# Uniform polyteron: Wikis

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# Encyclopedia

 5-simplex (hexateron) Truncated 5-simplex Rectified 5-simplex Birectified 5-simplex 5-orthoplex, 211 (Pentacross) Truncated 5-orthoplex Rectified 5-orthoplex 5-cube (Penteract) Truncated 5-cube Rectified 5-cube 5-demicube. 121 (Demipenteract) Truncated 5-demicube Rectified 5-demicube

In geometry, a uniform polyteron (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

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 convex and zero nonconvex regular polytopes.

There are exactly three such convex regular polytopes:

1. {3,3,3,3} - Hexateron (5-simplex)
2. {4,3,3,3} - Penteract (5-hypercube)
3. {3,3,3,4} - Pentacross (5-orthoplex)

There are no nonconvex regular 5-polytopes.

## 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 Coxeter-Dynkin diagram
1 A5 [34]
2 B5 [4,33]
3 D5 [32,1,1]

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

# Coxeter groups Coxeter graph
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 [31,1,1] × [ ]

There is one infinite family of 5-polytopes based on prisms of the 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 Coxeter graph
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

1. Simplex family: A5 [34] -
• 19 uniform 5-polytopes as permutations of rings in the group diagram, including one regular:
1. {34} - 5-simplex, hexateron, or hexa-5-tope,
2. Hypercube/orthoplex family: B5 [4,33] -
• 31 uniform 5-polytopes as permutations of rings in the group diagram, including two regular forms:
1. {4,33} — 5-cube, penteract, or deca-5-tope.
2. {33,4} — 5-orthoplex, pentacross, or triacontadi-5-tope.
3. Demihypercube D5/E5 family: [32,1,1] -
• 23 uniform 5-polytopes (8 unique) as permutations of rings in the group diagram, including:
1. 121 demipenteract - ; also as h{4,33},
2. 211 pentacross -
4. Prisms and duoprisms:
1. 56 uniform 5-polytope (46 unique) constructions based on prismatic families: [3,3,3]x[ ], [4,3,3]x[ ], [5,3,3]x[ ], [31,1,1]x[ ].
2. 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

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

### The A5 [3,3,3,3] family (5-simplex)

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).

# Coxeter-Dynkin diagram
Schläfli symbol
Name
Graph Vertex
figure
Facet counts by location: [3,3,3,3] Element counts
4 3 2 1 0

[3,3,3]
(6)

[3,3]×[ ]
(15)

[3]×[3]
(20)

[ ]×[3,3]
(15)

[3,3,3]
(6)
4-faces Cells Faces Edges Vertices
1
t0{3,3,3,3}
Hexateron (hix)

{3,3,3}
(5)

{3,3,3}
- - - - 6 15 20 15 6
2
t1{3,3,3,3}
Rectified hexateron (rix)

{3,3}x{ }
(4)

t1{3,3,3}
- - - (2)

{3,3,3}
12 45 80 60 15
3
t2{3,3,3,3}
Birectified hexateron (dot)

{3}x{3}
(3)

t1{3,3,3}
- - - (3)

t1{3,3,3}
12 60 120 90 20
4
t0,1{3,3,3,3}
Truncated hexateron (tix)

Tetrah.pyr
(4)

t0,1{3,3,3}
- - - (1)

{3,3,3}
12 45 80 75 30
5
t1,2{3,3,3,3}
Bitruncated hexateron (bittix)
(3)

t1,2{3,3,3}
- - - (2)

t0,1{3,3,3}
12 60 140 150 60
6
t0,2{3,3,3,3}
Cantellated hexateron (sarx)

prism-wedge
(3)

t0,2{3,3,3}
- - (1)

{ }×{3,3}
(1)

t1{3,3,3}
27 135 290 240 60
7
t1,3{3,3,3,3}
Bicantellated hexateron (sibrid)
(2)

t0,2{3,3,3}
- (8)

