5simplex (hexateron) 
Truncated 5simplex 
Rectified 5simplex 
Birectified 5simplex 

5orthoplex, 2_{11} (Pentacross) 
Truncated 5orthoplex 
Rectified 5orthoplex 

5cube (Penteract) 
Truncated 5cube 
Rectified 5cube 

5demicube. 1_{21} (Demipenteract) 
Truncated 5demicube 
Rectified 5demicube 
In geometry, a uniform polyteron (or uniform 5polytope) is a fivedimensional uniform polytope. By definition, a uniform polyteron is vertextransitive and constructed from uniform polychoron facets.
The complete set of convex uniform polytera has not been determined, but most 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 CoxeterDynkin diagrams.
Regular 5polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face. There are exactly three convex and zero nonconvex regular polytopes.
There are exactly three such convex regular polytopes:
There are no nonconvex regular 5polytopes.
There are 105 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.
The hexateron is the regular form in the A_{5} family. The penteract and pentacross are the regular forms in the B_{5} family. The bifurcating graph of the D_{6} family contains the pentacross, as well as a demipenteract which is an alternated penteract.
Fundamental families
#  Coxeter group  CoxeterDynkin diagram  

1  A_{5}  [3^{4}]  
2  B_{5}  [4,3^{3}]  
3  D_{5}  [3^{2,1,1}] 
Uniform prisms There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4polytopes:
#  Coxeter groups  Coxeter graph  

1  A_{4} × A_{1}  [3,3,3] × [ ]  
2  B_{4} × A_{1}  [4,3,3] × [ ]  
3  F_{4} × A_{1}  [3,4,3] × [ ]  
4  H_{4} × A_{1}  [5,3,3] × [ ]  
5  D_{4} × A_{1}  [3^{1,1,1}] × [ ] 
There is one infinite family of 5polytopes based on prisms of the the uniform duoprisms {p}×{q}×{ }:
Coxeter groups  Coxeter graph  

I_{2}(p) × I_{2}(q) × A_{1}  [p] × [q] × [ ] 
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 groups  Coxeter graph  

1  A_{3} × I_{2}(p)  [3,3] × [p]  
2  B_{3} × I_{2}(p)  [4,3] × [p]  
3.  H_{3} × I_{2}(p)  [5,3] × [p] 
That brings the tally to: 19+31+8+46+1=105
In addition there are:
There are 19 forms based on all permutations of the CoxeterDynkin 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).
#  CoxeterDynkin diagram Schläfli symbol Name 
Graph  Vertex figure 
Facet counts by location: [3,3,3,3]  Element counts  

4  3  2  1  0  
[3,3,3] (6) 
[3,3]×[ ] (15) 
[3]×[3] (20) 
[ ]×[3,3] (15) 
[3,3,3] (6) 
4faces  Cells  Faces  Edges  Vertices  
1  t_{0}{3,3,3,3} Hexateron (hix) 
{3,3,3} 
(5) {3,3,3} 
        6  15  20  15  6  
2  t_{1}{3,3,3,3} Rectified hexateron (rix) 
{3,3}x{ } 
(4) t_{1}{3,3,3} 
      (2) {3,3,3} 
12  45  80  60  15  
