# Uniform 5-polytope

Short description: Five-dimensional geometric shape
 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 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]
• 2000-2023: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes,[3] with a current count of 1297 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

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:

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](more unsolved problems in mathematics)

There are 104 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.

### 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.

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 [3,3,3,3] [3,3,3,3]+ 6 15
D5 1920 [3,3,31,1] [3,3,31,1]+ 8 20
B5 3840 [4,3,3,3] 10 5 20
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 [3,3,3,2] = [3,3,3]×[ ] [3,3,3]+ 10 1
D4A1 384 [31,1,1,2] = [31,1,1]×[ ] [31,1,1]+ 12 1
B4A1 768 [4,3,3,2] = [4,3,3]×[ ] 4 12 1
F4A1 2304 [3,4,3,2] = [3,4,3]×[ ] [3+,4,3+] 12 12 1
H4A1 28800 [5,3,3,2] = [3,4,3]×[ ] [5,3,3]+ 60 1
Duoprismatic prisms (use 2p and 2q for evens)
I2(p)I2(q)A1 8pq [p,2,q,2] = [p]×[q]×[ ] [p+,2,q+] p q 1
I2(2p)I2(q)A1 16pq [2p,2,q,2] = [2p]×[q]×[ ] p p q 1
I2(2p)I2(2q)A1 32pq [2p,2,2q,2] = [2p]×[2q]×[ ] p p q q 1
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 [3,3,2,p] = [3,3]×[p] [(3,3)+,2,p+] 6 p
A3I2(2p) 96p [3,3,2,2p] = [3,3]×[2p] 6 p p
B3I2(p) 96p [4,3,2,p] = [4,3]×[p] 3 6 p
B3I2(2p) 192p [4,3,2,2p] = [4,3]×[2p] 3 6 p p
H3I2(p) 240p [5,3,2,p] = [5,3]×[p] [(5,3)+,2,p+] 15 p
H3I2(2p) 480p [5,3,2,2p] = [5,3]×[2p] 15 p p

### 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: [32,1,1]
• 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]×[ ], [31,1,1]×[ ].
• 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

• 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

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]
4 3 2 1 0
[3,3,3]
(6)

[3,3,2]
(15)

[3,2,3]
(20)

[2,3,3]
(15)

[3,3,3]
(6)
Alt
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}

{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}×{ }

r{3,3,3}
- - -
{3,3,3}
3 (0,0,0,0,1,2) or (0,1,2,2,2,2) Truncated 5-simplex
truncated hexateron (tix)
12 45 80 75 30
Tetrah.pyr

t{3,3,3}
- - -
{3,3,3}
4 (0,0,0,1,1,2) or (0,1,1,2,2,2) Cantellated 5-simplex
small rhombated hexateron (sarx)
27 135 290 240 60
prism-wedge

rr{3,3,3}
- -
{ }×{3,3}

r{3,3,3}
5 (0,0,0,1,2,2) or (0,0,1,2,2,2) Bitruncated 5-simplex
bitruncated hexateron (bittix)
12 60 140 150 60
2t{3,3,3}
- - -
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)
27 135 290 300 120
tr{3,3,3}
- -
{ }×{3,3}

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)
47 255 420 270 60
t0,3{3,3,3}
-
{3}×{3}

{ }×r{3,3}

r{3,3,3}
8 (0,0,1,1,2,3) or (0,1,2,2,3,3) Runcitruncated 5-simplex
prismatotruncated hexateron (pattix)
47 315 720 630 180
t0,1,3{3,3,3}
-
{6}×{3}

{ }×r{3,3}

rr{3,3,3}
9 (0,0,1,2,2,3) or (0,1,1,2,3,3) Runcicantellated 5-simplex
prismatorhombated hexateron (pirx)
47 255 570 540 180
t0,1,3{3,3,3}
-
{3}×{3}

{ }×t{3,3}

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)
47 315 810 900 360
Irr.5-cell

t0,1,2,3{3,3,3}
-
{3}×{6}

{ }×t{3,3}

tr{3,3,3}
11 (0,1,1,1,2,3) or (0,1,2,2,2,3) Steritruncated 5-simplex
celliprismated hexateron (cappix)
62 330 570 420 120
t{3,3,3}