{3}×{3}
- (2)

t0,2{3,3,3}
32 180 420 360 90
8
t0,3{3,3,3,3}
Runcinated hexateron (spix)
(2)

t0,3{3,3,3}
- (3)

{3}×{3}
(3)

{ }×t1{3,3}
(1)

t1{3,3,3}
47 255 420 270 60
9
t0,4{3,3,3,3}

Irr.16-cell
(1)

{3,3,3}
(4)

{ }×{3,3}
(6)

{3}×{3}
(4)

{ }×{3,3}
(1)

{3,3,3}
62 180 210 120 30
10
t0,1,2{3,3,3,3}
Cantitruncated hexateron (garx)

t0,1,2{3,3,3}
- -
{ }×{3,3}

t0,1{3,3,3}
27 135 290 300 120
11
t1,2,3{3,3,3,3}
Bicantitruncated hexateron (gibrid)

t0,1,2{3,3,3}
-
{3}×{3}
-
t0,1,2{3,3,3}
32 180 420 450 180
12
t0,1,3{3,3,3,3}
Runcitruncated hexateron (pattix)

t0,1,3{3,3,3}
-
{6}×{3}

{ }×t1{3,3}

t0,2{3,3,3}
47 315 720 630 180
13
t0,2,3{3,3,3,3}
Runcicantellated hexateron (pirx)

t0,1,3{3,3,3}
-
{3}×{3}

{ }×t0,1{3,3}

t1,2{3,3,3}
47 255 570 540 180
14
t0,1,4{3,3,3,3}
Steritruncated hexateron (cappix)

t0,1{3,3,3}

{ }×t0,1{3,3}

{3}×{6}

{ }×{3,3}

t0,3{3,3,3}
62 330 570 420 120
15
t0,2,4{3,3,3,3}
Stericantellated hexateron (card)

t0,2{3,3,3}

{ }×t0,2{3,3}

{3}×{3}

{ }×t0,2{3,3}

t0,2{3,3,3}
62 420 900 720 180
16
t0,1,2,3{3,3,3,3}
Runcicantitruncated hexateron (gippix)

Irr.5-cell

t0,1,2,3{3,3,3}
-
{3}×{6}

{ }×t0,1{3,3}

t0,2{3,3,3}
47 315 810 900 360
17
t0,1,2,4{3,3,3,3}
Stericantitruncated hexateron (cograx)

t0,1,2{3,3,3}

{ }×t0,1,2{3,3}

{3}×{6}

{ }×t0,2{3,3}

t0,1,3{3,3,3}
62 480 1140 1080 360
18
t0,1,3,4{3,3,3,3}
Steriruncitruncated hexateron (captid)

t0,1,3{3,3,3}

{ }×t0,1{3,3}

{6}×{6}

{ }×t0,1,3{3,3}

t0,1,3{3,3,3}
62 450 1110 1080 360
19
t0,1,2,3,4{3,3,3,3}

Irr. {3,3,3}
(1)

t0,1,2,3{3,3,3}
(1)

{ }×t0,1,2{3,3}
(1)

{6}×{6}
(1)

{ }×t0,1,2{3,3}
(1)

t0,1,2,3{3,3,3}
62 540 1560 1800 720

### The B5 [4,3,3,3] family (penteract/pentacross)

This family has 31 Wythoffian uniform polyhedra, from 25-1 permutations of the Coxeter-Dynkin diagram with one or more rings.

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

The B5 family has symmetry of order 3840 (2^5*5!).

#### The penteract subfamily

There are 20 forms here, 7 shared with the pentacross family. Four are shared with the demipenteract family.