3  t_{2}{3,3,3,3} Birectified hexateron (dot) 
{3}x{3} 
(3) t_{1}{3,3,3} 
      (3) t_{1}{3,3,3} 
12  60  120  90  20  
4  t_{0,1}{3,3,3,3} Truncated hexateron (tix) 
Tetrah.pyr 
(4) t_{0,1}{3,3,3} 
      (1) {3,3,3} 
12  45  80  75  30  
5  t_{1,2}{3,3,3,3} Bitruncated hexateron (bittix) 
(3) t_{1,2}{3,3,3} 
      (2) t_{0,1}{3,3,3} 
12  60  140  150  60  
6  t_{0,2}{3,3,3,3} Cantellated hexateron (sarx) 
prismwedge 
(3) t_{0,2}{3,3,3} 
    (1) { }×{3,3} 
(1) t_{1}{3,3,3} 
27  135  290  240  60  
7  t_{1,3}{3,3,3,3} Bicantellated hexateron (sibrid) 
(2) t_{0,2}{3,3,3} 
  (8) {3}×{3} 
  (2) t_{0,2}{3,3,3} 
32  180  420  360  90  
8  t_{0,3}{3,3,3,3} Runcinated hexateron (spix) 
(2) t_{0,3}{3,3,3} 
  (3) {3}×{3} 
(3) { }×t_{1}{3,3} 
(1) t_{1}{3,3,3} 
47  255  420  270  60  
9  t_{0,4}{3,3,3,3} Stericated hexateron (scad) 
Irr.16cell 
(1) {3,3,3} 
(4) { }×{3,3} 
(6) {3}×{3} 
(4) { }×{3,3} 
(1) {3,3,3} 
62  180  210  120  30  
10  t_{0,1,2}{3,3,3,3} Cantitruncated hexateron (garx) 
t_{0,1,2}{3,3,3} 
    { }×{3,3} 
t_{0,1}{3,3,3} 
27  135  290  300  120  
11  t_{1,2,3}{3,3,3,3} Bicantitruncated hexateron (gibrid) 
t_{0,1,2}{3,3,3} 
  {3}×{3} 
  t_{0,1,2}{3,3,3} 
32  180  420  450  180  
12  t_{0,1,3}{3,3,3,3} Runcitruncated hexateron (pattix) 
t_{0,1,3}{3,3,3} 
  {6}×{3} 
{ }×t_{1}{3,3} 
t_{0,2}{3,3,3} 
47  315  720  630  180  
13  t_{0,2,3}{3,3,3,3} Runcicantellated hexateron (pirx) 
t_{0,1,3}{3,3,3} 
  {3}×{3} 
{ }×t_{0,1}{3,3} 
t_{1,2}{3,3,3} 
47  255  570  540  180  
14  t_{0,1,4}{3,3,3,3} Steritruncated hexateron (cappix) 
t_{0,1}{3,3,3} 
{ }×t_{0,1}{3,3} 
{3}×{6} 
{ }×{3,3} 
t_{0,3}{3,3,3} 
62  330  570  420  120  
15  t_{0,2,4}{3,3,3,3} Stericantellated hexateron (card) 
t_{0,2}{3,3,3} 
{ }×t_{0,2}{3,3} 
{3}×{3} 
{ }×t_{0,2}{3,3} 
t_{0,2}{3,3,3} 
62  420  900  720  180  
16  t_{0,1,2,3}{3,3,3,3} Runcicantitruncated hexateron (gippix) 
Irr.5cell 
t_{0,1,2,3}{3,3,3} 
  {3}×{6} 
{ }×t_{0,1}{3,3} 
t_{0,2}{3,3,3} 
47  315  810  900  360  
17  t_{0,1,2,4}{3,3,3,3} Stericantitruncated hexateron (cograx) 
t_{0,1,2}{3,3,3} 
{ }×t_{0,1,2}{3,3} 
{3}×{6} 
{ }×t_{0,2}{3,3} 
t_{0,1,3}{3,3,3} 
62  480  1140  1080  360  
18  t_{0,1,3,4}{3,3,3,3} Steriruncitruncated hexateron (captid) 
t_{0,1,3}{3,3,3} 
{ }×t_{0,1}{3,3} 
{6}×{6} 
{ }×t_{0,1,3}{3,3} 
t_{0,1,3}{3,3,3} 
62  450  1110  1080  360  
19  t_{0,1,2,3,4}{3,3,3,3} Omnitruncated hexateron (gocad) 
Irr. {3,3,3} 
(1) t_{0,1,2,3}{3,3,3} 
(1) { }×t_{0,1,2}{3,3} 
(1) {6}×{6} 
(1) { }×t_{0,1,2}{3,3} 
(1) t_{0,1,2,3}{3,3,3} 
62  540  1560  1800  720 
This family has 31 Wythoffian uniform polyhedra, from 2^{5}1 permutations of the CoxeterDynkin diagram with one or more rings.
