Graphs of regular and uniform 5-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

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 5-polytopes

Main article: List of regular polytopes § Five Dimensions

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.

Coxeter diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.
Fundamental families[7]
Group
symbol
Order Coxeter
graph
Bracket
notation
Commutator
subgroup
Coxeter
number

(h)
Reflections
m=5/2 h[8]
A5 720 [3,3,3,3] [3,3,3,3]+ 6 15
D5 1920 [3,3,31,1] [3,3,31,1]+ 8 20
B5 3840 [4,3,3,3] 10 5 20
Uniform prisms

There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. There is one infinite family of 5-polytopes based on prisms of the uniform duoprisms {p}×{q}×{ }.

Coxeter
group
Order Coxeter
diagram
Coxeter
notation
Commutator
subgroup
Reflections
A4A1 120 [3,3,3,2] = [3,3,3]×[ ] [3,3,3]+ 10 1
D4A1 384 [31,1,1,2] = [31,1,1]×[ ] [31,1,1]+ 12 1
B4A1 768 [4,3,3,2] = [4,3,3]×[ ] 4 12 1
F4A1 2304 [3,4,3,2] = [3,4,3]×[ ] [3+,4,3+] 12 12 1
H4A1 28800 [5,3,3,2] = [3,4,3]×[ ] [5,3,3]+ 60 1
Duoprismatic prisms (use 2p and 2q for evens)
I2(p)I2(q)A1 8pq [p,2,q,2] = [p]×[q]×[ ] [p+,2,q+] p q 1
I2(2p)I2(q)A1 16pq [2p,2,q,2] = [2p]×[q]×[ ] p p q 1
I2(2p)I2(2q)A1 32pq [2p,2,2q,2] = [2p]×[2q]×[ ] p p q q 1
Uniform duoprisms

There are 3 categorical uniform duoprismatic families of polytopes based on Cartesian products of the uniform polyhedra and regular polygons: {q,r}×{p}.

Coxeter
group
Order Coxeter
diagram
Coxeter
notation
Commutator
subgroup
Reflections
Prismatic groups (use 2p for even)
A3I2(p) 48p [3,3,2,p] = [3,3]×[p] [(3,3)+,2,p+] 6 p
A3I2(2p) 96p [3,3,2,2p] = [3,3]×[2p] 6 p p
B3I2(p) 96p [4,3,2,p] = [4,3]×[p] 3 6 p
B3I2(2p) 192p [4,3,2,2p] = [4,3]×[2p] 3 6 p p
H3I2(p) 240p [5,3,2,p] = [5,3]×[p] [(5,3)+,2,p+] 15 p
H3I2(2p) 480p [5,3,2,2p] = [5,3]×[2p] 15 p p

Enumerating the convex uniform 5-polytopes

That brings the tally to: 19+31+8+45+1=104

In addition there are:

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)
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
small cellated dodecateron (scad)
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
great cellated dodecateron (gocad)
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

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. 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
triacontaditeron (tac)
32 80 80 40 10
{3,3,4}
- - - -
{3,3,3}
21 (0,0,0,1,1)√2 Rectified 5-orthoplex
rectified triacontaditeron (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
truncated triacontaditeron (tot)
42 240 400 280 80
(Octah.pyr)

{3,3,4}
- - -
t{3,3,3}
23 (0,0,1,1,1)√2 Birectified 5-cube
penteractitriacontaditeron (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
small rhombated triacontaditeron (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
bitruncated triacontaditeron (bittit)
42 280 720 720 240
t{3,3,4}
- - -
2t{3,3,3}
26 (0,0,1,2,3)√2 Cantitruncated 5-orthoplex
great rhombated triacontaditeron (gart)
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
small prismated triacontaditeron (spat)
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
small birhombated penteractitriacontaditeron (sibrant)
(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
prismatotruncated triacontaditeron (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
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
prismatorhombated triacontaditeron (pirt)
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
great birhombated triacontaditeron (gibrant)
(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
great prismated triacontaditeron (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
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
small cellated penteractitriacontaditeron (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
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
celliprismated triacontaditeron (cappin)
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
cellirhombated penteractitriacontaditeron (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
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
celligreatorhombated triacontaditeron (cogart)
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
celliprismated triacontaditeron (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
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
celliprismatotruncated penteractitriacontaditeron (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
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
great cellated penteractitriacontaditeron (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}
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 prismatotriacontaditeron (snippit)
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

Further information: D5 polytope

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}×{ }
snub 24-cell prism (sadip)
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]