In geometry, a uniform 5polytope is a fivedimensional uniform polytope. By definition, a uniform 5polytope is vertextransitive and constructed from uniform 4polytope facets.
The complete set of convex uniform 5polytopes 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.
Regular 5polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} 4polytope facets around each face. There are exactly three such regular polytopes, all convex:
There are no nonconvex regular polytopes in 5 dimensions or above.
There are 104 known convex uniform 5polytopes, plus a number of infinite families of duoprism prisms, and polygonpolyhedron duoprisms. All except the grand antiprism prism are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.^{[citation needed]}
The 5simplex is the regular form in the A_{5} family. The 5cube and 5orthoplex are the regular forms in the B_{5} family. The bifurcating graph of the D_{5} family contains the 5orthoplex, as well as a 5demicube which is an alternated 5cube.
Each reflective uniform 5polytope 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 evenbranches. 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 5polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.
Group symbol 
Order  Coxeter graph 
Bracket notation 
Commutator subgroup 
Coxeter number (h) 
Reflections m=5/2 h^{[8]}  

A_{5}  720  [3,3,3,3]  [3,3,3,3]^{+}  6  15  
D_{5}  1920  [3,3,3^{1,1}]  [3,3,3^{1,1}]^{+}  8  20  
B_{5}  3840  [4,3,3,3]  10  5  20 
There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4polytopes. There is one infinite family of 5polytopes based on prisms of the uniform duoprisms {p}×{q}×{ }.
Coxeter group 
Order  Coxeter diagram 
Coxeter notation 
Commutator subgroup 
Reflections  

A_{4}A_{1}  120  [3,3,3,2] = [3,3,3]×[ ]  [3,3,3]^{+}  10  1  
D_{4}A_{1}  384  [3^{1,1,1},2] = [3^{1,1,1}]×[ ]  [3^{1,1,1}]^{+}  12  1  
B_{4}A_{1}  768  [4,3,3,2] = [4,3,3]×[ ]  4  12  1  
F_{4}A_{1}  2304  [3,4,3,2] = [3,4,3]×[ ]  [3^{+},4,3^{+}]  12  12  1  
H_{4}A_{1}  28800  [5,3,3,2] = [3,4,3]×[ ]  [5,3,3]^{+}  60  1  
Duoprismatic prisms (use 2p and 2q for evens)  
I_{2}(p)I_{2}(q)A_{1}  8pq  [p,2,q,2] = [p]×[q]×[ ]  [p^{+},2,q^{+}]  p  q  1  
I_{2}(2p)I_{2}(q)A_{1}  16pq  [2p,2,q,2] = [2p]×[q]×[ ]  p  p  q  1  
I_{2}(2p)I_{2}(2q)A_{1}  32pq  [2p,2,2q,2] = [2p]×[2q]×[ ]  p  p  q  q  1 
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)  
A_{3}I_{2}(p)  48p  [3,3,2,p] = [3,3]×[p]  [(3,3)^{+},2,p^{+}]  6  p  
A_{3}I_{2}(2p)  96p  [3,3,2,2p] = [3,3]×[2p]  6  p  p  
B_{3}I_{2}(p)  96p  [4,3,2,p] = [4,3]×[p]  3  6  p  
B_{3}I_{2}(2p)  192p  [4,3,2,2p] = [4,3]×[2p]  3  6  p  p  
H_{3}I_{2}(p)  240p  [5,3,2,p] = [5,3]×[p]  [(5,3)^{+},2,p^{+}]  15  p  
H_{3}I_{2}(2p)  480p  [5,3,2,2p] = [5,3]×[2p]  15  p  p 
That brings the tally to: 19+31+8+45+1=104
In addition there are:
There are 19 forms based on all permutations of the Coxeter diagrams with one or more rings. (16+41 cases)
They are named by Norman Johnson from the Wythoff construction operations upon regular 5simplex (hexateron).
The A_{5} 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 5polytopes with 5simplex symmetry can be generated as permutations of simple integers in 6space, all in hyperplanes with normal vector (1,1,1,1,1,1).
#  Base point  Johnson naming system Bowers name and (acronym) Coxeter diagram 
kface 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)  5simplex 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 5simplex 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 5simplex 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 5simplex small rhombated hexateron (sarx) 
27  135  290  240  60  prismwedge 
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 5simplex 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 5simplex 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 5simplex small prismated hexateron (spix) 
47  255  420  270  60  t_{0,3}{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 5simplex prismatotruncated hexateron (pattix) 
47  315  720  630  180  t_{0,1,3}{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 5simplex prismatorhombated hexateron (pirx) 
47  255  570  540  180  t_{0,1,3}{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 5simplex great prismated hexateron (gippix) 
47  315  810  900  360  Irr.5cell 
t_{0,1,2,3}{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 5simplex celliprismated hexateron (cappix) 
62  330  570  420  120  t{3,3,3} 
{ }×t{3,3} 
{3}×{6} 
{ }×{3,3} 
t_{0,3}{3,3,3} 