{ }×t{3,3}

{3}×{6}

{ }×{3,3}

t0,3{3,3,3}
12 (0,1,1,2,3,4) or (0,1,2,3,3,4) Stericantitruncated 5-simplex
celligreatorhombated hexateron (cograx)
62 480 1140 1080 360
tr{3,3,3}

{ }×tr{3,3}

{3}×{6}

{ }×rr{3,3}

t0,1,3{3,3,3}
13 (0,0,0,1,1,1) Birectified 5-simplex
dodecateron (dot)
12 60 120 90 20
{3}×{3}

r{3,3,3}
- - -
r{3,3,3}
14 (0,0,1,1,2,2) Bicantellated 5-simplex
small birhombated dodecateron (sibrid)
32 180 420 360 90
rr{3,3,3}
-
{3}×{3}
-
rr{3,3,3}
15 (0,0,1,2,3,3) Bicantitruncated 5-simplex
great birhombated dodecateron (gibrid)
32 180 420 450 180
tr{3,3,3}
-
{3}×{3}
-
tr{3,3,3}
16 (0,1,1,1,1,2) Stericated 5-simplex
62 180 210 120 30
Irr.16-cell

{3,3,3}

{ }×{3,3}

{3}×{3}

{ }×{3,3}

{3,3,3}
17 (0,1,1,2,2,3) Stericantellated 5-simplex
small cellirhombated dodecateron (card)
62 420 900 720 180
rr{3,3,3}

{ }×rr{3,3}

{3}×{3}

{ }×rr{3,3}

rr{3,3,3}
18 (0,1,2,2,3,4) Steriruncitruncated 5-simplex
celliprismatotruncated dodecateron (captid)
62 450 1110 1080 360
t0,1,3{3,3,3}

{ }×t{3,3}

{6}×{6}

{ }×t{3,3}

t0,1,3{3,3,3}
19 (0,1,2,3,4,5) Omnitruncated 5-simplex
62 540 1560 1800 720
Irr. {3,3,3}

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

{ }×tr{3,3}

{6}×{6}

{ }×tr{3,3}

t0,1,2,3{3,3,3}
Nonuniform Omnisnub 5-simplex
snub dodecateron (snod)
snub hexateron (snix)
422 2340 4080 2520 360 ht0,1,2,3{3,3,3} ht0,1,2,3{3,3,2} ht0,1,2,3{3,2,3} ht0,1,2,3{3,3,2} ht0,1,2,3{3,3,3} (360)

Irr. {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 diagram. Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D5 family as ... = ..... (There are more alternations that are not listed because they produce only repetitions, as ... = .... and ... = .... 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]
4 3 2 1 0
[4,3,3]
(10)

[4,3,2]
(40)

[4,2,3]
(80)

[2,3,3]
(80)

