Uniform 5-polytope

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.

Graphs of regular and uniform polytopes.

5-simplex
Rectified 5-simplex
Truncated 5-simplex

Cantellated 5-simplex
Runcinated 5-simplex
Stericated 5-simplex

5-orthoplex
Truncated 5-orthoplex
Rectified 5-orthoplex

Cantellated 5-orthoplex
Runcinated 5-orthoplex

Cantellated 5-cube
Runcinated 5-cube
Stericated 5-cube

5-cube
Truncated 5-cube
Rectified 5-cube

5-demicube
Truncated 5-demicube

Cantellated 5-demicube
Runcinated 5-demicube

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

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 5,6,7,8,9,10,11 and 12 dimensions.

Convex uniform 5-polytopes
Unsolved problem in mathematics:
What is the complete set of uniform 5-polytopes?
(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[2]
Group
symbol		OrderCoxeter
graph		Bracket
notation		Commutator
subgroup		Coxeter
number
(h)		Reflections
m=5/2 h[3]
A5 720[3,3,3,3][3,3,3,3]+615
D5 1920[3,3,31,1][3,3,31,1]+820
B5 3840[4,3,3,3]105 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 6p
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

In addition there are:

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

The A5 family
Further information: A5 polytope

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
small cellated dodecateron (scad)
 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
great cellated dodecateron (gocad)
 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
Further information: B5 polytope

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]
43210
[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)√25-orthoplex (tac)
3280804010
{3,3,4}
{3,3,3}		----
21 (0,0,0,1,1)√2Rectified 5-orthoplex (rat)
4224040024040
{ }×{3,4}

{3,3,4}		---
r{3,3,3}
22 (0,0,0,1,2)√2Truncated 5-orthoplex (tot)
4224040028080
(Octah.pyr)
t{3,3,3}
{3,3,3}		---
23 (0,0,1,1,1)√2Birectified 5-cube (nit)
(Birectified 5-orthoplex)
4228064048080
{4}×{3}
r{3,3,4}		---
r{3,3,3}
24 (0,0,1,1,2)√2Cantellated 5-orthoplex (sart)
8264015201200240
Prism-wedge		r{3,3,4}{ }×{3,4}--
rr{3,3,3}
25 (0,0,1,2,2)√2Bitruncated 5-orthoplex (bittit)
42280720720240 t{3,3,4}---
2t{3,3,3}
26 (0,0,1,2,3)√2Cantitruncated 5-orthoplex (gart)
8264015201440480 rr{3,3,4}{ }×r{3,4}
{6}×{4}		-
t0,1,3{3,3,3}
27 (0,1,1,1,1)√2Rectified 5-cube (rin)
4220040032080
{3,3}×{ }
r{4,3,3}		---
{3,3,3}
28 (0,1,1,1,2)√2Runcinated 5-orthoplex (spat)
162120021601440320 r{4,3,3}-
{3}×{4}
t0,3{3,3,3}
29 (0,1,1,2,2)√2Bicantellated 5-cube (sibrant)
(Bicantellated 5-orthoplex)
12284021601920480
rr{4,3,3}		-
{4}×{3}		-
rr{3,3,3}
30 (0,1,1,2,3)√2Runcitruncated 5-orthoplex (pattit)
162144036803360960 rr{3,3,4}{ }×r{3,4}
{6}×{4}		-
t0,1,3{3,3,3}
31 (0,1,2,2,2)√2Bitruncated 5-cube (tan)
42280720800320
2t{4,3,3}		---
t{3,3,3}
32 (0,1,2,2,3)√2Runcicantellated 5-orthoplex (pirt)
162120029602880960 { }×t{3,4}2t{3,3,4}
{3}×{4}		-
t0,1,3{3,3,3}
33 (0,1,2,3,3)√2Bicantitruncated 5-cube (gibrant)
(Bicantitruncated 5-orthoplex)
12284021602400960
rr{4,3,3}		-
{4}×{3}		-
rr{3,3,3}
34 (0,1,2,3,4)√2Runcicantitruncated 5-orthoplex (gippit)
1621440416048001920 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)
1040808032
{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)
2428001040640160
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)
202124021601440320
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)
242152028802240640 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)
12268015201280320
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)
242208047203840960
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)
202124029602880960
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)
2422320592057601920
{ }×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)
42200400400160
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)
242160029602240640
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)
202156037603360960
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)
2422160576057601920
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)
12268015201600640
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)
2422400600057601920
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)
2021560424048001920
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)
2422640816096003840
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
Further information: D5 polytope

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
FacetsCellsFacesEdgesVertices
59 = {3,3,3}×{ }
5-cell prism		 720302510
60 = r{3,3,3}×{ }
Rectified 5-cell prism		 1250907020
61 = t{3,3,3}×{ }
Truncated 5-cell prism		 125010010040
62 = rr{3,3,3}×{ }
Cantellated 5-cell prism		 2212025021060
63 = t0,3{3,3,3}×{ }
Runcinated 5-cell prism		 3213020014040
64 = 2t{3,3,3}×{ }
Bitruncated 5-cell prism		 126014015060
65 = tr{3,3,3}×{ }
Cantitruncated 5-cell prism		 22120280300120
66 = t0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism		 32180390360120
67 = t0,1,2,3{3,3,3}×{ }
Omnitruncated 5-cell prism		 32210540600240