# Coxeter-Dynkin
andSchläfli
symbols
Name
Graph Vertex
figure
Facet counts by location: [4,3,3,3] Element counts
4 3 2 1 0

[4,3,3]
(10)

[4,3]×[ ]
(40)

[4]×[3]
(80)

[ ]×[3,3]
(80)

[3,3,3]
(32)
Facets Cells Faces Edges Vertices
20
t0{4,3,3,3}
Penteract
Pent

{3,3,3}

{4,3,3}
- - - - 10 40 80 80 32
21
t1{4,3,3,3}
Rectified penteract
Rin

{3,3}x{ }

t1{4,3,3}
- - -
{3,3,3}
42 200 400 320 80
22
t2{4,3,3,3}
Birectified penteract
Nit

{4}×{3}

t1{4,3,3}
- - -
t1{3,3,3}
42 280 640 480 80
23
t0,1{4,3,3,3}
Truncated penteract
Tan

Tetrah.pyr

t0,1{4,3,3}
- - -
{3,3,3}
42 200 400 400 160
24
t1,2{4,3,3,3}
Bitruncated penteract
Bittin

t1,2{4,3,3}
- - -
t0,1{3,3,3}
42 280 720 800 320
25
t0,2{4,3,3,3}
Cantellated penteract
Sirn

Prism-wedge

t0,2{4,3,3}
- -
{ }×{3,3}

t1{3,3,3}
122 680 1520 1280 320
26
t1,3{4,3,3,3}
Bicantellated penteract
Sibrant

t0,2{4,3,3}
-
{4}×{3}
-
t0,2{3,3,3}
122 840 2160 1920 480
27
t0,3{4,3,3,3}
Runcinated penteract
Span

t0,3{4,3,3}
-
{4}×{3}

{ }×t1{3,3}

{3,3,3}
202 1240 2160 1440 320
28
t0,4{4,3,3,3}
Stericated penteract
Scant

Tetr.antiprm

{4,3,3}

{4,3}×{ }

{4}×{3}

{ }×{3,3}

{3,3,3}
242 800 1040 640 160
29
t0,1,2{4,3,3,3}
Cantitruncated penteract
Girn

t0,1,2{4,3,3}
- -
{ }×{3,3}

t0,1{3,3,3}
122 680 1520 1600 640
30
t1,2,3{4,3,3,3}
Bicantitruncated penteract
Gibrant

t0,1,2{4,3,3}
-
{4}×{3}
-
t0,1,2{3,3,3}
122 840 2160 2400 960
31
t0,1,3{4,3,3,3}
Runcitruncated penteract
Pattin

t0,1,3{4,3,3}
- {8}×{3}
{ }×t1{3,3}

t0,2{3,3,3}
202 1560 3760 3360 960
32
t0,2,3{4,3,3,3}
Runcicantellated penteract
Prin

t0,1,3{4,3,3}
-
{4}×{3}

{ }×t0,1{3,3}

t1,2{3,3,3}
202 1240 2960 2880 960
33
t0,1,4{4,3,3,3}
Steritruncated penteract
Capt

t0,1{4,3,3}

t0,1{4,3}×{ }
{8}×{3}
{ }×{3,3}

t0,3{3,3,3}
242 1600 2960 2240 640
34
t0,2,4{4,3,3,3}
Stericantellated penteract
Carnit

t0,2{4,3,3}

t0,2{4,3}×{ }

{4}×{3}

{ }×t0,2{3,3}

t0,2{3,3,3}
242 2080 4720 3840 960
35
t0,1,2,3{4,3,3,3}
Runcicantitruncated penteract
Gippin

t0,1,2,3{4,3,3}
- {8}×{3}
{ }×t0,1{3,3}

t0,1,2{3,3,3}
202 1560 4240 4800 1920
36
t0,1,2,4{4,3,3,3}
Stericantitruncated penteract
Cogrin

t0,1,2{4,3,3}

t0,1,2{4,3}×{ }
{8}×{3}
{ }×t0,2{3,3}

t0,1,3{3,3,3}
242 2400 6000 5760 1920
37
t0,1,3,4{4,3,3,3}
Steriruncitruncated penteract
Captint

t0,1,3{4,3,3}

t0,1{4,3}×{ }
{8}×{6}
{ }×t0,1{3,3}

t0,1,3{3,3,3}
242 2160 5760 5760 1920
38
t0,1,2,3,4{4,3,3,3}
Omnitruncated penteract
Gacnet