[3,3,3]
(32)
Alt
20 (0,0,0,0,1)√2 5-orthoplex
32 80 80 40 10
{3,3,4}
- - - - 60px
{3,3,3}
21 (0,0,0,1,1)√2 Rectified 5-orthoplex
42 240 400 240 40
{ }×{3,4}
60px
{3,3,4}
- - - 60px
r{3,3,3}
22 (0,0,0,1,2)√2 Truncated 5-orthoplex
42 240 400 280 80
(Octah.pyr)
60px
{3,3,4}
- - - 60px
t{3,3,3}
23 (0,0,1,1,1)√2 Birectified 5-cube
(Birectified 5-orthoplex)
42 280 640 480 80
{4}×{3}
60px
r{3,3,4}
- - - 60px
r{3,3,3}
24 (0,0,1,1,2)√2 Cantellated 5-orthoplex
82 640 1520 1200 240
Prism-wedge
60px
r{3,3,4}
60px
{ }×{3,4}
- - 60px
rr{3,3,3}
25 (0,0,1,2,2)√2 Bitruncated 5-orthoplex
42 280 720 720 240 60px
t{3,3,4}
- - -
2t{3,3,3}
26 (0,0,1,2,3)√2 Cantitruncated 5-orthoplex
82 640 1520 1440 480 60px
t{3,3,4}
60px
{ }×{3,4}
- -
t0,1,3{3,3,3}
27 (0,1,1,1,1)√2 Rectified 5-cube
rectified penteract (rin)
42 200 400 320 80
{3,3}×{ }
60px
r{4,3,3}
- - - 60px
{3,3,3}
28 (0,1,1,1,2)√2 Runcinated 5-orthoplex
162 1200 2160 1440 320 60px
r{4,3,3}
60px
{ }×r{3,4}
60px
{3}×{4}
60px
t0,3{3,3,3}
29 (0,1,1,2,2)√2 Bicantellated 5-cube
(Bicantellated 5-orthoplex)
122 840 2160 1920 480 60px
rr{3,3,4}
- 60px
{4}×{3}
- 60px
rr{3,3,3}
30 (0,1,1,2,3)√2 Runcitruncated 5-orthoplex
162 1440 3680 3360 960 60px
rr{3,3,4}
60px
{ }×r{3,4}
60px
{6}×{4}
-
t0,1,3{3,3,3}
31 (0,1,2,2,2)√2 Bitruncated 5-cube
bitruncated penteract (bittin)
42 280 720 800 320 60px
2t{4,3,3}
- - - 60px
t{3,3,3}
32 (0,1,2,2,3)√2 Runcicantellated 5-orthoplex
162 1200 2960 2880 960 60px
2t{4,3,3}
60px
{ }×t{3,4}
60px
{3}×{4}
-
t0,1,3{3,3,3}
33 (0,1,2,3,3)√2 Bicantitruncated 5-cube
(Bicantitruncated 5-orthoplex)
122 840 2160 2400 960 60px
tr{3,3,4}
- 60px
{4}×{3}
- 60px
rr{3,3,3}
34 (0,1,2,3,4)√2 Runcicantitruncated 5-orthoplex
162 1440 4160 4800 1920 60px
tr{3,3,4}
60px
{ }×t{3,4}
60px
{6}×{4}
-
t0,1,2,3{3,3,3}
35 (1,1,1,1,1) 5-cube
penteract (pent)
10 40 80 80 32
{3,3,3}
60px
{4,3,3}
- - - -
36 (1,1,1,1,1)
+ (0,0,0,0,1)√2
Stericated 5-cube
(Stericated 5-orthoplex)
242 800 1040 640 160
Tetr.antiprm
60px
{4,3,3}
60px
{4,3}×{ }
60px
{4}×{3}
60px
{ }×{3,3}
60px
{3,3,3}
37 (1,1,1,1,1)
+ (0,0,0,1,1)√2
Runcinated 5-cube
small prismated penteract (span)
202 1240 2160 1440 320 60px
t0,3{4,3,3}
- 60px
{4}×{3}
60px
{ }×r{3,3}
60px
r{3,3,3}
38 (1,1,1,1,1)
+ (0,0,0,1,2)√2
Steritruncated 5-orthoplex
242 1520 2880 2240 640 60px
t0,3{4,3,3}
60px
{4,3}×{ }
60px
{6}×{4}
60px
{ }×t{3,3}
60px
t{3,3,3}
39 (1,1,1,1,1)
+ (0,0,1,1,1)√2
Cantellated 5-cube
small rhombated penteract (sirn)
122 680 1520 1280 320
Prism-wedge
60px
rr{4,3,3}
- - 60px
{ }×{3,3}
60px
r{3,3,3}
40 (1,1,1,1,1)
+ (0,0,1,1,2)√2
Stericantellated 5-cube
(Stericantellated 5-orthoplex)
242 2080 4720 3840 960 60px
rr{4,3,3}
60px
rr{4,3}×{ }
60px
{4}×{3}
60px
{ }×rr{3,3}
60px
rr{3,3,3}
41 (1,1,1,1,1)
+ (0,0,1,2,2)√2
Runcicantellated 5-cube
prismatorhombated penteract (prin)
202 1240 2960 2880 960 60px
t0,2,3{4,3,3}
- 60px
{4}×{3}
60px
{ }×t{3,3}