For simplicity it divided into two subfamilies, each with 12 forms, and 7 "middle" forms which equally belong in both subfamilies.
The B_{5} family has symmetry of order 3840 (2^5*5!).
There are 20 forms here, 7 shared with the pentacross family. Four are shared with the demipenteract family.
#  CoxeterDynkin andSchläfli symbols Name 
Graph  Vertex figure 
Facet counts by location: [4,3,3,3]  Element counts  

4  3  2  1  0  
[4,3,3] (10) 
[4,3]×[ ] (40) 
[4]×[3] (80) 
[ ]×[3,3] (80) 
[3,3,3] (32) 
Facets  Cells  Faces  Edges  Vertices  
20  t_{0}{4,3,3,3} Penteract Pent 
{3,3,3} 
{4,3,3} 
        10  40  80  80  32  
21  t_{1}{4,3,3,3} Rectified penteract Rin 
{3,3}x{ } 
t_{1}{4,3,3} 
      {3,3,3} 
42  200  400  320  80  
22  t_{2}{4,3,3,3} Birectified penteract Nit 
{4}×{3} 
t_{1}{4,3,3} 
      t_{1}{3,3,3} 
42  280  640  480  80  
23  t_{0,1}{4,3,3,3} Truncated penteract Tan 
Tetrah.pyr 
t_{0,1}{4,3,3} 
      {3,3,3} 
42  200  400  400  160  
24  t_{1,2}{4,3,3,3} Bitruncated penteract Bittin 
t_{1,2}{4,3,3} 
      t_{0,1}{3,3,3} 
42  280  720  800  320  
25  t_{0,2}{4,3,3,3} Cantellated penteract Sirn 
Prismwedge 
t_{0,2}{4,3,3} 
    { }×{3,3} 
t_{1}{3,3,3} 
122  680  1520  1280  320  
26  t_{1,3}{4,3,3,3} Bicantellated penteract Sibrant 
t_{0,2}{4,3,3} 
  {4}×{3} 
  t_{0,2}{3,3,3} 
122  840  2160  1920  480  
27  t_{0,3}{4,3,3,3} Runcinated penteract Span 
t_{0,3}{4,3,3} 
  {4}×{3} 
{ }×t_{1}{3,3} 
{3,3,3} 
202  1240  2160  1440  320  
28  t_{0,4}{4,3,3,3} Stericated penteract Scant 
Tetr.antiprm 
{4,3,3} 
{4,3}×{ } 
{4}×{3} 
{ }×{3,3} 
{3,3,3} 
242  800  1040  640  160  
29  t_{0,1,2}{4,3,3,3} Cantitruncated penteract Girn 
t_{0,1,2}{4,3,3} 
    { }×{3,3} 
t_{0,1}{3,3,3} 
122  680  1520  1600  640  
30  t_{1,2,3}{4,3,3,3} Bicantitruncated penteract Gibrant 
t_{0,1,2}{4,3,3} 
  {4}×{3} 
  t_{0,1,2}{3,3,3} 
122  840  2160  2400  960  
31  t_{0,1,3}{4,3,3,3} Runcitruncated penteract Pattin 
t_{0,1,3}{4,3,3} 
  {8}×{3}  { }×t_{1}{3,3} 
t_{0,2}{3,3,3} 
202  1560  3760  3360  960  
32  t_{0,2,3}{4,3,3,3} Runcicantellated penteract Prin 
t_{0,1,3}{4,3,3} 
  {4}×{3} 
{ }×t_{0,1}{3,3} 
t_{1,2}{3,3,3} 