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
FacetsCellsFacesEdgesVertices
[16] = {4,3,3}×{ }
Tesseractic prism
(Same as 5-cube)		 1040808032
68 = r{4,3,3}×{ }
Rectified tesseractic prism		 2613627222464
69 = t{4,3,3}×{ }
Truncated tesseractic prism		 26136304320128
70 = rr{4,3,3}×{ }
Cantellated tesseractic prism		 58360784672192
71 = t0,3{4,3,3}×{ }
Runcinated tesseractic prism		 82368608448128
72 = 2t{4,3,3}×{ }
Bitruncated tesseractic prism		 26168432480192
73 = tr{4,3,3}×{ }
Cantitruncated tesseractic prism		 58360880960384
74 = t0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism		 8252812161152384
75 = t0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism		 8262416961920768
76 = {3,3,4}×{ }
16-cell prism		 1864885616
77 = r{3,3,4}×{ }
Rectified 16-cell prism
(Same as 24-cell prism)		 2614428821648
78 = t{3,3,4}×{ }
Truncated 16-cell prism		 2614431228896
79 = rr{3,3,4}×{ }
Cantellated 16-cell prism
(Same as rectified 24-cell prism)		 50336768672192
80 = tr{3,3,4}×{ }
Cantitruncated 16-cell prism
(Same as truncated 24-cell prism)		 50336864960384
81 = t0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism		 8252812161152384
82 = sr{3,3,4}×{ }
snub 24-cell prism		 1467681392960192

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
FacetsCellsFacesEdgesVertices
[77] = {3,4,3}×{ }
24-cell prism		 2614428821648
[79] = r{3,4,3}×{ }
rectified 24-cell prism		 50336768672192
[80] = t{3,4,3}×{ }
truncated 24-cell prism		 50336864960384
83 = rr{3,4,3}×{ }
cantellated 24-cell prism		 146100823042016576
84 = t0,3{3,4,3}×{ }
runcinated 24-cell prism		 242115219201296288
85 = 2t{3,4,3}×{ }
bitruncated 24-cell prism		 5043212481440576
86 = tr{3,4,3}×{ }
cantitruncated 24-cell prism		 1461008259228801152
87 = t0,1,3{3,4,3}×{ }
runcitruncated 24-cell prism		 2421584364834561152
88 = t0,1,2,3{3,4,3}×{ }
omnitruncated 24-cell prism		 2421872508857602304
[82] = s{3,4,3}×{ }
snub 24-cell prism		 1467681392960192

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
FacetsCellsFacesEdgesVertices
89 = {5,3,3}×{ }
120-cell prism		 122960264030001200
90 = r{5,3,3}×{ }
Rectified 120-cell prism		 7224560984084002400
91 = t{5,3,3}×{ }
Truncated 120-cell prism		 722456011040120004800
92 = rr{5,3,3}×{ }
Cantellated 120-cell prism		 19221296029040252007200
93 = t0,3{5,3,3}×{ }
Runcinated 120-cell prism		 26421272022080168004800
94 = 2t{5,3,3}×{ }
Bitruncated 120-cell prism		 722576015840180007200
95 = tr{5,3,3}×{ }
Cantitruncated 120-cell prism		 192212960326403600014400
96 = t0,1,3{5,3,3}×{ }
Runcitruncated 120-cell prism		 264218720448804320014400
97 = t0,1,2,3{5,3,3}×{ }
Omnitruncated 120-cell prism		 264222320628807200028800
98 = {3,3,5}×{ }
600-cell prism		 602240031201560240
99 = r{3,3,5}×{ }
Rectified 600-cell prism		 72250401080079201440
100 = t{3,3,5}×{ }
Truncated 600-cell prism		 722504011520100802880
101 = rr{3,3,5}×{ }
Cantellated 600-cell prism		 14421152028080252007200
102 = tr{3,3,5}×{ }
Cantitruncated 600-cell prism		 144211520316803600014400
103 = t0,1,3{3,3,5}×{ }
Runcitruncated 600-cell prism		 264218720448804320014400

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
FacetsCellsFacesEdgesVertices
104grand antiprism prism
Gappip		322136019401100200

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)
Quadrirectified t4{p,q,r,s}4r{p,q,r,s} Quadrirectification reduces 4-faces to points. (Dual)
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[3[5]][(3,3,3,3,3)]7
2[4,3,3,4]19
3[4,3,31,1][4,3,3,4,1+] = 23 (8 new)
4[31,1,1,1][1+,4,3,3,4,1+] = 9 (0 new)
5[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 , family, all new, including:
    • 4-simplex honeycomb
    • Truncated 4-simplex honeycomb
    • Omnitruncated 4-simplex honeycomb
  • There are 9 uniquely ringed forms in the : [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×[4,3,4,2,∞]
2×[4,31,1,2,∞]
3×[3[4],2,∞]
4×x[4,4,2,∞,2,∞]
5×x[6,3,2,∞,2,∞]
6×x[3[3],2,∞,2,∞]
7×xx[∞,2,∞,2,∞,2,∞]
8x[3[3],2,3[3]]
9×[3[3],2,4,4]
10×[3[3],2,6,3]
11×[4,4,2,4,4]
12×[4,4,2,6,3]
13×[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.

Compact hyperbolic groups

= [(3,3,3,3,4)]:

= [5,3,31,1]:

= [3,3,3,5]:

= [4,3,3,5]:
= [5,3,3,5]:

Paracompact hyperbolic groups

= [3,3[4]]:

= [4,3[4]]:
= [(3,3,4,3,4)]:
= [3[3]×[]]:

= [4,/3\,3,4]:
= [3,4,31,1]:
= [4,32,1]:
= [4,31,1,1]:

= [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
    • (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)
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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