2t{3,3,3}
42 (1,1,1,1,1)
+ (0,0,1,2,3)√2
Stericantitruncated 5-orthoplex
242 2320 5920 5760 1920 60px
t0,2,3{4,3,3}
60px
rr{4,3}×{ }
60px
{6}×{4}
60px
{ }×tr{3,3}

tr{3,3,3}
43 (1,1,1,1,1)
+ (0,1,1,1,1)√2
Truncated 5-cube
truncated penteract (tan)
42 200 400 400 160
Tetrah.pyr
60px
t{4,3,3}
- - - 60px
{3,3,3}
44 (1,1,1,1,1)
+ (0,1,1,1,2)√2
Steritruncated 5-cube
242 1600 2960 2240 640 60px
t{4,3,3}
60px
t{4,3}×{ }
60px
{8}×{3}
60px
{ }×{3,3}
60px
t0,3{3,3,3}
45 (1,1,1,1,1)
+ (0,1,1,2,2)√2
Runcitruncated 5-cube
prismatotruncated penteract (pattin)
202 1560 3760 3360 960 60px
t0,1,3{4,3,3}
- 60px
{8}×{3}
60px
{ }×r{3,3}
60px
rr{3,3,3}
46 (1,1,1,1,1)
+ (0,1,1,2,3)√2
Steriruncitruncated 5-cube
(Steriruncitruncated 5-orthoplex)
242 2160 5760 5760 1920 60px
t0,1,3{4,3,3}
60px
t{4,3}×{ }
60px
{8}×{6}
60px
{ }×t{3,3}

t0,1,3{3,3,3}
47 (1,1,1,1,1)
+ (0,1,2,2,2)√2
Cantitruncated 5-cube
great rhombated penteract (girn)
122 680 1520 1600 640 60px
tr{4,3,3}
- - 60px
{ }×{3,3}
60px
t{3,3,3}
48 (1,1,1,1,1)
+ (0,1,2,2,3)√2
Stericantitruncated 5-cube
celligreatorhombated penteract (cogrin)
242 2400 6000 5760 1920 60px
tr{4,3,3}
60px
tr{4,3}×{ }
60px
{8}×{3}
60px
{ }×rr{3,3}

t0,1,3{3,3,3}
49 (1,1,1,1,1)
+ (0,1,2,3,3)√2
Runcicantitruncated 5-cube
great prismated penteract (gippin)
202 1560 4240 4800 1920 60px
t0,1,2,3{4,3,3}
- 60px
{8}×{3}
60px
{ }×t{3,3}

tr{3,3,3}
50 (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}
60px
tr{4,3}×{ }
60px
tr{4,3}×{ }
60px
{8}×{6}
60px
{ }×tr{3,3}

t0,1,2,3{3,3,3}
51 5-demicube
hemipenteract (hin)
=
26 120 160 80 16
r{3,3,3}

h{4,3,3}
- - - - (16)

{3,3,3}
52 Cantic 5-cube
Truncated hemipenteract (thin)
=
42 280 640 560 160
h2{4,3,3}
- - - (16)

r{3,3,3}
(16)

t{3,3,3}
53 Runcic 5-cube
Small rhombated hemipenteract (sirhin)
=
42 360 880 720 160
h3{4,3,3}
- - - (16)

r{3,3,3}
(16)

rr{3,3,3}
54 Steric 5-cube
Small prismated hemipenteract (siphin)
=
82 480 720 400 80
h{4,3,3}

h{4,3}×{}
- - (16)

{3,3,3}
(16)

t0,3{3,3,3}
55 Runcicantic 5-cube
Great rhombated hemipenteract (girhin)
=
42 360 1040 1200 480
h2,3{4,3,3}
- - - (16)