Irr. {3,3,3}

t0,1,2{4,3}×{ }

t0,1,2{4,3}×{ }
{8}×{6}
{ }×t0,1,2{3,3}

t0,1,2,3{3,3,3}
242 2640 8160 9600 3840
[51]
h0{4,3,3,3}
Demipenteract
Hin

t1{3,3,3}

(16) {3,3,3}
- - -
{3,3,4}
26 120 160 80 16

#### Pentacross subfamily

There are 19 forms, 12 new ones. 7 are shared from the penteract family, and 10 shared with the demipenteract family.

# Coxeter-Dynkin
andSchläfli
symbols
Name
Graph Vertex
figure
Facet counts by location: [4,3,3,3] Element counts
4 3 2 1 0

[3,3,3]
(32)

[3,3]×[ ]
(80)

[3]×[4]
(80)

[ ]×[3,4]
(40)

[3,3,4]
(10)
Facets Cells Faces Edges Vertices
39
t0{3,3,3,4}
Pentacross
Tac

{3,3,4}

{3,3,3}
- - - - 32 80 80 40 10
40
t1{3,3,3,4}
Rectified pentacross
Rat

{ }×{3,4}

t1{3,3,3}
- - -

{3,3,4}
42 240 400 240 40
[22]
t2{3,3,3,4}
Birectified pentacross
Nit

{4}×{3}

t1{3,3,3}
- - -
t1{3,3,4}
42 280 640 480 80
41
t0,1{3,3,3,4}
Truncated pentacross
Tot

(Octah.pyr)

t0,1{3,3,3}
- - - {3,3,3} 42 240 400 280 80
42
t1,2{3,3,3,4}
Bitruncated pentacross
Bittit

t1,2{3,3,3}
- - - t0,1{3,3,4} 42 280 720 720 240
43
t0,2{3,3,3,4}
Cantellated pentacross
Sart

Prism-wedge

t0,2{3,3,3}
- - { }×{3,4} t1{3,3,4} 82 640 1520 1200 240
[26]
t1,3{3,3,3,4}
Bicantellated pentacross
Sibrant

t1,3{3,3,3}
- {3}×{4} - t0,2{3,3,4} 122 840 2160 1920 480
44
t0,3{3,3,3,4}
Runcinated pentacross
Spat

t0,3{3,3,3}
- {3}×{4} t1{4,3,3} 162 1200 2160 1440 320
[28]
t0,4{3,3,3,4}
Stericated pentacross
Scant

Tetr.antiprm

{3,3,3}
- - - {4,3,3} 242 800 1040 640 160
45
t0,1,2{3,3,3,4}
Cantitruncated pentacross
Gart

t0,1,2{3,3,3}
- - - t0,1{3,3,4} 82 640 1520 1440 480
[30]
t1,2,3{3,3,3,4}
Bicantitruncated pentacross
Gibrant

t1,2,3{3,3,3}
- {3}×{4} - t0,1,2{3,3,4} 122 840 2160 2400 960
46
t0,1,3{3,3,3,4}
Runcitruncated pentacross
Pattit

t0,1,3{3,3,3}
- {6}×{4} { }×t1{3,4} t0,2{3,3,4} 162 1440 3680 3360 960
47
t0,2,3{3,3,3,4}
Runcicantellated pentacross
Pirt

t0,1,3{3,3,3}
- {3}×{4} { }×t0,1{3,4} t1,2{3,3,4} 162 1200 2960 2880 960
48
t0,1,4{3,3,3,4}
Steritruncated pentacross
Cappin

t0,1{3,3,3}
- - { }×{4,3} t0,3{3,3,4} 242 1520 2880 2240 640
[34]
t0,2,4{3,3,3,4}
Stericantellated pentacross
Carnit