12  (0,1,1,2,3,4) or (0,1,2,3,3,4)  Stericantitruncated 5simplex celligreatorhombated hexateron (cograx) 
62  480  1140  1080  360  tr{3,3,3} 
{ }×tr{3,3} 
{3}×{6} 
{ }×rr{3,3} 
t_{0,1,3}{3,3,3} 

13  (0,0,0,1,1,1)  Birectified 5simplex 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 5simplex 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 5simplex 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 5simplex small cellated dodecateron (scad) 
62  180  210  120  30  Irr.16cell 
{3,3,3} 
{ }×{3,3} 
{3}×{3} 
{ }×{3,3} 
{3,3,3} 

17  (0,1,1,2,2,3)  Stericantellated 5simplex 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 5simplex celliprismatotruncated dodecateron (captid) 
62  450  1110  1080  360  t_{0,1,3}{3,3,3} 
{ }×t{3,3} 
{6}×{6} 
{ }×t{3,3} 
t_{0,1,3}{3,3,3} 

19  (0,1,2,3,4,5)  Omnitruncated 5simplex great cellated dodecateron (gocad) 
62  540  1560  1800  720  Irr. {3,3,3} 
t_{0,1,2,3}{3,3,3} 
{ }×tr{3,3} 
{6}×{6} 
{ }×tr{3,3} 
t_{0,1,2,3}{3,3,3} 

Nonuniform  Omnisnub 5simplex snub dodecateron (snod) snub hexateron (snix) 
422  2340  4080  2520  360  ht_{0,1,2,3}{3,3,3}  ht_{0,1,2,3}{3,3,2}  ht_{0,1,2,3}{3,2,3}  ht_{0,1,2,3}{3,3,2}  ht_{0,1,2,3}{3,3,3}  (360) Irr. {3,3,3} 
The B_{5} family has symmetry of order 3840 (5!×2^{5}).
This family has 2^{5}−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 D_{5} 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 5polytopes 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 5cube family of 5polytopes 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 5polytope. All coordinates correspond with uniform 5polytopes 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  5orthoplex triacontaditeron (tac) 
32  80  80  40  10  {3,3,4} 
        {3,3,3} 

21  (0,0,0,1,1)√2  Rectified 5orthoplex 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 5orthoplex 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 5cube penteractitriacontaditeron (nit) (Birectified 5orthoplex) 
42  280  640  480  80  {4}×{3} 
r{3,3,4} 
      r{3,3,3} 

24  (0,0,1,1,2)√2  Cantellated 5orthoplex small rhombated triacontaditeron (sart) 
82  640  1520  1200  240  Prismwedge 
r{3,3,4} 
{ }×{3,4} 
    rr{3,3,3} 

25  (0,0,1,2,2)√2  Bitruncated 5orthoplex bitruncated triacontaditeron (bittit) 
42  280  720  720  240  t{3,3,4} 
      2t{3,3,3} 