202  1240  2960  2880  960  
33  t_{0,1,4}{4,3,3,3} Steritruncated penteract Capt 
t_{0,1}{4,3,3} 
t_{0,1}{4,3}×{ } 
{8}×{3}  { }×{3,3} 
t_{0,3}{3,3,3} 
242  1600  2960  2240  640  
34  t_{0,2,4}{4,3,3,3} Stericantellated penteract Carnit 
t_{0,2}{4,3,3} 
t_{0,2}{4,3}×{ } 
{4}×{3} 
{ }×t_{0,2}{3,3} 
t_{0,2}{3,3,3} 
242  2080  4720  3840  960  
35  t_{0,1,2,3}{4,3,3,3} Runcicantitruncated penteract Gippin 
t_{0,1,2,3}{4,3,3} 
  {8}×{3}  { }×t_{0,1}{3,3} 
t_{0,1,2}{3,3,3} 
202  1560  4240  4800  1920  
36  t_{0,1,2,4}{4,3,3,3} Stericantitruncated penteract Cogrin 
t_{0,1,2}{4,3,3} 
t_{0,1,2}{4,3}×{ } 
{8}×{3}  { }×t_{0,2}{3,3} 
t_{0,1,3}{3,3,3} 
242  2400  6000  5760  1920  
37  t_{0,1,3,4}{4,3,3,3} Steriruncitruncated penteract Captint 
t_{0,1,3}{4,3,3} 
t_{0,1}{4,3}×{ } 
{8}×{6}  { }×t_{0,1}{3,3} 
t_{0,1,3}{3,3,3} 
242  2160  5760  5760  1920  
38  t_{0,1,2,3,4}{4,3,3,3} Omnitruncated penteract Gacnet 
Irr. {3,3,3} 
t_{0,1,2}{4,3}×{ } 
t_{0,1,2}{4,3}×{ } 
{8}×{6}  { }×t_{0,1,2}{3,3} 
t_{0,1,2,3}{3,3,3} 
242  2640  8160  9600  3840  
[51]  h_{0}{4,3,3,3} Demipenteract Hin 
t_{1}{3,3,3} 
(16) {3,3,3} 
      {3,3,4} 
26  120  160  80  16 
There are 19 forms, 12 new ones. 7 are shared from the penteract family, and 10 shared with the demipenteract family.
#  CoxeterDynkin andSchläfli symbols Name 
Graph  Vertex figure 
Facet counts by location: [4,3,3,3]  Element counts  

4  3  2  1  0  
[3,3,3] (32) 
[3,3]×[ ] (80) 
[3]×[4] (80) 
[ ]×[3,4] (40) 
[3,3,4] (10) 
Facets  Cells  Faces  Edges  Vertices  
39  t_{0}{3,3,3,4} Pentacross Tac 
{3,3,4} 
{3,3,3} 
        32  80  80  40  10  
40  t_{1}{3,3,3,4} Rectified pentacross Rat 
{ }×{3,4} 
t_{1}{3,3,3} 
      {3,3,4} 
42  240  400  240  40  
[22]  t_{2}{3,3,3,4} Birectified pentacross Nit 
{4}×{3} 
t_{1}{3,3,3} 
      t_{1}{3,3,4} 
42  280  640  480  80  
41  t_{0,1}{3,3,3,4} Truncated pentacross Tot 
(Octah.pyr) 
t_{0,1}{3,3,3} 
      {3,3,3}  42  240  400  280  80  
42  t_{1,2}{3,3,3,4} Bitruncated pentacross Bittit 
t_{1,2}{3,3,3} 
      t_{0,1}{3,3,4}  42  280  720  720  240  
43  t_{0,2}{3,3,3,4} Cantellated pentacross Sart 
Prismwedge 