2t{3,3,3}
(16)

tr{3,3,3}
56 Stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
=
82 720 1840 1680 480
h2{4,3,3}

h2{4,3}×{}
- - (16)

rr{3,3,3}
(16)

t0,1,3{3,3,3}
57 Steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
=
82 560 1280 1120 320
h3{4,3,3}

h{4,3}×{}
- - (16)

t{3,3,3}
(16)

t0,1,3{3,3,3}
58 Steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
=
82 720 2080 2400 960
h2,3{4,3,3}

h2{4,3}×{}
- - (16)

tr{3,3,3}
(16)

t0,1,2,3{3,3,3}
Nonuniform Alternated runcicantitruncated 5-orthoplex
Snub hemipenteract (snahin)
=
1122 6240 10880 6720 960
sr{3,3,4}
sr{2,3,4} sr{3,2,4} - ht0,1,2,3{3,3,3} (960)

Irr. {3,3,3}
Nonuniform Edge-snub 5-orthoplex
Pyritosnub penteract (pysnan)
1202 7920 15360 10560 1920 sr3{3,3,4} sr3{2,3,4} sr3{3,2,4}
s{3,3}×{ }
ht0,1,2,3{3,3,3} (960)

Irr. {3,3}×{ }
Nonuniform Snub 5-cube
Snub penteract (snan)
2162 12240 21600 13440 960 ht0,1,2,3{3,3,4} ht0,1,2,3{2,3,4} ht0,1,2,3{3,2,4} ht0,1,2,3{3,3,2} ht0,1,2,3{3,3,3} (1920)

Irr. {3,3,3}

### The D5 family

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 element are identical and the symmetry doubles: the relations are ... = .... and ... = ..., 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 ... = ... duplicating uniform 5-polytopes 51 through 58 above.

# Coxeter diagram
Schläfli symbol symbols
Johnson and Bowers names
Element counts Vertex
figure
Facets by location: [31,2,1]
4 3 2 1 0
[3,3,3]
(16)

[31,1,1]
(10)

[3,3]×[ ]
(40)

[ ]×[3]×[ ]
(80)

[3,3,3]
(16)
Alt
[51] =
h{4,3,3,3}, 5-demicube
Hemipenteract (hin)
26 120 160 80 16
r{3,3,3}

{3,3,3}

h{4,3,3}
- - -
[52] =
h2{4,3,3,3}, cantic 5-cube
Truncated hemipenteract (thin)
42 280 640 560 160
t{3,3,3}

h2{4,3,3}
- -
r{3,3,3}
[53] =
h3{4,3,3,3}, runcic 5-cube
Small rhombated hemipenteract (sirhin)
42 360 880 720 160
rr{3,3,3}

h3{4,3,3}
- -
r{3,3,3}
[54] =
h4{4,3,3,3}, steric 5-cube
Small prismated hemipenteract (siphin)
82 480 720 400 80
t0,3{3,3,3}

h{4,3,3}

h{4,3}×{}
-
{3,3,3}
[55] =
h2,3{4,3,3,3}, runcicantic 5-cube
Great rhombated hemipenteract (girhin)
42 360 1040 1200 480
2t{3,3,3}

h2,3{4,3,3}
- -
tr{3,3,3}
[56] =
h2,4{4,3,3,3}, stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
82 720 1840 1680 480
t0,1,3{3,3,3}

h2{4,3,3}

h2{4,3}×{}
-
rr{3,3,3}
[57] =
h3,4{4,3,3,3}, steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
82 560 1280 1120 320
t0,1,3{3,3,3}

h3{4,3,3}

h{4,3}×{}
-
t{3,3,3}
[58] =
h2,3,4{4,3,3,3}, steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
82 720 2080 2400 960
t0,1,2,3{3,3,3}

h2,3{4,3,3}

h2{4,3}×{}
-
tr{3,3,3}
Nonuniform =
ht0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplex
Snub hemipenteract (snahin)
1122 6240 10880 6720 960 ht0,1,2,3{3,3,3}
sr{3,3,4}
sr{2,3,4} sr{3,2,4} ht0,1,2,3{3,3,3} (960)

