# Uniform 5-polytope

Last updated
 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.

## Contents

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]
• 2000-2022: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes, [3] with a current count of 1294 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]

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

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 [7]
Group
symbol
Order Coxeter
graph
Bracket
notation
Commutator
subgroup
Coxeter
number

(h)
Reflections
m=5/2 h [8]
A5720[3,3,3,3][3,3,3,3]+615
D51920[3,3,31,1][3,3,31,1]+820
B53840[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
A4A1120[3,3,3,2] = [3,3,3]×[ ][3,3,3]+10 1
D4A1384[31,1,1,2] = [31,1,1]×[ ][31,1,1]+12 1
B4A1768[4,3,3,2] = [4,3,3]×[ ]4 12 1
F4A12304[3,4,3,2] = [3,4,3]×[ ][3+,4,3+]12 12 1
H4A128800[5,3,3,2] = [3,4,3]×[ ][5,3,3]+60 1
Duoprismatic prisms (use 2p and 2q for evens)
I2(p)I2(q)A18pq[p,2,q,2] = [p]×[q]×[ ][p+,2,q+]pq1
I2(2p)I2(q)A116pq[2p,2,q,2] = [2p]×[q]×[ ]p pq1
I2(2p)I2(2q)A132pq[2p,2,2q,2] = [2p]×[2q]×[ ]ppqq1
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 pp
B3I2(p)96p[4,3,2,p] = [4,3]×[p]3 6p
B3I2(2p)192p[4,3,2,2p] = [4,3]×[2p]3 6 pp
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 pp

### 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]
43210
[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)
61520156
{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)
1245806015
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)
1245807530
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)
2713529024060
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)
126014015060
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)
27135290300120
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)
4725542027060
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)
47315720630180
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)
47255570540180
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)
47315810900360
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)
62330570420120
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)
6248011401080360
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)
12601209020
{3}×{3}

r{3,3,3}
---
r{3,3,3}
14(0,0,1,1,2,2) Bicantellated 5-simplex
small birhombated dodecateron (sibrid)
3218042036090
rr{3,3,3}
-
{3}×{3}
-
rr{3,3,3}
15(0,0,1,2,3,3) Bicantitruncated 5-simplex
great birhombated dodecateron (gibrid)
32180420450180
tr{3,3,3}
-
{3}×{3}
-
tr{3,3,3}
16(0,1,1,1,1,2) Stericated 5-simplex
6218021012030
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)
62420900720180
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)
6245011101080360
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
6254015601800720
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)
422234040802520360 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 251=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 pointName
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)
Alt
20(0,0,0,0,1)√2 5-orthoplex
3280804010
{3,3,4}
----
{3,3,3}
21(0,0,0,1,1)√2 Rectified 5-orthoplex
4224040024040
{ }×{3,4}

{3,3,4}
---
r{3,3,3}
22(0,0,0,1,2)√2 Truncated 5-orthoplex
4224040028080
(Octah.pyr)

{3,3,4}
---
t{3,3,3}
23(0,0,1,1,1)√2 Birectified 5-cube
(Birectified 5-orthoplex)
4228064048080
{4}×{3}

r{3,3,4}
---
r{3,3,3}
24(0,0,1,1,2)√2 Cantellated 5-orthoplex
8264015201200240
Prism-wedge

r{3,3,4}

{ }×{3,4}
--
rr{3,3,3}
25(0,0,1,2,2)√2 Bitruncated 5-orthoplex
42280720720240
t{3,3,4}
---
2t{3,3,3}
26(0,0,1,2,3)√2 Cantitruncated 5-orthoplex
8264015201440480
t{3,3,4}

{ }×{3,4}
--
t0,1,3{3,3,3}
27(0,1,1,1,1)√2 Rectified 5-cube
rectified penteract (rin)
4220040032080
{3,3}×{ }

r{4,3,3}
---
{3,3,3}
28(0,1,1,1,2)√2 Runcinated 5-orthoplex
162120021601440320
r{4,3,3}

{ }×r{3,4}

{3}×{4}

t0,3{3,3,3}
29(0,1,1,2,2)√2 Bicantellated 5-cube
(Bicantellated 5-orthoplex)
12284021601920480
rr{3,3,4}
-
{4}×{3}
-
rr{3,3,3}
30(0,1,1,2,3)√2 Runcitruncated 5-orthoplex
162144036803360960
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
bitruncated penteract (bittin)
42280720800320
2t{4,3,3}
---
t{3,3,3}
32(0,1,2,2,3)√2 Runcicantellated 5-orthoplex
162120029602880960
2t{4,3,3}

{ }×t{3,4}

{3}×{4}
-
t0,1,3{3,3,3}
33(0,1,2,3,3)√2 Bicantitruncated 5-cube
(Bicantitruncated 5-orthoplex)
12284021602400960
tr{3,3,4}
-
{4}×{3}
-
rr{3,3,3}
34(0,1,2,3,4)√2 Runcicantitruncated 5-orthoplex
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
penteract (pent)
1040808032
{3,3,3}

{4,3,3}
----
36(1,1,1,1,1)
+ (0,0,0,0,1)√2
Stericated 5-cube
(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
small prismated penteract (span)
202124021601440320
t0,3{4,3,3}
-
{4}×{3}

{ }×r{3,3}

r{3,3,3}
38(1,1,1,1,1)
+ (0,0,0,1,2)√2
Steritruncated 5-orthoplex
242152028802240640
t0,3{4,3,3}