t_{0,2}{3,3,3} 
    { }×{3,4}  t_{1}{3,3,4}  82  640  1520  1200  240  
[26]  t_{1,3}{3,3,3,4} Bicantellated pentacross Sibrant 
t_{1,3}{3,3,3} 
  {3}×{4}    t_{0,2}{3,3,4}  122  840  2160  1920  480  
44  t_{0,3}{3,3,3,4} Runcinated pentacross Spat 
t_{0,3}{3,3,3} 
  {3}×{4}  t_{1}{4,3,3}  162  1200  2160  1440  320  
[28]  t_{0,4}{3,3,3,4} Stericated pentacross Scant 
Tetr.antiprm 
{3,3,3} 
      {4,3,3}  242  800  1040  640  160  
45  t_{0,1,2}{3,3,3,4} Cantitruncated pentacross Gart 
t_{0,1,2}{3,3,3} 
      t_{0,1}{3,3,4}  82  640  1520  1440  480  
[30]  t_{1,2,3}{3,3,3,4} Bicantitruncated pentacross Gibrant 
t_{1,2,3}{3,3,3} 
  {3}×{4}    t_{0,1,2}{3,3,4}  122  840  2160  2400  960  
46  t_{0,1,3}{3,3,3,4} Runcitruncated pentacross Pattit 
t_{0,1,3}{3,3,3} 
  {6}×{4}  { }×t_{1}{3,4}  t_{0,2}{3,3,4}  162  1440  3680  3360  960  
47  t_{0,2,3}{3,3,3,4} Runcicantellated pentacross Pirt 
t_{0,1,3}{3,3,3} 
  {3}×{4}  { }×t_{0,1}{3,4}  t_{1,2}{3,3,4}  162  1200  2960  2880  960  
48  t_{0,1,4}{3,3,3,4} Steritruncated pentacross Cappin 
t_{0,1}{3,3,3} 
    { }×{4,3}  t_{0,3}{3,3,4}  242  1520  2880  2240  640  
[34]  t_{0,2,4}{3,3,3,4} Stericantellated pentacross Carnit 
t_{0,2}{3,3,3} 
{ }×t_{0,2}{3,3}  {3}×{4}  { }×t_{0,2}{3,4}  t_{0,2}{4,3,3}  242  2080  4720  3840  960  
49  t_{0,1,2,3}{3,3,3,4} Runcicantitruncated pentacross Gippit 
t_{0,1,2,3}{3,3,3} 
  {6}×{4}  { }×t_{0,1}{3,4}  t_{0,1,2}{3,3,4}  162  1440  4160  4800  1920  
50  t_{0,1,2,4}{3,3,3,4} Stericantitruncated pentacross Cogart 
t_{0,1,2}{3,3,3} 
{ }×t_{0,1,}{3,3}  {6}×{4}  { }×t_{0,2}{3,4}  t_{0,1,3}{3,3,4}  242  2320  5920  5760  1920  
[37]  t_{0,1,3,4}{3,3,3,4} Steriruncitruncated pentacross Captint 
t_{0,1,3}{3,3,3} 
{ }×t_{0,1}{3,3}  {6}×{8}  { }×t_{0,1}{4,3}  t_{0,1,3}{4,3,3}  242  2160  5760  5760  1920  
[38]  t_{0,1,2,3,4}{3,3,3,4} Omnitruncated pentacross Gacnet 
Irr. {3,3,3} 
t_{0,1,2,3}{3,3,3} 
{ }×t_{0,1,2}{3,3}  {6}×{8}  { }×t_{0,1,2}{3,4}  t_{0,1,2,3}{3,3,4}  242  2640  8160  9600  3840 
There are 23 forms. 16 are repeated from the [4,3,3,3] family and 7 are new ones.
The D_{5} family has symmetry of order 1920 (2^4*5!).