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
Facets Cells Faces Edges Vertices
59 = {3,3,3}×{ }
5-cell prism (penp)
7 20 30 25 10
60 = r{3,3,3}×{ }
Rectified 5-cell prism (rappip)
12 50 90 70 20
61 = t{3,3,3}×{ }
Truncated 5-cell prism (tippip)
12 50 100 100 40
62 = rr{3,3,3}×{ }
Cantellated 5-cell prism (srippip)
22 120 250 210 60
63 = t0,3{3,3,3}×{ }
Runcinated 5-cell prism (spiddip)
32 130 200 140 40
64 = 2t{3,3,3}×{ }
Bitruncated 5-cell prism (decap)
12 60 140 150 60
65 = tr{3,3,3}×{ }
Cantitruncated 5-cell prism (grippip)
22 120 280 300 120
66 = t0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism (prippip)
32 180 390 360 120
67 = t0,1,2,3{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
Facets Cells Faces Edges Vertices
[16] = {4,3,3}×{ }
Tesseractic prism (pent)
(Same as 5-cube)
10 40 80 80 32
68 = r{4,3,3}×{ }
Rectified tesseractic prism (rittip)
26 136 272 224 64
69 = t{4,3,3}×{ }
Truncated tesseractic prism (tattip)
26 136 304 320 128
70 = rr{4,3,3}×{ }
Cantellated tesseractic prism (srittip)
58 360 784 672 192
71 = t0,3{4,3,3}×{ }
Runcinated tesseractic prism (sidpithip)
82 368 608 448 128
72 = 2t{4,3,3}×{ }
Bitruncated tesseractic prism (tahp)
26 168 432 480 192
73 = tr{4,3,3}×{ }
Cantitruncated tesseractic prism (grittip)
58 360 880 960 384
74 = t0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism (prohp)
82 528 1216 1152 384
75 = t0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism (gidpithip)
82 624 1696 1920 768
76 = {3,3,4}×{ }
16-cell prism (hexip)
18 64 88 56 16
77 = r{3,3,4}×{ }
Rectified 16-cell prism (icope)
(Same as 24-cell prism)
26 144 288 216 48
78 = t{3,3,4}×{ }
Truncated 16-cell prism (thexip)
26 144 312 288 96
79 = rr{3,3,4}×{ }
Cantellated 16-cell prism (ricope)
(Same as rectified 24-cell prism)
50 336 768 672 192
80 = tr{3,3,4}×{ }
Cantitruncated 16-cell prism (ticope)
(Same as truncated 24-cell prism)
50 336 864 960 384
81 = t0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism (prittip)
82 528 1216 1152 384
82 = sr{3,3,4}×{ }
146 768 1392 960 192
Nonuniform
rectified tesseractic alterprism (rita)
50 288 464 288 64
Nonuniform
truncated 16-cell alterprism (thexa)
26 168 384 336 96
Nonuniform
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
Facets Cells Faces Edges Vertices
[77] = {3,4,3}×{ }
24-cell prism (icope)
26 144 288 216 48
[79] = r{3,4,3}×{ }
rectified 24-cell prism (ricope)
50 336 768 672 192
[80] = t{3,4,3}×{ }
truncated 24-cell prism (ticope)
50 336 864 960 384
83 = rr{3,4,3}×{ }
cantellated 24-cell prism (sricope)
146 1008 2304 2016 576
84 = t0,3{3,4,3}×{ }
runcinated 24-cell prism (spiccup)
242 1152 1920 1296 288
85 = 2t{3,4,3}×{ }
bitruncated 24-cell prism (contip)
50 432 1248 1440 576
86 = tr{3,4,3}×{ }
cantitruncated 24-cell prism (gricope)
146 1008 2592 2880 1152
87 = t0,1,3{3,4,3}×{ }
runcitruncated 24-cell prism (pricope)
242 1584 3648 3456 1152
88 = t0,1,2,3{3,4,3}×{ }
omnitruncated 24-cell prism (gippiccup)
242 1872 5088 5760 2304
[82] = s{3,4,3}×{ }
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
Facets Cells Faces Edges Vertices
89 = {5,3,3}×{ }
120-cell prism (hipe)
122 960 2640 3000 1200
90 = r{5,3,3}×{ }
Rectified 120-cell prism (rahipe)
722 4560 9840 8400 2400
91 = t{5,3,3}×{ }
Truncated 120-cell prism (thipe)
722 4560 11040 12000 4800
92 = rr{5,3,3}×{ }
Cantellated 120-cell prism (srahip)
1922 12960 29040 25200 7200
93 = t0,3{5,3,3}×{ }
Runcinated 120-cell prism (sidpixhip)
2642 12720 22080 16800 4800
94 = 2t{5,3,3}×{ }
Bitruncated 120-cell prism (xhip)
722 5760 15840 18000 7200
95 = tr{5,3,3}×{ }
Cantitruncated 120-cell prism (grahip)
1922 12960 32640 36000 14400
96 = t0,1,3{5,3,3}×{ }
Runcitruncated 120-cell prism (prixip)
2642 18720 44880 43200 14400
97 = t0,1,2,3{5,3,3}×{ }
Omnitruncated 120-cell prism (gidpixhip)
2642 22320 62880 72000 28800
98 = {3,3,5}×{ }
600-cell prism (exip)
602 2400 3120 1560 240
99 = r{3,3,5}×{ }
Rectified 600-cell prism (roxip)
722 5040 10800 7920 1440
100 = t{3,3,5}×{ }
Truncated 600-cell prism (texip)
722 5040 11520 10080 2880
101 = rr{3,3,5}×{ }
Cantellated 600-cell prism (srixip)
1442 11520 28080 25200 7200
102 = tr{3,3,5}×{ }
Cantitruncated 600-cell prism (grixip)
1442 11520 31680 36000 14400
103 = t0,1,3{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.