26  (0,0,1,2,3)√2  Cantitruncated 5orthoplex great rhombated triacontaditeron (gart) 
82  640  1520  1440  480  t{3,3,4} 
{ }×{3,4} 
    t_{0,1,3}{3,3,3} 

27  (0,1,1,1,1)√2  Rectified 5cube 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 5orthoplex small prismated triacontaditeron (spat) 
162  1200  2160  1440  320  r{4,3,3} 
{ }×r{3,4} 
{3}×{4} 
t_{0,3}{3,3,3} 

29  (0,1,1,2,2)√2  Bicantellated 5cube small birhombated penteractitriacontaditeron (sibrant) (Bicantellated 5orthoplex) 
122  840  2160  1920  480  rr{3,3,4} 
  {4}×{3} 
  rr{3,3,3} 

30  (0,1,1,2,3)√2  Runcitruncated 5orthoplex prismatotruncated triacontaditeron (pattit) 
162  1440  3680  3360  960  rr{3,3,4} 
{ }×r{3,4} 
{6}×{4} 
  t_{0,1,3}{3,3,3} 

31  (0,1,2,2,2)√2  Bitruncated 5cube bitruncated penteract (bittin) 
42  280  720  800  320  2t{4,3,3} 
      t{3,3,3} 

32  (0,1,2,2,3)√2  Runcicantellated 5orthoplex prismatorhombated triacontaditeron (pirt) 
162  1200  2960  2880  960  2t{4,3,3} 
{ }×t{3,4} 
{3}×{4} 
  t_{0,1,3}{3,3,3} 

33  (0,1,2,3,3)√2  Bicantitruncated 5cube great birhombated triacontaditeron (gibrant) (Bicantitruncated 5orthoplex) 
122  840  2160  2400  960  tr{3,3,4} 
  {4}×{3} 
  rr{3,3,3} 

34  (0,1,2,3,4)√2  Runcicantitruncated 5orthoplex great prismated triacontaditeron (gippit) 
162  1440  4160  4800  1920  tr{3,3,4} 
{ }×t{3,4} 
{6}×{4} 
  t_{0,1,2,3}{3,3,3} 

35  (1,1,1,1,1)  5cube 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 5cube small cellated penteractitriacontaditeron (scant) (Stericated 5orthoplex) 
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 5cube small prismated penteract (span) 
202  1240  2160  1440  320  t_{0,3}{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 5orthoplex celliprismated triacontaditeron (cappin) 
242  1520  2880  2240  640  t_{0,3}{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 5cube small rhombated penteract (sirn) 
122  680  1520  1280  320  Prismwedge 
rr{4,3,3} 
    { }×{3,3} 
r{3,3,3} 

40  (1,1,1,1,1) + (0,0,1,1,2)√2 
Stericantellated 5cube cellirhombated penteractitriacontaditeron (carnit) (Stericantellated 5orthoplex) 
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 5cube prismatorhombated penteract (prin) 
202  1240  2960  2880  960  t_{0,2,3}{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 5orthoplex celligreatorhombated triacontaditeron (cogart) 
242  2320  5920  5760  1920  t_{0,2,3}{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 5cube 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 5cube celliprismated triacontaditeron (capt) 
242  1600  2960  2240  640  t{4,3,3} 
t{4,3}×{ } 
{8}×{3} 
{ }×{3,3} 
t_{0,3}{3,3,3} 

45  (1,1,1,1,1) + (0,1,1,2,2)√2 
Runcitruncated 5cube prismatotruncated penteract (pattin) 
202  1560  3760  3360  960  t_{0,1,3}{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 5cube celliprismatotruncated penteractitriacontaditeron (captint) (Steriruncitruncated 5orthoplex) 
242  2160  5760  5760  1920  t_{0,1,3}{4,3,3} 
t{4,3}×{ } 
{8}×{6} 
{ }×t{3,3} 
t_{0,1,3}{3,3,3} 