#  CoxeterDynkin and Schläfli symbols Name 
Graph  Vertex figure 
Facets by location: [3^{1,2,1}]  Element counts  

4  3  2  1  0  
[3,3,3] (16) 
[3^{1,1,1}] (10) 
[3,3]×[ ] (40) 
[ ]×[3]×[ ] (80) 
[3,3,3] (16) 
Facets  Cells  Faces  Edges  Vertices  
51  (1_{21}) Demipenteract Hin 
t_{1}{3,3,3} 
{3,3,3}  t_{0}(1_{11})        26  120  160  80  16  
[22]  t_{1}(1_{21}) Rectified demipenteract (Same as birectified penteract) Nit 
{ }×{ }×{3} 
t_{1}{3,3,3}  t_{1}(1_{11})      t_{1}{3,3,3}  42  280  640  480  80  
[40]  t_{2}(1_{21}) Birectified demipenteract (Same as rectified pentacross) Rat 
{ }×t_{1}{3,3} 
t_{1}{3,3,3}  t_{0}(1_{11})      t_{1}{3,3,3}  42  240  400  240  40  
[39]  t_{3}(1_{21}) Trirectified demipenteract (Same as pentacross) Tac 
(1_{11}) 
{3,3,3}        {3,3,3}  32  80  80  40  10  
52  t_{0,1}(1_{21}) Truncated demipenteract Thin 
42  280  640  560  160  
53  t_{0,2}(1_{21}) Cantellated demipenteract Sirhin 
42  360  880  720  160  
54  t_{0,3}(1_{21}) Runcinated demipenteract Siphin 
82  480  720  400  80  
[21]  t_{0,4}(1_{21}) Stericated demipenteract (Same as rectified penteract) Rin 
{3,3}x{ } 
42  200  400  320  80  
[42]  t_{1,2}(1_{21}) Bitruncated demipenteract (Same as bitruncated pentacross) Bittit 
42  280  720  720  240  
[43]  t_{1,3}(1_{21}) Bicantellated demipenteract (Same as cantellated pentacross) Sart 
Prismwedge 
82  640  1520  1200  240  
[41]  t_{2,3}(1_{21}) Tritruncated demipenteract (Same as truncated pentacross) Tot 
(Octah.pyr) 
42  240  400  280  80  
[24]  t_{0,1,4}(1_{21}) Steritruncated demipenteract (Same as bitruncated penteract) Bittin 
42  280  720  800  320  
55  t_{0,1,2}(1_{21}) Cantitruncated demipenteract Girhin 
42  360  1040  1200  480  
56  t_{0,1,3}(1_{21}) Runcitruncated demipenteract Pithin 
82  720  1840  1680  480  
[26]  t_{0,2,4}(1_{21}) Stericantellated demipenteract (Same as bicantellated penteract) Sibrant 
122  840  2160  1920  480  
[44]  t_{0,3,4}(1_{21}) Steriruncinated demipenteract (Same as runcinated pentacross) Spat 
162  1200  2160  1440  320  
57  t_{0,2,3}(1_{21}) Runcicantellated demipenteract Pirhin 
82  560  1280  1120  320  
[45]  t_{1,2,3}(1_{21}) Bicantitruncated demipenteract (Same as cantitruncated pentacross) Gart 
82  640  1520  1440  480  
[30]  t_{0,1,2,4}(1_{21}) Stericantitruncated demipenteract (Same as bicantitruncated pentacross) Gibrant 
122  840  2160  2400  960  
[46]  t_{0,1,3,4}(1_{21}) Steriruncitruncated demipenteract (Same as runcicantellated pentacross) Pirt 
162  1440  3680  3360  960  
58  t_{0,1,2,3}(1_{21}) Runcicantitruncated demipenteract Giphin 
82  720  2080  2400  960  
[47]  t_{0,2,3,4}(1_{21}) Steriruncicantellated demipenteract (Same as runcitruncated pentacross) Pattit 
162  1200  2960  2880  960  
[49]  t_{0,1,2,3,4}(1_{21}) Omnitruncated demipenteract (Same as runcicantitruncated pentacross) Gippit 
Irr. {3,3,3} 
162  1440  4160  4800  1920 
There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4polytopes:
This prismatic family has 9 forms:
The A_{1} x A_{4} family has symmetry of order 240 (2*5!).