Coxeter diagram Names Element counts
4-faces Cells Faces Edges Vertices
{p}×{q}×{ }[9] p+q+2 3pq+3p+3q 4pq+2p+2q 5pq 2pq
{p}2×{ } 2(p+1) 3p(p+1) 4p(p+1) 5p2 2p2
{3}2×{ } 8 36 48 45 18
{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 , 20 pentagonal antiprism prisms 50px, and 300 tetrahedral prisms ).

# Name Element counts
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 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 diagram Description
Parent t0{p,q,r,s} {p,q,r,s} Any regular 5-polytope
Rectified t1{p,q,r,s} r{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} 2r{p,q,r,s} Birectification reduces faces to points, cells to their duals.
Trirectified t3{p,q,r,s} 3r{p,q,r,s} Trirectification reduces cells to points. (Dual rectification)
Truncated t0,1{p,q,r,s} t{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} rr{p,q,r,s} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place.
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} 2r2r{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 5-polytopes.)
Omnitruncated t0,1,2,3,4{p,q,r,s} All four operators, truncation, cantellation, runcination, and sterication are applied.
Half h{2p,3,q,r} Alternation, same as
Cantic h2{2p,3,q,r} Same as
Runcic h3{2p,3,q,r} Same as
Runcicantic h2,3{2p,3,q,r} Same as
Steric h4{2p,3,q,r} Same as
Steriruncic h3,4{2p,3,q,r} Same as
Stericantic h2,4{2p,3,q,r} Same as
Steriruncicantic h2,3,4{2p,3,q,r} Same as
Snub s{p,2q,r,s} Alternated truncation
Snub rectified sr{p,q,2r,s} Alternated truncated rectification
ht0,1,2,3{p,q,r,s} Alternated runcicantitruncation
Full snub ht0,1,2,3,4{p,q,r,s} Alternated omnitruncation

## Regular and uniform honeycombs

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 $\displaystyle{ {\tilde{A}}_4 }$ [3[5]] [(3,3,3,3,3)] 7
2 $\displaystyle{ {\tilde{C}}_4 }$ [4,3,3,4] 19
3 $\displaystyle{ {\tilde{B}}_4 }$ [4,3,31,1] [4,3,3,4,1+] = 23 (8 new)
4 $\displaystyle{ {\tilde{D}}_4 }$ [31,1,1,1] [1+,4,3,3,4,1+] = 9 (0 new)
5 $\displaystyle{ {\tilde{F}}_4 }$ [3,4,3,3] 31 (21 new)

There are three regular honeycombs of Euclidean 4-space:

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, =
• There are 7 uniquely ringed forms from the $\displaystyle{ {\tilde{A}}_4 }$, family, all new, including:
• 4-simplex honeycomb
• Truncated 4-simplex honeycomb
• Omnitruncated 4-simplex honeycomb
• There are 9 uniquely ringed forms in the $\displaystyle{ {\tilde{D}}_4 }$: [31,1,1,1] family, two new ones, including the quarter tesseractic honeycomb, = , and the bitruncated tesseractic honeycomb, = .