47  (1,1,1,1,1) + (0,1,2,2,2)√2 
Cantitruncated 5cube 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 5cube celligreatorhombated penteract (cogrin) 
242  2400  6000  5760  1920  tr{4,3,3} 
tr{4,3}×{ } 
{8}×{3} 
{ }×rr{3,3} 
t_{0,1,3}{3,3,3} 

49  (1,1,1,1,1) + (0,1,2,3,3)√2 
Runcicantitruncated 5cube great prismated penteract (gippin) 
202  1560  4240  4800  1920  t_{0,1,2,3}{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 5cube great cellated penteractitriacontaditeron (gacnet) (omnitruncated 5orthoplex) 
242  2640  8160  9600  3840  Irr. {3,3,3} 
tr{4,3}×{ } 
tr{4,3}×{ } 
{8}×{6} 
{ }×tr{3,3} 
t_{0,1,2,3}{3,3,3} 

51  5demicube hemipenteract (hin) = 
26  120  160  80  16  r{3,3,3} 
h{4,3,3} 
        (16) {3,3,3}  
52  Cantic 5cube Truncated hemipenteract (thin) = 
42  280  640  560  160  h_{2}{4,3,3} 
      (16) r{3,3,3} 
(16) t{3,3,3}  
53  Runcic 5cube Small rhombated hemipenteract (sirhin) = 
42  360  880  720  160  h_{3}{4,3,3} 
      (16) r{3,3,3} 
(16) rr{3,3,3}  
54  Steric 5cube Small prismated hemipenteract (siphin) = 
82  480  720  400  80  h{4,3,3} 
h{4,3}×{} 
    (16) {3,3,3} 
(16) t_{0,3}{3,3,3}  
55  Runcicantic 5cube Great rhombated hemipenteract (girhin) = 
42  360  1040  1200  480  h_{2,3}{4,3,3} 
      (16) 2t{3,3,3} 
(16) tr{3,3,3}  
56  Stericantic 5cube Prismatotruncated hemipenteract (pithin) = 
82  720  1840  1680  480  h_{2}{4,3,3} 
h_{2}{4,3}×{} 
    (16) rr{3,3,3} 
(16) t_{0,1,3}{3,3,3}  
57  Steriruncic 5cube Prismatorhombated hemipenteract (pirhin) = 
82  560  1280  1120  320  h_{3}{4,3,3} 
h{4,3}×{} 
    (16) t{3,3,3} 
(16) t_{0,1,3}{3,3,3}  
58  Steriruncicantic 5cube Great prismated hemipenteract (giphin) = 
82  720  2080  2400  960  h_{2,3}{4,3,3} 
h_{2}{4,3}×{} 
    (16) tr{3,3,3} 
(16) t_{0,1,2,3}{3,3,3}  
Nonuniform  Alternated runcicantitruncated 5orthoplex Snub prismatotriacontaditeron (snippit) Snub hemipenteract (snahin) = 
1122  6240  10880  6720  960  sr{3,3,4} 
sr{2,3,4}  sr{3,2,4}    ht_{0,1,2,3}{3,3,3}  (960) Irr. {3,3,3}  
Nonuniform  Edgesnub 5orthoplex Pyritosnub penteract (pysnan) 
1202  7920  15360  10560  1920  sr_{3}{3,3,4}  sr_{3}{2,3,4}  sr_{3}{3,2,4}  s{3,3}×{ } 
ht_{0,1,2,3}{3,3,3}  (960) Irr. {3,3}×{ }  
Nonuniform  Snub 5cube Snub penteract (snan) 
2162  12240  21600  13440  960  ht_{0,1,2,3}{3,3,4}  ht_{0,1,2,3}{2,3,4}  ht_{0,1,2,3}{3,2,4}  ht_{0,1,2,3}{3,3,2}  ht_{0,1,2,3}{3,3,3}  (1920) Irr. {3,3,3} 
The D_{5} family has symmetry of order 1920 (5! x 2^{4}).
This family has 23 Wythoffian uniform polytopes, from 3×81 permutations of the D_{5} Coxeter diagram with one or more rings. 15 (2×81) are repeated from the B_{5} family and 8 are unique to this family, though even those 8 duplicate the alternations from the B_{5} family.
In the 15 repeats, both of the nodes terminating the length1 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 5polytopes 20 through 34 above. The 8 new forms have one such node ringed and one not, with the relation ... = ... duplicating uniform 5polytopes 51 through 58 above.
#  Coxeter diagram Schläfli symbol symbols Johnson and Bowers names 
Element counts  Vertex figure 
Facets by location: [3^{1,2,1}]  