#  CoxeterDynkin andSchläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
59  {3,3,3}x{ } 5cell prism 
7  20  30  25  10 
60  t_{1}{3,3,3}x{ } Rectified 5cell prism 
12  50  90  70  20 
61  t_{0,1}{3,3,3}x{ } Truncated 5cell prism 
12  50  100  100  40 
62  t_{0,2}{3,3,3}x{ } Cantellated 5cell prism 
22  120  250  210  60 
63  t_{0,3}{3,3,3}x{ } Runcinated 5cell prism 
32  130  200  140  40 
64  t_{1,2}{3,3,3}x{ } Bitruncated 5cell prism 
12  60  140  150  60 
65  t_{0,1,2}{3,3,3}x{ } Cantitruncated 5cell prism 
22  120  280  300  120 
66  t_{0,1,3}{3,3,3}x{ } Runcitruncated 5cell prism 
32  180  390  360  120 
67  t_{0,1,2,3}{3,3,3}x{ } Omnitruncated 5cell prism 
32  210  540  600  240 
This prismatic family has 16 forms. (Three are shared with [3,4,3]×[ ] family)
The A_{1} x B_{4} family has symmetry of order 768 (2*2^4*4!).
#  CoxeterDynkin andSchläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
68  {4,3,3}x{ } Tesseractic prism 
10  40  80  80  32 
69  t_{1}{4,3,3}x{ } Rectified tesseractic prism 
26  136  272  224  64 
70  t_{0,1}{4,3,3}x{ } Truncated tesseractic prism 
26  136  304  320  128 
71  t_{0,2}{4,3,3}x{ } Cantellated tesseractic prism 
58  360  784  672  192 
72  t_{0,3}{4,3,3}x{ } Runcinated tesseractic prism 
82  368  608  448  128 
73  t_{1,2}{4,3,3}x{ } Bitruncated tesseractic prism 
26  168  432  480  192 
74  t_{0,1,2}{4,3,3}x{ } Cantitruncated tesseractic prism 
58  360  880  960  384 
75  t_{0,1,3}{4,3,3}x{ } Runcitruncated tesseractic prism 
82  528  1216  1152  384 
76  t_{0,1,2,3}{4,3,3}x{ } Omnitruncated tesseractic prism 
82  624  1696  1920  768 
#  CoxeterDynkin andSchläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
77  {3,3,4}x{ } 16cell prism 
18  64  88  56  16 
78  t_{1}{3,3,4}x{ } Rectified 16cell prism (Same as 24cell prism) 
26  144  288  216  48 
79  t_{0,1}{3,3,4}x{ } Truncated 16cell prism 
26  144  312  288  96 
80  t_{0,2}{3,3,4}x{ } Cantellated 16cell prism (Same as rectified 24cell prism) 
50  336  768  672  192 
[72]  t_{0,3}{4,3,3}x{ } Runcinated 16cell prism (Same as Runcinated tesseractic prism) 
82  368  608  448  128 
[73]  t_{1,2}{4,3,3}x{ } Bitruncated 16cell prism (Same as bitruncated tesseractic prism) 
26  168  432  480  192 
81  t_{0,1,2}{3,3,4}x{ } Cantitruncated 16cell prism (Same as truncated 24cell prism) 
50  336  864  960  384 
82  t_{0,1,3}{3,3,4}x{ } Runcitruncated 16cell prism 
82  528  1216  1152  384 
[76]  t_{0,1,2,3}{3,3,4}x{ } Omnitruncated 16cell prism (Same as omnitruncated tesseractic prism) 
82  624  1696  1920  768 
83  h_{0,1,2}{3,3,4}x{ } Snub 24cell prism 
146  768  1392  960  192 
This prismatic family has 10 forms.
The A_{1} x F_{4} family has symmetry of order 2304 (2*1152).