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 $\displaystyle{ {\tilde{C}}_3 }$×$\displaystyle{ {\tilde{I}}_1 }$ [4,3,4,2,∞]
2 $\displaystyle{ {\tilde{B}}_3 }$×$\displaystyle{ {\tilde{I}}_1 }$ [4,31,1,2,∞]
3 $\displaystyle{ {\tilde{A}}_3 }$×$\displaystyle{ {\tilde{I}}_1 }$ [3[4],2,∞]
4 $\displaystyle{ {\tilde{C}}_2 }$×$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [4,4,2,∞,2,∞]
5 $\displaystyle{ {\tilde{H}}_2 }$×$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [6,3,2,∞,2,∞]
6 $\displaystyle{ {\tilde{A}}_2 }$×$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [3[3],2,∞,2,∞]
7 $\displaystyle{ {\tilde{I}}_1 }$×$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [∞,2,∞,2,∞,2,∞]
8 $\displaystyle{ {\tilde{A}}_2 }$x$\displaystyle{ {\tilde{A}}_2 }$ [3[3],2,3[3]]
9 $\displaystyle{ {\tilde{A}}_2 }$×$\displaystyle{ {\tilde{B}}_2 }$ [3[3],2,4,4]
10 $\displaystyle{ {\tilde{A}}_2 }$×$\displaystyle{ {\tilde{G}}_2 }$ [3[3],2,6,3]
11 $\displaystyle{ {\tilde{B}}_2 }$×$\displaystyle{ {\tilde{B}}_2 }$ [4,4,2,4,4]
12 $\displaystyle{ {\tilde{B}}_2 }$×$\displaystyle{ {\tilde{G}}_2 }$ [4,4,2,6,3]
13 $\displaystyle{ {\tilde{G}}_2 }$×$\displaystyle{ {\tilde{G}}_2 }$ [6,3,2,6,3]

### 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.

 $\displaystyle{ {\widehat{AF}}_4 }$ = [(3,3,3,3,4)]: $\displaystyle{ {\bar{DH}}_4 }$ = [5,3,31,1]: $\displaystyle{ {\bar{H}}_4 }$ = [3,3,3,5]: $\displaystyle{ {\bar{BH}}_4 }$ = [4,3,3,5]: $\displaystyle{ {\bar{K}}_4 }$ = [5,3,3,5]:

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} {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} {5,3,3} {5,3} {5} {3} {3,3} {3,3,3} {3,3,3,5}
Order-5 tesseractic (pitest) {4,3,3,5} {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} {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} {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} {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} {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} {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} {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.

 $\displaystyle{ {\bar{P}}_4 }$ = [3,3[4]]: $\displaystyle{ {\bar{BP}}_4 }$ = [4,3[4]]: $\displaystyle{ {\bar{FR}}_4 }$ = [(3,3,4,3,4)]: $\displaystyle{ {\bar{DP}}_4 }$ = [3[3]×[]]: $\displaystyle{ {\bar{N}}_4 }$ = [4,/3\,3,4]: $\displaystyle{ {\bar{O}}_4 }$ = [3,4,31,1]: $\displaystyle{ {\bar{S}}_4 }$ = [4,32,1]: $\displaystyle{ {\bar{M}}_4 }$ = [4,31,1,1]: $\displaystyle{ {\bar{R}}_4 }$ = [3,4,3,4]:

## Notes

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. Bowers, Jonathan (2000). "Uniform Polychora". in Reza Sarhagi. Bridges Conference. pp. 239–246.
4. Uniform Polytera, Jonathan Bowers
5. Uniform polytope
6. ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, retrieved 2016-10-04
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. Regular polytopes, p.297. Table IV, Fundamental regions for irreducible groups generated by reflections.
10. Regular and Semiregular polytopes, II, pp.298-302 Four-dimensional honeycombs
11. 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, edited 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, 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]