Uniform 5-polytope

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 Unsolved problem in mathematics:Find the complete set of uniform 5-polytopes(more unsolved problems in mathematics)

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 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 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

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 or more dimensions.

Convex uniform 5-polytopes

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.[citation needed]

Symmetry of uniform 5-polytopes in four dimensions

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.

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.

Fundamental families[2]
Group
symbol
Order Coxeter
graph
Bracket
notation
Commutator
subgroup
Coxeter
number

(h)
Reflections
m=5/2 h[3]
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 (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: BC5 [4,33]
• 31 uniform 5-polytopes
• Demihypercube D5/E5 family: [32,1,1]
• 23 uniform 5-polytopes (8 unique)
• Prisms and duoprisms:
• 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)
1 (0,0,0,0,0,1) or (0,1,1,1,1,1) 5-simplex
hexateron (hix)
6 15 20 15 6
{3,3,3}
(5)

{3,3,3}
- - - -
2 (0,0,0,0,1,1) or (0,0,1,1,1,1) Rectified 5-simplex
rectified hexateron (rix)
12 45 80 60 15
t{3,3}×{ }
(4)

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

{3,3,3}
3 (0,0,0,0,1,2) or (0,1,2,2,2,2) Truncated 5-simplex
truncated hexateron (tix)
12 45 80 75 30
Tetrah.pyr
(4)

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

{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
(3)

rr{3,3,3}
- - (1)
×
{ }×{3,3}
(1)

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 (3)

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

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 (2)

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

{3}×{3}
(3)
×
{ }×r{3,3}
(1)

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}

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

r{3,3,3}
- - - (3)

r{3,3,3}
14 (0,0,1,1,2,2) Bicantellated 5-simplex
small birhombated dodecateron (sibrid)
32 180 420 360 90 (2)

rr{3,3,3}
- (8)

{3}×{3}
- (2)

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
(1)

{3,3,3}
(4)
×
{ }×{3,3}
(6)

{3}×{3}
(4)
×
{ }×{3,3}
(1)

{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}
(1)

t0,1,2,3{3,3,3}
(1)
×
{ }×tr{3,3}
(1)

{6}×{6}
(1)
×
{ }×tr{3,3}
(1)

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

The B5 family

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

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

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)
20 (0,0,0,0,1)√2 5-orthoplex (tac)
32 80 80 40 10
{3,3,4}

{3,3,3}
- - - -
21 (0,0,0,1,1)√2 Rectified 5-orthoplex (rat)
42 240 400 240 40
{ }×{3,4}

{3,3,4}
- - -
r{3,3,3}
22 (0,0,0,1,2)√2 Truncated 5-orthoplex (tot)
42 240 400 280 80
(Octah.pyr)

t{3,3,3}

{3,3,3}
- - -
23 (0,0,1,1,1)√2 Birectified 5-cube (nit)
(Birectified 5-orthoplex)
42 280 640 480 80
{4}×{3}

r{3,3,4}
- - -
r{3,3,3}
24 (0,0,1,1,2)√2 Cantellated 5-orthoplex (sart)
82 640 1520 1200 240
Prism-wedge
r{3,3,4} { }×{3,4} - -
rr{3,3,3}
25 (0,0,1,2,2)√2 Bitruncated 5-orthoplex (bittit)
42 280 720 720 240 t{3,3,4} - - -
2t{3,3,3}
26 (0,0,1,2,3)√2 Cantitruncated 5-orthoplex (gart)
82 640 1520 1440 480 rr{3,3,4} { }×r{3,4}
{6}×{4}
-
t0,1,3{3,3,3}
27 (0,1,1,1,1)√2 Rectified 5-cube (rin)
42 200 400 320 80
{3,3}×{ }

r{4,3,3}
- - -
{3,3,3}
28 (0,1,1,1,2)√2 Runcinated 5-orthoplex (spat)
162 1200 2160 1440 320 r{4,3,3} -
{3}×{4}