4  3  2  1  0  [3,3,3] (16) 
[3^{1,1,1}] (10) 
[3,3]×[ ] (40) 
[ ]×[3]×[ ] (80) 
[3,3,3] (16) 
Alt  
[51]  = h{4,3,3,3}, 5demicube Hemipenteract (hin) 
26  120  160  80  16  r{3,3,3} 
{3,3,3} 
h{4,3,3} 
      
[52]  = h_{2}{4,3,3,3}, cantic 5cube Truncated hemipenteract (thin) 
42  280  640  560  160  t{3,3,3} 
h_{2}{4,3,3} 
    r{3,3,3} 

[53]  = h_{3}{4,3,3,3}, runcic 5cube Small rhombated hemipenteract (sirhin) 
42  360  880  720  160  rr{3,3,3} 
h_{3}{4,3,3} 
    r{3,3,3} 

[54]  = h_{4}{4,3,3,3}, steric 5cube Small prismated hemipenteract (siphin) 
82  480  720  400  80  t_{0,3}{3,3,3} 
h{4,3,3} 
h{4,3}×{} 
  {3,3,3} 

[55]  = h_{2,3}{4,3,3,3}, runcicantic 5cube Great rhombated hemipenteract (girhin) 
42  360  1040  1200  480  2t{3,3,3} 
h_{2,3}{4,3,3} 
    tr{3,3,3} 

[56]  = h_{2,4}{4,3,3,3}, stericantic 5cube Prismatotruncated hemipenteract (pithin) 
82  720  1840  1680  480  t_{0,1,3}{3,3,3} 
h_{2}{4,3,3} 
h_{2}{4,3}×{} 
  rr{3,3,3} 

[57]  = h_{3,4}{4,3,3,3}, steriruncic 5cube Prismatorhombated hemipenteract (pirhin) 
82  560  1280  1120  320  t_{0,1,3}{3,3,3} 
h_{3}{4,3,3} 
h{4,3}×{} 
  t{3,3,3} 

[58]  = h_{2,3,4}{4,3,3,3}, steriruncicantic 5cube Great prismated hemipenteract (giphin) 
82  720  2080  2400  960  t_{0,1,2,3}{3,3,3} 
h_{2,3}{4,3,3} 
h_{2}{4,3}×{} 
  tr{3,3,3} 

