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]
• 2000-2024: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes,[3] with a current count of 1348 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]
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}
- - - -
{3,3,3}
21 (0,0,0,1,1)√2 Rectified 5-orthoplex
42 240 400 240 40
{ }×{3,4}

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

{3,3,4}
- - -
t{3,3,3}
23 (0,0,1,1,1)√2 Birectified 5-cube
(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
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
42 280 720 720 240
t{3,3,4}
- - -
2t{3,3,3}
26 (0,0,1,2,3)√2 Cantitruncated 5-orthoplex
82 640 1520 1440 480
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)
42 200 400 320 80
{3,3}×{ }

r{4,3,3}
- - -
{3,3,3}
28 (0,1,1,1,2)√2 Runcinated 5-orthoplex
162 1200 2160 1440 320
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)
122 840 2160 1920 480
rr{3,3,4}
-
{4}×{3}
-
rr{3,3,3}
30 (0,1,1,2,3)√2 Runcitruncated 5-orthoplex
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
bitruncated penteract (bittin)
42 280 720 800 320
2t{4,3,3}
- - -
t{3,3,3}
32 (0,1,2,2,3)√2 Runcicantellated 5-orthoplex
162 1200 2960 2880 960
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)
122 840 2160 2400 960
tr{3,3,4}
-
{4}×{3}
-
rr{3,3,3}
34 (0,1,2,3,4)√2 Runcicantitruncated 5-orthoplex
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
penteract (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
(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
small prismated penteract (span)
202 1240 2160 1440 320
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
242 1520 2880 2240 640
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)
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
(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
prismatorhombated penteract (prin)
202 1240 2960 2880 960
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
242 2320 5920 5760 1920
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)
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
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
prismatotruncated penteract (pattin)
202 1560 3760 3360 960
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)
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
great rhombated penteract (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
celligreatorhombated penteract (cogrin)
242 2400 6000 5760 1920
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)
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
(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}
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]