{4,3}×{ }

{6}×{4}

{ }×t{3,3}

t{3,3,3}
39(1,1,1,1,1)
+ (0,0,1,1,1)√2
Cantellated 5-cube
small rhombated penteract (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
(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
prismatorhombated penteract (prin)
202124029602880960
t0,2,3{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
2422320592057601920
t0,2,3{4,3,3}

rr{4,3}×{ }

{6}×{4}

{ }×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)
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
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
prismatotruncated penteract (pattin)
202156037603360960
t0,1,3{4,3,3}
-
{8}×{3}

{ }×r{3,3}

rr{3,3,3}
46(1,1,1,1,1)
+ (0,1,1,2,3)√2
Steriruncitruncated 5-cube
(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
great rhombated penteract (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
celligreatorhombated penteract (cogrin)
2422400600057601920
tr{4,3,3}

tr{4,3}×{ }

{8}×{3}

{ }×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)
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
(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}
51 5-demicube
hemipenteract (hin)
=
261201608016
r{3,3,3}

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

{3,3,3}
52 Cantic 5-cube
Truncated hemipenteract (thin)
=
42280640560160
h2{4,3,3}
---(16)

r{3,3,3}
(16)

t{3,3,3}
53 Runcic 5-cube
Small rhombated hemipenteract (sirhin)
=
42360880720160
h3{4,3,3}
---(16)

r{3,3,3}
(16)

rr{3,3,3}
54 Steric 5-cube
Small prismated hemipenteract (siphin)
=
8248072040080
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)
=
4236010401200480
h2,3{4,3,3}
---(16)

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

tr{3,3,3}
56 Stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
=
8272018401680480
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)
=
8256012801120320
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)
=
8272020802400960
h2,3{4,3,3}

h2{4,3}×{}
--(16)

tr{3,3,3}
(16)

t0,1,2,3{3,3,3}
NonuniformAlternated runcicantitruncated 5-orthoplex
Snub hemipenteract (snahin)
=
11226240108806720960
sr{3,3,4}
sr{2,3,4} sr{3,2,4} - ht0,1,2,3{3,3,3} (960)

Irr. {3,3,3}
NonuniformEdge-snub 5-orthoplex
Pyritosnub penteract (pysnan)
1202792015360105601920sr3{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}×{ }
NonuniformSnub 5-cube
Snub penteract (snan)
2162122402160013440960 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 polyhedra, 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]
43210
[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)
261201608016
r{3,3,3}

{3,3,3}

h{4,3,3}
---
[52] =
h2{4,3,3,3}, cantic 5-cube
Truncated hemipenteract (thin)
42280640560160
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)
42360880720160
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)
8248072040080
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)
4236010401200480
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)
8272018401680480
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)
8256012801120320
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)
8272020802400960
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)
11226240108806720960 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
FacetsCellsFacesEdgesVertices
59 = {3,3,3}×{ }
5-cell prism (penp)
720302510
60 = r{3,3,3}×{ }
Rectified 5-cell prism (rappip)
1250907020
61 = t{3,3,3}×{ }
Truncated 5-cell prism (tippip)
125010010040
62 = rr{3,3,3}×{ }
Cantellated 5-cell prism (srippip)
2212025021060
63 = t0,3{3,3,3}×{ }
Runcinated 5-cell prism (spiddip)
3213020014040
64 = 2t{3,3,3}×{ }
Bitruncated 5-cell prism (decap)
126014015060
65 = tr{3,3,3}×{ }
Cantitruncated 5-cell prism (grippip)
22120280300120
66 = t0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism (prippip)
32180390360120
67 = t0,1,2,3{3,3,3}×{ }
Omnitruncated 5-cell prism (gippiddip)
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!).

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
FacetsCellsFacesEdgesVertices
[16] = {4,3,3}×{ }
Tesseractic prism (pent)
(Same as 5-cube)
1040808032
68 = r{4,3,3}×{ }
Rectified tesseractic prism (rittip)
2613627222464
69 = t{4,3,3}×{ }
Truncated tesseractic prism (tattip)
26136304320128
70 = rr{4,3,3}×{ }
Cantellated tesseractic prism (srittip)
58360784672192
71 = t0,3{4,3,3}×{ }
Runcinated tesseractic prism (sidpithip)
82368608448128
72 = 2t{4,3,3}×{ }
Bitruncated tesseractic prism (tahp)
26168432480192
73 = tr{4,3,3}×{ }
Cantitruncated tesseractic prism (grittip)
58360880960384
74 = t0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism (prohp)
8252812161152384
75 = t0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism (gidpithip)
8262416961920768
76 = {3,3,4}×{ }
16-cell prism (hexip)
1864885616
77 = r{3,3,4}×{ }
Rectified 16-cell prism (icope)
(Same as 24-cell prism)
2614428821648
78 = t{3,3,4}×{ }
Truncated 16-cell prism (thexip)
2614431228896
79 = rr{3,3,4}×{ }
Cantellated 16-cell prism (ricope)
(Same as rectified 24-cell prism)
50336768672192
80 = tr{3,3,4}×{ }
Cantitruncated 16-cell prism (ticope)
(Same as truncated 24-cell prism)
50336864960384
81 = t0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism (prittip)
8252812161152384
82 = sr{3,3,4}×{ }
1467681392960192
Nonuniform
rectified tesseractic alterprism (rita)
5028846428864
Nonuniform
truncated 16-cell alterprism (thexa)
2616838433696
Nonuniform
bitruncated tesseractic alterprism (taha)
50288624576192

#### 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 (icope)
2614428821648
[79] = r{3,4,3}×{ }
rectified 24-cell prism (ricope)
50336768672192
[80] = t{3,4,3}×{ }
truncated 24-cell prism (ticope)
50336864960384
83 = rr{3,4,3}×{ }
cantellated 24-cell prism (sricope)
146100823042016576
84