Nonuniform  = ht_{0,1,2,3}{3,3,3,4}, alternated runcicantitruncated 5orthoplex Snub hemipenteract (snahin) 
1122  6240  10880  6720  960  ht_{0,1,2,3}{3,3,3}  sr{3,3,4} 
sr{2,3,4}  sr{3,2,4}  ht_{0,1,2,3}{3,3,3}  (960) Irr. {3,3,3} 
There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4polytopes. For simplicity, most alternations are not shown.
This prismatic family has 9 forms:
The A_{1} x A_{4} 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}×{ } 5cell prism (penp) 
7  20  30  25  10 
60  = r{3,3,3}×{ } Rectified 5cell prism (rappip) 
12  50  90  70  20 
61  = t{3,3,3}×{ } Truncated 5cell prism (tippip) 
12  50  100  100  40 
62  = rr{3,3,3}×{ } Cantellated 5cell prism (srippip) 
22  120  250  210  60 
63  = t_{0,3}{3,3,3}×{ } Runcinated 5cell prism (spiddip) 
32  130  200  140  40 
64  = 2t{3,3,3}×{ } Bitruncated 5cell prism (decap) 
12  60  140  150  60 
65  = tr{3,3,3}×{ } Cantitruncated 5cell prism (grippip) 
22  120  280  300  120 
66  = t_{0,1,3}{3,3,3}×{ } Runcitruncated 5cell prism (prippip) 
32  180  390  360  120 
67  = t_{0,1,2,3}{3,3,3}×{ } Omnitruncated 5cell prism (gippiddip) 
32  210  540  600  240 
This prismatic family has 16 forms. (Three are shared with [3,4,3]×[ ] family)
The A_{1}×B_{4} family has symmetry of order 768 (2^{5}4!).
The last three snubs can be realised with equallength edges, but turn out nonuniform anyway because some of their 4faces are not uniform 4polytopes.
#  Coxeter diagram and Schläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
[16]  = {4,3,3}×{ } Tesseractic prism (pent) (Same as 5cube) 
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  = t_{0,3}{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  = t_{0,1,3}{4,3,3}×{ } Runcitruncated tesseractic prism (prohp) 
82  528  1216  1152  384 
75  = t_{0,1,2,3}{4,3,3}×{ } Omnitruncated tesseractic prism (gidpithip) 
82  624  1696  1920  768 
76  = {3,3,4}×{ } 16cell prism (hexip) 
18  64  88  56  16 
77  = r{3,3,4}×{ } Rectified 16cell prism (icope) (Same as 24cell prism) 
26  144  288  216  48 
78  = t{3,3,4}×{ } Truncated 16cell prism (thexip) 
26  144  312  288  96 
79  = rr{3,3,4}×{ } Cantellated 16cell prism (ricope) (Same as rectified 24cell prism) 
50  336  768  672  192 
80  = tr{3,3,4}×{ } Cantitruncated 16cell prism (ticope) (Same as truncated 24cell prism) 
50  336  864  960  384 
81  = t_{0,1,3}{3,3,4}×{ } Runcitruncated 16cell prism (prittip) 
82  528  1216  1152  384 
82  = sr{3,3,4}×{ } snub 24cell prism (sadip) 
146  768  1392  960  192 
Nonuniform  rectified tesseractic alterprism (rita) 
50  288  464  288  64 
Nonuniform  truncated 16cell alterprism (thexa) 
26  168  384  336  96 
Nonuniform  bitruncated tesseractic alterprism (taha) 
50  288  624  576  192 
This prismatic family has 10 forms.