t0,2{3,3,3}
{ }×t0,2{3,3} {3}×{4} { }×t0,2{3,4} t0,2{4,3,3} 242 2080 4720 3840 960
49
t0,1,2,3{3,3,3,4}
Runcicantitruncated pentacross
Gippit

t0,1,2,3{3,3,3}
- {6}×{4} { }×t0,1{3,4} t0,1,2{3,3,4} 162 1440 4160 4800 1920
50
t0,1,2,4{3,3,3,4}
Stericantitruncated pentacross
Cogart

t0,1,2{3,3,3}
{ }×t0,1,{3,3} {6}×{4} { }×t0,2{3,4} t0,1,3{3,3,4} 242 2320 5920 5760 1920
[37]
t0,1,3,4{3,3,3,4}
Steriruncitruncated pentacross
Captint

t0,1,3{3,3,3}
{ }×t0,1{3,3} {6}×{8} { }×t0,1{4,3} t0,1,3{4,3,3} 242 2160 5760 5760 1920
[38]
t0,1,2,3,4{3,3,3,4}
Omnitruncated pentacross
Gacnet

Irr. {3,3,3}

t0,1,2,3{3,3,3}
{ }×t0,1,2{3,3} {6}×{8} { }×t0,1,2{3,4} t0,1,2,3{3,3,4} 242 2640 8160 9600 3840

### The D5 [31,2,1] family (demipenteract)

There are 23 forms. 16 are repeated from the [4,3,3,3] family and 7 are new ones.

The D5 family has symmetry of order 1920 (2^4*5!).

# Coxeter-Dynkin
and Schläfli symbols
Name
Graph Vertex
figure
Facets by location: [31,2,1] Element counts
4 3 2 1 0

[3,3,3]
(16)

[31,1,1]
(10)

[3,3]×[ ]
(40)

[ ]×[3]×[ ]
(80)

[3,3,3]
(16)
Facets Cells Faces Edges Vertices
51 (121)
Demipenteract
Hin

t1{3,3,3}
{3,3,3} t0(111) - - - 26 120 160 80 16
[22] t1(121)
Rectified demipenteract
(Same as birectified penteract)
Nit

{ }×{ }×{3}
t1{3,3,3} t1(111) - - t1{3,3,3} 42 280 640 480 80
[40] t2(121)
Birectified demipenteract
(Same as rectified pentacross)
Rat

{ }×t1{3,3}
t1{3,3,3} t0(111) - - t1{3,3,3} 42 240 400 240 40
[39] t3(121)
Trirectified demipenteract
(Same as pentacross)
Tac

(111)
{3,3,3} - - - {3,3,3} 32 80 80 40 10
52 t0,1(121)
Truncated demipenteract
Thin
42 280 640 560 160
53 t0,2(121)
Cantellated demipenteract
Sirhin
42 360 880 720 160
54 t0,3(121)
Runcinated demipenteract
Siphin
82 480 720 400 80
[21] t0,4(121)
Stericated demipenteract
(Same as rectified penteract)
Rin

{3,3}x{ }
42 200 400 320 80
[42] t1,2(121)
Bitruncated demipenteract
(Same as bitruncated pentacross)
Bittit
42 280 720 720 240
[43] t1,3(121)
Bicantellated demipenteract
(Same as cantellated pentacross)
Sart

Prism-wedge
82 640 1520 1200 240
[41] t2,3(121)
Tritruncated demipenteract
(Same as truncated pentacross)
Tot