t0,3{3,3,3}
29 (0,1,1,2,2)√2 Bicantellated 5-cube (sibrant)
(Bicantellated 5-orthoplex)
122 840 2160 1920 480
rr{4,3,3}
-
{4}×{3}
-
rr{3,3,3}
30 (0,1,1,2,3)√2 Runcitruncated 5-orthoplex (pattit)
162 1440 3680 3360 960 rr{3,3,4} { }×r{3,4}
{6}×{4}
-
t0,1,3{3,3,3}
31 (0,1,2,2,2)√2 Bitruncated 5-cube (tan)
42 280 720 800 320
2t{4,3,3}
- - -
t{3,3,3}
32 (0,1,2,2,3)√2 Runcicantellated 5-orthoplex (pirt)
162 1200 2960 2880 960 { }×t{3,4} 2t{3,3,4}
{3}×{4}
-
t0,1,3{3,3,3}
33 (0,1,2,3,3)√2 Bicantitruncated 5-cube (gibrant)
(Bicantitruncated 5-orthoplex)
122 840 2160 2400 960
rr{4,3,3}
-
{4}×{3}
-
rr{3,3,3}
34 (0,1,2,3,4)√2 Runcicantitruncated 5-orthoplex (gippit)
162 1440 4160 4800 1920 tr{3,3,4} { }×t{3,4}
{6}×{4}
-
t0,1,2,3{3,3,3}
35 (1,1,1,1,1) 5-cube (pent)
10 40 80 80 32
{3,3,3}

{4,3,3}
- - - -
36 (1,1,1,1,1)
+ (0,0,0,0,1)√2
Stericated 5-cube (scant)
(Stericated 5-orthoplex)
242 800 1040 640 160
Tetr.antiprm

{4,3,3}

{4,3}×{ }

{4}×{3}

{ }×{3,3}

{3,3,3}
37 (1,1,1,1,1)
+ (0,0,0,1,1)√2
Runcinated 5-cube (span)
202 1240 2160 1440 320
t0,3{4,3,3}
-
{4}×{3}

{ }×r{3,3}

{3,3,3}
38 (1,1,1,1,1)
+ (0,0,0,1,2)√2
Steritruncated 5-orthoplex (cappin)
242 1520 2880 2240 640 t0,3{3,3,4} { }×{4,3} - -
t{3,3,3}
39 (1,1,1,1,1)
+ (0,0,1,1,1)√2
Cantellated 5-cube (sirn)
122 680 1520 1280 320
Prism-wedge

rr{4,3,3}
- -
{ }×{3,3}

r{3,3,3}
40 (1,1,1,1,1)
+ (0,0,1,1,2)√2
Stericantellated 5-cube (carnit)
(Stericantellated 5-orthoplex)
242 2080 4720 3840 960
rr{4,3,3}

rr{4,3}×{ }

{4}×{3}

{ }×rr{3,3}

rr{3,3,3}
41 (1,1,1,1,1)
+ (0,0,1,2,2)√2
Runcicantellated 5-cube (prin)
202 1240 2960 2880 960
t0,1,3{4,3,3}
-
{4}×{3}

{ }×t{3,3}

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

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

{6}×{4}

{ }×t{3,3}

tr{3,3,3}
43 (1,1,1,1,1)
+ (0,1,1,1,1)√2
Truncated 5-cube (tan)
42 200 400 400 160
Tetrah.pyr

t{4,3,3}
- - -
{3,3,3}
44 (1,1,1,1,1)
+ (0,1,1,1,2)√2
Steritruncated 5-cube (capt)
242 1600 2960 2240 640
t{4,3,3}

t{4,3}×{ }

{8}×{3}

{ }×{3,3}

t0,3{3,3,3}
45 (1,1,1,1,1)
+ (0,1,1,2,2)√2
Runcitruncated 5-cube (pattin)
202 1560 3760 3360 960
t0,1,3{4,3,3}
{ }×t{4,3}
{6}×{8}
{ }×t{3,3} t0,1,3{3,3,3}]]
46 (1,1,1,1,1)
+ (0,1,1,2,3)√2
Steriruncitruncated 5-cube (captint)
(Steriruncitruncated 5-orthoplex)
242 2160 5760 5760 1920
t0,1,3{4,3,3}

t{4,3}×{ }

{8}×{6}

{ }×t{3,3}

t0,1,3{3,3,3}
47 (1,1,1,1,1)
+ (0,1,2,2,2)√2
Cantitruncated 5-cube (girn)
122 680 1520 1600 640
tr{4,3,3}
- -
{ }×{3,3}

