Uniform 5-polytope

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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

[edit]
  • 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

[edit]
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

[edit]
Unsolved problem in mathematics:
What is the complete set of convex uniform 5-polytopes?[6]
(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.[citation needed]

Symmetry of uniform 5-polytopes in four dimensions

[edit]

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

[edit]
  • 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

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

[edit]
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

[edit]
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)

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