(Octah.pyr)
42 240 400 280 80
[24] t0,1,4(121)
Steritruncated demipenteract
(Same as bitruncated penteract)
Bittin
42 280 720 800 320
55 t0,1,2(121)
Cantitruncated demipenteract
Girhin
42 360 1040 1200 480
56 t0,1,3(121)
Runcitruncated demipenteract
Pithin
82 720 1840 1680 480
[26] t0,2,4(121)
Stericantellated demipenteract
(Same as bicantellated penteract)
Sibrant
122 840 2160 1920 480
[44] t0,3,4(121)
Steriruncinated demipenteract
(Same as runcinated pentacross)
Spat
162 1200 2160 1440 320
57 t0,2,3(121)
Runcicantellated demipenteract
Pirhin
82 560 1280 1120 320
[45] t1,2,3(121)
Bicantitruncated demipenteract
(Same as cantitruncated pentacross)
Gart
82 640 1520 1440 480
[30] t0,1,2,4(121)
Stericantitruncated demipenteract
(Same as bicantitruncated pentacross)
Gibrant
122 840 2160 2400 960
[46] t0,1,3,4(121)
Steriruncitruncated demipenteract
(Same as runcicantellated pentacross)
Pirt
162 1440 3680 3360 960
58 t0,1,2,3(121)
Runcicantitruncated demipenteract
Giphin
82 720 2080 2400 960
[47] t0,2,3,4(121)
Steriruncicantellated demipenteract
(Same as runcitruncated pentacross)
Pattit
162 1200 2960 2880 960
[49] t0,1,2,3,4(121)
Omnitruncated demipenteract
(Same as runcicantitruncated pentacross)
Gippit

Irr. {3,3,3}
162 1440 4160 4800 1920

### Uniform prismatic forms

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

#### [3,3,3] × [ ]

This prismatic family has 9 forms:

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

# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
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
t0,1{3,3,3}x{ }
Truncated 5-cell prism
12 50 100 100 40
62
t0,2{3,3,3}x{ }
Cantellated 5-cell prism
22 120 250 210 60
63
t0,3{3,3,3}x{ }
Runcinated 5-cell prism
32 130 200 140 40
64
t1,2{3,3,3}x{ }
Bitruncated 5-cell prism
12 60 140 150 60
65
t0,1,2{3,3,3}x{ }
Cantitruncated 5-cell prism
22 120 280 300 120
66
t0,1,3{3,3,3}x{ }
Runcitruncated 5-cell prism
32 180 390 360 120
67
t0,1,2,3{3,3,3}x{ }
Omnitruncated 5-cell prism
32 210 540 600 240

#### [4,3,3] × [ ]

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!).

##### Tesseractic prism family {4,3,3}x{ }
# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
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
t0,1{4,3,3}x{ }
Truncated tesseractic prism
26 136 304 320 128
71
t0,2{4,3,3}x{ }
Cantellated tesseractic prism
58 360 784 672 192
72
t0,3{4,3,3}x{ }
Runcinated tesseractic prism
82 368 608 448 128
73
t1,2{4,3,3}x{ }
Bitruncated tesseractic prism
26 168 432 480 192
74
t0,1,2{4,3,3}x{ }
Cantitruncated tesseractic prism
58 360 880 960 384
75
t0,1,3{4,3,3}x{ }
Runcitruncated tesseractic prism
82 528 1216 1152 384
76
t0,1,2,3{4,3,3}x{ }
Omnitruncated tesseractic prism
82 624 1696 1920 768
##### 16-cell prism family [3,3,4]x[ ]
# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
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
t0,1{3,3,4}x{ }
Truncated 16-cell prism
26 144 312 288 96
80
t0,2{3,3,4}x{ }
Cantellated 16-cell prism
(Same as rectified 24-cell prism)
50 336 768 672 192
[72]
t0,3{4,3,3}x{ }
Runcinated 16-cell prism
(Same as Runcinated tesseractic prism)
82 368 608 448 128
[73]
t1,2{4,3,3}x{ }
Bitruncated 16-cell prism
(Same as bitruncated tesseractic prism)
26 168 432 480 192
81
t0,1,2{3,3,4}x{ }
Cantitruncated 16-cell prism
(Same as truncated 24-cell prism)
50 336 864 960 384
82
t0,1,3{3,3,4}x{ }
Runcitruncated 16-cell prism
82 528 1216 1152 384
[76]
t0,1,2,3{3,3,4}x{ }
Omnitruncated 16-cell prism
(Same as omnitruncated tesseractic prism)
82 624 1696 1920 768
83
h0,1,2{3,3,4}x{ }
Snub 24-cell prism
146 768 1392 960 192

#### [3,4,3] × [ ]

This prismatic family has 10 forms.