t{3,3,3}
48 (1,1,1,1,1)
+ (0,1,2,2,3)√2
Stericantitruncated 5-cube (cogrin)
242 2400 6000 5760 1920
tr{4,3,3}

tr{4,3}×{ }

{8}×{3}

{ }×t0,2{3,3}

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

{ }×t{3,3}

tr{3,3,3}
50 (1,1,1,1,1)
+ (0,1,2,3,4)√2
Omnitruncated 5-cube (gacnet)
(omnitruncated 5-orthoplex)
242 2640 8160 9600 3840
Irr. {3,3,3}

tr{4,3}×{ }

tr{4,3}×{ }

{8}×{6}

{ }×tr{3,3}

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

The D5 family

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

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

# 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)
51 =
h{4,3,3,3}, 5-demicube
Hemipenteract (hin)
26 120 160 80 16
t1{3,3,3}
{3,3,3} t0(111) - - -
52 =
h2{4,3,3,3}, cantic 5-cube
Truncated hemipenteract (thin)
42 280 640 560 160
53 =
h3{4,3,3,3}, runcic 5-cube
Small rhombated hemipenteract (sirhin)
42 360 880 720 160
54 =
h4{4,3,3,3}, steric 5-cube
Small prismated hemipenteract (siphin)
82 480 720 400 80
55 =
h2,3{4,3,3,3}, runcicantic 5-cube
Great rhombated hemipenteract (girhin)
42 360 1040 1200 480
56 =
h2,4{4,3,3,3}, stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
82 720 1840 1680 480
57 =
h3,4{4,3,3,3}, steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
82 560 1280 1120 320
58 =
h2,3,4{4,3,3,3}, steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
82 720 2080 2400 960

Uniform prismatic forms

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

A4 × A1

This prismatic family has 9 forms:

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

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
59 = {3,3,3}×{ }
5-cell prism
7 20 30 25 10
60 = r{3,3,3}×{ }
Rectified 5-cell prism
12 50 90 70 20
61 = t{3,3,3}×{ }
Truncated 5-cell prism
12 50 100 100 40
62 = rr{3,3,3}×{ }
Cantellated 5-cell prism
22 120 250 210 60
63 = t0,3{3,3,3}×{ }
Runcinated 5-cell prism
32 130 200 140 40
64 = 2t{3,3,3}×{ }
Bitruncated 5-cell prism
12 60 140 150 60
65 = tr{3,3,3}×{ }
Cantitruncated 5-cell prism
22 120 280 300 120
66 = t0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism
32 180 390 360 120
67 = t0,1,2,3{3,3,3}×{ }
Omnitruncated 5-cell prism
32 210 540 600 240

B4 × A1

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

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

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
[16] = {4,3,3}×{ }
Tesseractic prism
(Same as 5-cube)
10 40 80 80 32
68 = r{4,3,3}×{ }
Rectified tesseractic prism
26 136 272 224 64
69 = t{4,3,3}×{ }
Truncated tesseractic prism
26 136 304 320 128
70 = rr{4,3,3}×{ }
Cantellated tesseractic prism
58 360 784 672 192
71 = t0,3{4,3,3}×{ }
Runcinated tesseractic prism
82 368 608 448 128
72 = 2t{4,3,3}×{ }
Bitruncated tesseractic prism
26 168 432 480 192
73 = tr{4,3,3}×{ }
Cantitruncated tesseractic prism
58 360 880 960 384
74 = t0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism
82 528 1216 1152 384
75 = t0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism
82 624 1696 1920 768
76 = {3,3,4}×{ }
16-cell prism
18 64 88 56 16
77 = r{3,3,4}×{ }
Rectified 16-cell prism
(Same as 24-cell prism)
26 144 288 216 48
78 = t{3,3,4}×{ }
Truncated 16-cell prism
26 144 312 288 96
79 = rr{3,3,4}×{ }
Cantellated 16-cell prism
(Same as rectified 24-cell prism)
50 336 768 672 192
80 = tr{3,3,4}×{ }
Cantitruncated 16-cell prism
(Same as truncated 24-cell prism)
50 336 864 960 384
81 = t0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism
82 528 1216 1152 384
82 = sr{3,3,4}×{ }
snub 24-cell prism
146 768 1392 960 192

