 new ones in the 16-cell honeycomb family. With symbols h{4,3²,4} it is geometrically identical to the 16-cell honeycomb, Template:CDD = Template:CDD
There are 7 uniquely ringed forms from the A~4, Template:CDD family, all new, including:
There are 9 uniquely ringed forms in the ${\tilde{D}}_{4}$ : [3^1,1,1,1] Template:CDD family, two new ones, including the quarter tesseractic honeycomb, Template:CDD = Template:CDD, and the bitruncated tesseractic honeycomb, Template:CDD = Template:CDD.

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

Prismatic groups
#	Coxeter group		Coxeter diagram
1	${\tilde{C}}_{3}$ × ${\tilde{I}}_{1}$	[4,3,4,2,∞]	Template:CDD
2	${\tilde{B}}_{3}$ × ${\tilde{I}}_{1}$	[4,3^1,1,2,∞]	Template:CDD
3	${\tilde{A}}_{3}$ × ${\tilde{I}}_{1}$	[3^[4],2,∞]	Template:CDD
4	${\tilde{C}}_{2}$ × ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,4,2,∞,2,∞]	Template:CDD
5	${\tilde{H}}_{2}$ × ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[6,3,2,∞,2,∞]	Template:CDD
6	${\tilde{A}}_{2}$ × ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[3^[3],2,∞,2,∞]	Template:CDD
7	${\tilde{I}}_{1}$ × ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[∞,2,∞,2,∞,2,∞]	Template:CDD
8	${\tilde{A}}_{2}$ x ${\tilde{A}}_{2}$	[3^[3],2,3^[3]]	Template:CDD
9	${\tilde{A}}_{2}$ × ${\tilde{B}}_{2}$	[3^[3],2,4,4]	Template:CDD
10	${\tilde{A}}_{2}$ × ${\tilde{G}}_{2}$	[3^[3],2,6,3]	Template:CDD
11	${\tilde{B}}_{2}$ × ${\tilde{B}}_{2}$	[4,4,2,4,4]	Template:CDD
12	${\tilde{B}}_{2}$ × ${\tilde{G}}_{2}$	[4,4,2,6,3]	Template:CDD
13	${\tilde{G}}_{2}$ × ${\tilde{G}}_{2}$	[6,3,2,6,3]	Template:CDD

Regular and uniform hyperbolic honeycombs

Hyperbolic compact groups

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.

${\hat{A F}}_{4}$ = [(3,3,3,3,4)]: Template:CDD	${\bar{D H}}_{4}$ = [5,3,3^1,1]: Template:CDD	${\bar{H}}_{4}$ = [3,3,3,5]: Template:CDD ${\bar{B H}}_{4}$ = [4,3,3,5]: Template:CDD ${\bar{K}}_{4}$ = [5,3,3,5]: Template:CDD

There are 5 regular compact convex hyperbolic honeycombs in H⁴ space:^[13]

Compact regular convex hyperbolic honeycombs
Honeycomb name	Schläfli Symbol {p,q,r,s}	Coxeter diagram	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 5-cell (pente)	{3,3,3,5}	Template:CDD	{3,3,3}	{3,3}	{3}	{5}	{3,5}	{3,3,5}	{5,3,3,3}
Order-3 120-cell (hitte)	{5,3,3,3}	Template:CDD	{5,3,3}	{5,3}	{5}	{3}	{3,3}	{3,3,3}	{3,3,3,5}
Order-5 tesseractic (pitest)	{4,3,3,5}	Template:CDD	{4,3,3}	{4,3}	{4}	{5}	{3,5}	{3,3,5}	{5,3,3,4}
Order-4 120-cell (shitte)	{5,3,3,4}	Template:CDD	{5,3,3}	{5,3}	{5}	{4}	{3,4}	{3,3,4}	{4,3,3,5}
Order-5 120-cell (phitte)	{5,3,3,5}	Template:CDD	{5,3,3}	{5,3}	{5}	{5}	{3,5}	{3,3,5}	Self-dual

There are also 4 regular compact hyperbolic star-honeycombs in H⁴ space:

Compact regular hyperbolic star-honeycombs
Honeycomb name	Schläfli Symbol {p,q,r,s}	Coxeter diagram	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 small stellated 120-cell	{5/2,5,3,3}	Template:CDD	{5/2,5,3}	{5/2,5}	{5}	{5}	{3,3}	{5,3,3}	{3,3,5,5/2}
Order-5/2 600-cell	{3,3,5,5/2}	Template:CDD	{3,3,5}	{3,3}	{3}	{5/2}	{5,5/2}	{3,5,5/2}	{5/2,5,3,3}
Order-5 icosahedral 120-cell	{3,5,5/2,5}	Template:CDD	{3,5,5/2}	{3,5}	{3}	{5}	{5/2,5}	{5,5/2,5}	{5,5/2,5,3}
Order-3 great 120-cell	{5,5/2,5,3}	Template:CDD	{5,5/2,5}	{5,5/2}	{5}	{3}	{5,3}	{5/2,5,3}	{3,5,5/2,5}

Hyperbolic paracompact groups

There are 9 paracompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Paracompact groups generate honeycombs with infinite facets or vertex figures.

${\bar{P}}_{4}$ = [3,3^[4]]: Template:CDD ${\bar{B P}}_{4}$ = [4,3^[4]]: Template:CDD ${\bar{F R}}_{4}$ = [(3,3,4,3,4)]: Template:CDD ${\bar{D P}}_{4}$ = [3^[3]×[]]: Template:CDD	${\bar{N}}_{4}$ = [4,/3\,3,4]: Template:CDD ${\bar{O}}_{4}$ = [3,4,3^1,1]: Template:CDD ${\bar{S}}_{4}$ = [4,3^2,1]: Template:CDD ${\bar{M}}_{4}$ = [4,3^1,1,1]: Template:CDD	${\bar{R}}_{4}$ = [3,4,3,4]: Template:CDD

Notes

↑ T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
↑ Multidimensional Glossary, George Olshevsky
↑ Script error: No such module "citation/CS1".
↑ Uniform Polytera, Jonathan Bowers
↑ Uniform polytope
↑ Script error: No such module "citation/CS1".
↑ Regular and semi-regular polytopes III, p.315 Three finite groups of 5-dimensions
↑ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Regular polytopes, p.297. Table IV, Fundamental regions for irreducible groups generated by reflections.
↑ Regular and Semiregular polytopes, II, pp.298-302 Four-dimensional honeycombs
↑ Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213

Script error: No such module "Check for unknown parameters".

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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Template:Isbn [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, p. 298 Four-dimensionsal honeycombs)
- (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
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

Template:KlitzingPolytopes – includes nonconvex forms as well as the duplicate constructions from the B₅ and D₅ families

Template:Polytopes Template:Honeycombs

eo:5-hiperpluredro

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

[2] Multidimensional Glossary, George Olshevsky

[3] Script error: No such module "citation/CS1".

[4] Uniform Polytera, Jonathan Bowers

[5] Uniform polytope

[6] Script error: No such module "citation/CS1".

[7] Regular and semi-regular polytopes III, p.315 Three finite groups of 5-dimensions

[8] Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61

[9] Script error: No such module "citation/CS1".

[10] Script error: No such module "citation/CS1".

[11] Regular polytopes, p.297. Table IV, Fundamental regions for irreducible groups generated by reflections.

[12] Regular and Semiregular polytopes, II, pp.298-302 Four-dimensional honeycombs

[13] Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

Uniform 5-polytope

Contents

History of discovery

Regular 5-polytopes

Convex uniform 5-polytopes

Symmetry of uniform 5-polytopes in four dimensions

Enumerating the convex uniform 5-polytopes

The A₅ family

The B₅ family

The D₅ family

Uniform prismatic forms

A₄ × A₁

B₄ × A₁

F₄ × A₁

H₄ × A₁

Duoprism prisms

Grand antiprism prism

Notes on the Wythoff construction for the uniform 5-polytopes

Regular and uniform honeycombs

Regular and uniform hyperbolic honeycombs

Notes

References

External links

Navigation menu

Uniform 5-polytope

History of discovery

Regular 5-polytopes

Convex uniform 5-polytopes

Symmetry of uniform 5-polytopes in four dimensions

Enumerating the convex uniform 5-polytopes

The A5 family

The B5 family

The D5 family

Uniform prismatic forms

A4 × A1

B4 × A1

F4 × A1

H4 × A1

Duoprism prisms

Grand antiprism prism

Notes on the Wythoff construction for the uniform 5-polytopes

Regular and uniform honeycombs

Regular and uniform hyperbolic honeycombs

Notes

References

External links

Navigation menu

Search

The A₅ family

The B₅ family

The D₅ family

A₄ × A₁

B₄ × A₁

F₄ × A₁

H₄ × A₁