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

# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
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]
t0,1{3,4,3}x{ }
truncated 24-cell prism
50 336 864 960 384
84
t0,2{3,4,3}x{ }
cantellated 24-cell prism
146 1008 2304 2016 576
85
t0,3{3,4,3}x{ }
runcinated 24-cell prism
242 1152 1920 1296 288
86
t1,2{3,4,3}x{ }
bitruncated 24-cell prism
50 432 1248 1440 576
87
t0,1,2{3,4,3}x{ }
cantitruncated 24-cell prism
146 1008 2592 2880 1152
88
t0,1,3{3,4,3}x{ }
runcitruncated 24-cell prism
242 1584 3648 3456 1152
89
t0,1,2,3{3,4,3}x{ }
omnitruncated 24-cell prism
242 1872 5088 5760 2304
[83]
h0,1{3,4,3}x{ }
snub 24-cell prism
146 768 1392 960 192

#### [5,3,3] × [ ]

This prismatic family has 15 forms:

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

##### Dodecaplex prism family {5,3,3}x{ }
# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
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
t0,1{5,3,3}x{ }
Truncated 120-cell prism
722 4560 11040 12000 4800
93
t0,2{5,3,3}x{ }
Cantellated 120-cell prism
1922 12960 29040 25200 7200
94
t0,3{5,3,3}x{ }
Runcinated 120-cell prism
2642 12720 22080 16800 4800
95
t1,2{5,3,3}x{ }
Bitruncated 120-cell prism
722 5760 15840 18000 7200
96
t0,1,2{5,3,3}x{ }
Cantitruncated 120-cell prism
1922 12960 32640 36000 14400
97
t0,1,3{5,3,3}x{ }
Runcitruncated 120-cell prism
2642 18720 44880 43200 14400
98
t0,1,2,3{5,3,3}x{ }
Omnitruncated 120-cell prism
2642 22320 62880 72000 28800
##### Tetraplex prism family [3,3,5]x[ ]
# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
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
t0,1{3,3,5}x{ }
Truncated 600-cell prism
722 5040 11520 10080 2880
102
t0,2{3,3,5}x{ }
Cantellated 600-cell prism
1442 11520 28080 25200 7200
[94]
t0,3{3,3,5}x{ }
Runcinated 600-cell prism
(Same as runcinated 120-cell prism)
2642 12720 22080 16800 4800
[95]
t1,2{3,3,5}x{ }
Bitruncated 600-cell prism
(Same as bitruncated 120-cell prism)
722 5760 15840 18000 7200
103
t0,1,2{3,3,5}x{ }
Cantitruncated 600-cell prism
1442 11520 31680 36000 14400
104
t0,1,3{3,3,5}x{ }
Runcitruncated 600-cell prism
2642 18720 44880 43200 14400
[98]
t0,1,2,3{3,3,5}x{ }
Omnitruncated 600-cell prism
(Same as omnitruncated 120-cell prism)
2642 22320 62880 72000 28800

#### 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
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
Coxeter-
Dynkin
diagram
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 t0,1{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 t0,2{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 t0,3{p,q,r,s} Runcination reduces cells and creates new cells at the vertices and edges.
Stericated t0,4{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 t0,1,2,3,4{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 that generate regular and uniform tessellations in 4-space. [1]

# Coxeter group Coxeter-Dynkin diagram
1 A~4 (3 3 3 3 3) [3[5]]
2 B~4 [4,3,3,4]
3 C~4 [4,3,31,1] h[4,3,3,4]
4 D~4 [31,1,1,1] q[4,3,3,4]
5 F~4 [3,4,3,3] h[4,3,3,4]

There are three regular honeycomb of Euclidean 4-space:

1. tesseractic honeycomb, with symbols {4,3,3,4}, = . There are 19 uniform honeycombs in this family.
2. Icositetrachoric honeycomb, with symbols {3,4,3,3}, . There are 31 uniform honeycombs in this family.
3. Hexadecachoric 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 A~4, family, all unique.
• There are 7 uniform honeycombs in the D~4: [31,1,1,1] 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.