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
26 144 288 216 48
[79] = r{3,4,3}×{ }
rectified 24-cell prism
50 336 768 672 192
[80] = t{3,4,3}×{ }
truncated 24-cell prism
50 336 864 960 384
83 = rr{3,4,3}×{ }
cantellated 24-cell prism
146 1008 2304 2016 576
84 = t0,3{3,4,3}×{ }
runcinated 24-cell prism
242 1152 1920 1296 288
85 = 2t{3,4,3}×{ }
bitruncated 24-cell prism
50 432 1248 1440 576
86 = tr{3,4,3}×{ }
cantitruncated 24-cell prism
146 1008 2592 2880 1152
87 = t0,1,3{3,4,3}×{ }
runcitruncated 24-cell prism
242 1584 3648 3456 1152
88 = t0,1,2,3{3,4,3}×{ }
omnitruncated 24-cell prism
242 1872 5088 5760 2304
[82] = s{3,4,3}×{ }
snub 24-cell prism
146 768 1392 960 192

H4 × A1

This prismatic family has 15 forms:

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

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
89 = {5,3,3}×{ }
120-cell prism
122 960 2640 3000 1200
90 = r{5,3,3}×{ }
Rectified 120-cell prism
722 4560 9840 8400 2400
91 = t{5,3,3}×{ }
Truncated 120-cell prism
722 4560 11040 12000 4800
92 = rr{5,3,3}×{ }
Cantellated 120-cell prism
1922 12960 29040 25200 7200
93 = t0,3{5,3,3}×{ }
Runcinated 120-cell prism
2642 12720 22080 16800 4800
94 = 2t{5,3,3}×{ }
Bitruncated 120-cell prism
722 5760 15840 18000 7200
95 = tr{5,3,3}×{ }
Cantitruncated 120-cell prism
1922 12960 32640 36000 14400
96 = t0,1,3{5,3,3}×{ }
Runcitruncated 120-cell prism
2642 18720 44880 43200 14400
97 = t0,1,2,3{5,3,3}×{ }
Omnitruncated 120-cell prism
2642 22320 62880 72000 28800
98 = {3,3,5}×{ }
600-cell prism
602 2400 3120 1560 240
99 = r{3,3,5}×{ }
Rectified 600-cell prism
722 5040 10800 7920 1440
100 = t{3,3,5}×{ }
Truncated 600-cell prism
722 5040 11520 10080 2880
101 = rr{3,3,5}×{ }
Cantellated 600-cell prism
1442 11520 28080 25200 7200
102 = tr{3,3,5}×{ }
Cantitruncated 600-cell prism
1442 11520 31680 36000 14400
103 = t0,1,3{3,3,5}×{ }
Runcitruncated 600-cell prism
2642 18720 44880 43200 14400

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 , and 300 tetrahedral prisms ).

# Name Element counts
Facets Cells Faces Edges Vertices
104 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 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
Runcisteric 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.[4][5]

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:
• 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]

Compact regular tessellations of hyperbolic 4-space

There are five kinds of convex regular honeycombs and four kinds of star-honeycombs in H4 space:[6]

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 {3,3,3,5} {3,3,3} {3,3} {3} {5} {3,5} {3,3,5} {5,3,3,3}
Order-3 120-cell {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-4 120-cell {5,3,3,4} {5,3,3} {5,3} {5} {4} {3,4} {3,3,4} {4,3,3,5}
Order-5 120-cell {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 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}

Regular and uniform hyperbolic honeycombs

There are 5 compact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams. There are also 9 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.

 ${\displaystyle {\widehat {AF}}_{4}}$ = [(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]:
 ${\displaystyle {\bar {P}}_{4}}$ = [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. ^ Regular and semi-regular polytopes III, p.315 Three finite groups of 5-dimensions
3. ^ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
4. ^ Regular polytopes, p.297. Table IV, Fundamental regions for irreducible groups generated by reflections.
5. ^ Regular and Semiregular polytopes, II, pp.298-302 Four-dimensional honeycombs
6. ^ 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]