The A_{1} x F_{4} 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 24cell 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}×{ } 24cell prism (icope) 
26  144  288  216  48 
[79]  = r{3,4,3}×{ } rectified 24cell prism (ricope) 
50  336  768  672  192 
[80]  = t{3,4,3}×{ } truncated 24cell prism (ticope) 
50  336  864  960  384 
83  = rr{3,4,3}×{ } cantellated 24cell prism (sricope) 
146  1008  2304  2016  576 
84  = t_{0,3}{3,4,3}×{ } runcinated 24cell prism (spiccup) 
242  1152  1920  1296  288 
85  = 2t{3,4,3}×{ } bitruncated 24cell prism (contip) 
50  432  1248  1440  576 
86  = tr{3,4,3}×{ } cantitruncated 24cell prism (gricope) 
146  1008  2592  2880  1152 
87  = t_{0,1,3}{3,4,3}×{ } runcitruncated 24cell prism (pricope) 
242  1584  3648  3456  1152 
88  = t_{0,1,2,3}{3,4,3}×{ } omnitruncated 24cell prism (gippiccup) 
242  1872  5088  5760  2304 
[82]  = s{3,4,3}×{ } snub 24cell prism (sadip) 
146  768  1392  960  192 
This prismatic family has 15 forms:
The A_{1} x H_{4} family has symmetry of order 28800 (2*14400).
#  Coxeter diagram and Schläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
89  = {5,3,3}×{ } 120cell prism (hipe) 
122  960  2640  3000  1200 
90  = r{5,3,3}×{ } Rectified 120cell prism (rahipe) 
722  4560  9840  8400  2400 
91  = t{5,3,3}×{ } Truncated 120cell prism (thipe) 
722  4560  11040  12000  4800 
92  = rr{5,3,3}×{ } Cantellated 120cell prism (srahip) 
1922  12960  29040  25200  7200 
93  = t_{0,3}{5,3,3}×{ } Runcinated 120cell prism (sidpixhip) 
2642  12720  22080  16800  4800 
94  = 2t{5,3,3}×{ } Bitruncated 120cell prism (xhip) 
722  5760  15840  18000  7200 
95  = tr{5,3,3}×{ } Cantitruncated 120cell prism (grahip) 
1922  12960  32640  36000  14400 
96  = t_{0,1,3}{5,3,3}×{ } Runcitruncated 120cell prism (prixip) 
2642  18720  44880  43200  14400 
97  = t_{0,1,2,3}{5,3,3}×{ } Omnitruncated 120cell prism (gidpixhip) 
2642  22320  62880  72000  28800 
98  = {3,3,5}×{ } 600cell prism (exip) 
602  2400  3120  1560  240 
99  = r{3,3,5}×{ } Rectified 600cell prism (roxip) 
722  5040  10800  7920  1440 
100  = t{3,3,5}×{ } Truncated 600cell prism (texip) 
722  5040  11520  10080  2880 
101  = rr{3,3,5}×{ } Cantellated 600cell prism (srixip) 
1442  11520  28080  25200  7200 
102  = tr{3,3,5}×{ } Cantitruncated 600cell prism (grixip) 
1442  11520  31680  36000  14400 
103  = t_{0,1,3}{3,3,5}×{ } Runcitruncated 600cell prism (prahip) 
2642  18720  44880  43200  14400 
Uniform duoprism prisms, {p}×{q}×{ }, form an infinite class for all integers p,q>2. {4}×{4}×{ } makes a lower symmetry form of the 5cube.
The extended fvector of {p}×{q}×{ } is computed as (p,p,1)*(q,q,1)*(2,1) = (2pq,5pq,4pq+2p+2q,3pq+3p+3q,p+q+2,1).
Coxeter diagram  Names  Element counts  