The single-ringed tessellations are given below, indexed by Olshevsky's listing.

### B~4 [4,3,3,4] family (Tesseractic honeycombs)

# Coxeter-Dynkin
andSchläfli
symbols
Name Facets by location: [4,3,3,4]
4 3 2 1 0

[4,3,3]

[4,3]×[ ]

[4]×[4]

[ ]×[3,4]

[3,3,4]
1
t0{4,3,3,4}
Tesseractic honeycomb {4,3,3} - - - -
87
t1{4,3,3,4}
Rectified tesseractic honeycomb t1{4,3,3} - - - {3,3,4}
88
t2{4,3,3,4}
Birectified tesseractic honeycomb
(Same as Icositetrachoric honeycomb {3,4,3,3})
t1{3,3,4}
or {3,4,3}
- - - t1{3,3,4}
or {3,4,3}

### C~4 [31,1,3,4] family (Demitesseractic honeycombs)

[2]

# Coxeter-Dynkin
andSchläfli
symbols
Name Facets by location: [31,1,3,4]
4 3 2 1 0

[3,3,4]

[31,1,1]

[3,3]×[ ]

[ ]×[3]×[ ]

[3,3,4]
104
{31,1,3,4}
Demitesseractic honeycomb
[88]
t1{31,1,3,4}
Rectified demitesseractic honeycomb
(Same as icositetrachoric honeycomb, {3,4,3,3})
(Also birectified tesseractic honeycomb)
[87]
t2{31,1,3,4}
Birectified demitesseractic honeycomb
(Same as Rectified tesseractic honeycomb)
[1]
t3{31,1,3,4}
Trirectified demitesseractic honeycomb
(Same as tesseractic honeycomb)

### F~4 (V5) [3,4,3,3] family (Icositetrachoric-hexadecachoric honeycombs)

[3]

# Coxeter-Dynkin
andSchläfli
symbols
Name Facets by location: [3,4,3,3]
4 3 2 1 0

[3,4,3]

[3,4]×[ ]

[3]×[3]

[ ]×[3,3]

[4,3,3]
104
t0{3,4,3,3}
Icositetrachoric honeycomb
[88]
t1{3,4,3,3}
Rectified icositetrachoric honeycomb
106
t2{3,4,3,3}
Birectified icositetrachoric honeycomb
?
t1{3,3,4,3}
88
t0{3,3,4,3}

### A~4 (P5) (3 3 3 3) family (Simplectic honeycombs)

# Coxeter-Dynkin
andSchläfli
symbols
Name Facets by location
4 3 2 1 0

[3,3,3]

[]x[3,3]

[3]x[3]

[3,3]x[]

[3,3,3]
134 Pentachoric-dispentachoric honeycomb

# Coxeter-Dynkin
andSchläfli
symbols
?
q[4,3,3,4]
?
q1[4,3,3,4]

## Regular and uniform hyperbolic honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in hyperbolic 4-space. They generate 5 regular hyperbolic honeycombs and numerous uniform ones.

# Coxeter group Coxeter-Dynkin diagram Uniform tessellations
1 (4 3 3 3 3) 25 forms
2 [5,3,3,3] 31 forms
3 [5,3,3,4] 31 forms
4 [5,3,31,1] 23 forms, most overlapping [5,3,3,4] family
5 [5,3,3,5] 19 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}
Coxeter-Dynkin
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 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}
Coxeter-Dynkin
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 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}

## Notes

1. ^ George Olshevsky, (2006, Uniform Panoploid Tetracombs, Manuscript (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
2. ^ Olshevsky section V
3. ^ Olshevsky section VI
4. ^ 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 quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons [2]
• 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) [3]