4faces  Cells  Faces  Edges  Vertices  
{p}×{q}×{ }^{[9]}  p+q+2  3pq+3p+3q  4pq+2p+2q  5pq  2pq  
{p}^{2}×{ }  2(p+1)  3p(p+1)  4p(p+1)  5p^{2}  2p^{2}  
{3}^{2}×{ }  8  36  48  45  18  
{4}^{2}×{ } = 5cube  10  40  80  80  32 
The grand antiprism prism is the only known convex nonWythoffian uniform 5polytope. 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  

Facets  Cells  Faces  Edges  Vertices  
104  grand antiprism prism (gappip)^{[10]}  322  1360  1940  1100  200 
Construction of the reflective 5dimensional 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 5polytopes 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 5polytopes.
The last operation, the snub, and more generally the alternation, are the operations 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  t_{0}{p,q,r,s}  {p,q,r,s}  Any regular 5polytope  
Rectified  t_{1}{p,q,r,s}  r{p,q,r,s}  The edges are fully truncated into single points. The 5polytope now has the combined faces of the parent and dual.  
Birectified  t_{2}{p,q,r,s}  2r{p,q,r,s}  Birectification reduces faces to points, cells to their duals.  
Trirectified  t_{3}{p,q,r,s}  3r{p,q,r,s}  Trirectification reduces cells to points. (Dual rectification)  
Quadrirectified  t_{4}{p,q,r,s}  4r{p,q,r,s}  Quadrirectification reduces 4faces to points. (Dual)  
Truncated  t_{0,1}{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 5polytope. The 5polytope has its original faces doubled in sides, and contains the faces of the dual.  
Cantellated  t_{0,2}{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  t_{0,3}{p,q,r,s}  Runcination reduces cells and creates new cells at the vertices and edges.  
Stericated  t_{0,4}{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 5polytopes.)  
Omnitruncated  t_{0,1,2,3,4}{p,q,r,s}  All four operators, truncation, cantellation, runcination, and sterication are applied.  
Half  h{2p,3,q,r}  Alternation, same as  
Cantic  h_{2}{2p,3,q,r}  Same as  
Runcic  h_{3}{2p,3,q,r}  Same as  
Runcicantic  h_{2,3}{2p,3,q,r}  Same as  
Steric  h_{4}{2p,3,q,r}  Same as  
Steriruncic  h_{3,4}{2p,3,q,r}  Same as  
Stericantic  h_{2,4}{2p,3,q,r}  Same as  
Steriruncicantic  h_{2,3,4}{2p,3,q,r}  Same as  
Snub  s{p,2q,r,s}  Alternated truncation  
Snub rectified  sr{p,q,2r,s}  Alternated truncated rectification  
ht_{0,1,2,3}{p,q,r,s}  Alternated runcicantitruncation  
Full snub  ht_{0,1,2,3,4}{p,q,r,s}  Alternated omnitruncation 
There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4space.^{[11]}^{[12]}
#  Coxeter group  Coxeter diagram  Forms  

1  [3^{[5]}]  [(3,3,3,3,3)]  7  
2  [4,3,3,4]  19  
3  [4,3,3^{1,1}]  [4,3,3,4,1^{+}]  =  23 (8 new)  
4  [3^{1,1,1,1}]  [1^{+},4,3,3,4,1^{+}]  =  9 (0 new)  
5  [3,4,3,3]  31 (21 new) 
There are three regular honeycombs of Euclidean 4space:
Other families that generate uniform honeycombs:
NonWythoffian uniform tessellations in 4space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.
#  Coxeter group  Coxeter diagram  

1  ×  [4,3,4,2,∞]  
2  ×  [4,3^{1,1},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,∞]  
8  x  [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] 
There are 5 compact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in hyperbolic 4space as permutations of rings of the Coxeter diagrams.
= [(3,3,3,3,4)]: 
= [5,3,3^{1,1}]: 
= [3,3,3,5]: = [4,3,3,5]: 
There are 5 regular compact convex hyperbolic honeycombs in H^{4} space:^{[13]}
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 

Order5 5cell (pente)  {3,3,3,5}  {3,3,3}  {3,3}  {3}  {5}  {3,5}  {3,3,5}  {5,3,3,3}  
Order3 120cell (hitte)  {5,3,3,3}  {5,3,3}  {5,3}  {5}  {3}  {3,3}  {3,3,3}  {3,3,3,5}  
Order5 tesseractic (pitest)  {4,3,3,5}  {4,3,3}  {4,3}  {4}  {5}  {3,5}  {3,3,5}  {5,3,3,4}  
Order4 120cell (shitte)  {5,3,3,4}  {5,3,3}  {5,3}  {5}  {4}  {3,4}  {3,3,4}  {4,3,3,5}  
Order5 120cell (phitte)  {5,3,3,5}  {5,3,3}  {5,3}  {5}  {5}  {3,5}  {3,3,5}  Selfdual 
There are also 4 regular compact hyperbolic starhoneycombs in H^{4} 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 

Order3 small stellated 120cell  {5/2,5,3,3}  {5/2,5,3}  {5/2,5}  {5}  {5}  {3,3}  {5,3,3}  {3,3,5,5/2}  
Order5/2 600cell  {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}  
Order5 icosahedral 120cell  {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}  
Order3 great 120cell  {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} 
There are 9 paracompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4space as permutations of rings of the Coxeter diagrams. Paracompact groups generate honeycombs with infinite facets or vertex figures.