#  CoxeterDynkin andSchläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
[79]  {3,4,3}x{ } 24cell prism 
26  144  288  216  48 
[80]  t_{1}{3,4,3}x{ } rectified 24cell prism 
50  336  768  672  192 
[81]  t_{0,1}{3,4,3}x{ } truncated 24cell prism 
50  336  864  960  384 
84  t_{0,2}{3,4,3}x{ } cantellated 24cell prism 
146  1008  2304  2016  576 
85  t_{0,3}{3,4,3}x{ } runcinated 24cell prism 
242  1152  1920  1296  288 
86  t_{1,2}{3,4,3}x{ } bitruncated 24cell prism 
50  432  1248  1440  576 
87  t_{0,1,2}{3,4,3}x{ } cantitruncated 24cell prism 
146  1008  2592  2880  1152 
88  t_{0,1,3}{3,4,3}x{ } runcitruncated 24cell prism 
242  1584  3648  3456  1152 
89  t_{0,1,2,3}{3,4,3}x{ } omnitruncated 24cell prism 
242  1872  5088  5760  2304 
[83]  h_{0,1}{3,4,3}x{ } snub 24cell prism 
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).
#  CoxeterDynkin andSchläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
90  {5,3,3}x{ } 120cell prism 
122  960  2640  3000  1200 
91  t_{1}{5,3,3}x{ } Rectified 120cell prism 
722  4560  9840  8400  2400 
92  t_{0,1}{5,3,3}x{ } Truncated 120cell prism 
722  4560  11040  12000  4800 
93  t_{0,2}{5,3,3}x{ } Cantellated 120cell prism 
1922  12960  29040  25200  7200 
94  t_{0,3}{5,3,3}x{ } Runcinated 120cell prism 
2642  12720  22080  16800  4800 
95  t_{1,2}{5,3,3}x{ } Bitruncated 120cell prism 
722  5760  15840  18000  7200 
96  t_{0,1,2}{5,3,3}x{ } Cantitruncated 120cell prism 
1922  12960  32640  36000  14400 
97  t_{0,1,3}{5,3,3}x{ } Runcitruncated 120cell prism 
2642  18720  44880  43200  14400 
98  t_{0,1,2,3}{5,3,3}x{ } Omnitruncated 120cell prism 
2642  22320  62880  72000  28800 
#  CoxeterDynkin andSchläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
99  {3,3,5}x{ } 600cell prism 
602  2400  3120  1560  240 
100  t_{1}{3,3,5}x{ } Rectified 600cell prism 
722  5040  10800  7920  1440 
101  t_{0,1}{3,3,5}x{ } Truncated 600cell prism 
722  5040  11520  10080  2880 
102  t_{0,2}{3,3,5}x{ } Cantellated 600cell prism 
1442  11520  28080  25200  7200 
[94]  t_{0,3}{3,3,5}x{ } Runcinated 600cell prism (Same as runcinated 120cell prism) 
2642  12720  22080  16800  4800 
[95]  t_{1,2}{3,3,5}x{ } Bitruncated 600cell prism (Same as bitruncated 120cell prism) 
722  5760  15840  18000  7200 
103  t_{0,1,2}{3,3,5}x{ } Cantitruncated 600cell prism 
1442  11520  31680  36000  14400 
104  t_{0,1,3}{3,3,5}x{ } Runcitruncated 600cell prism 
2642  18720  44880  43200  14400 
[98]  t_{0,1,2,3}{3,3,5}x{ } Omnitruncated 600cell prism (Same as omnitruncated 120cell prism) 
2642  22320  62880  72000  28800 
The grand antiprism prism is the only known convex nonwythoffian uniform polyteron. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (300 tetrahedrons, 20 pentagonal antiprisms, 700 triangular prisms, 20 pentagonal prisms), 322 hypercells (2 grand antiprisms , 20 pentagonal antiprism prisms , and 300 tetrahedral prisms ).
#  Name  Element counts  

Facets  Cells  Faces  Edges  Vertices  
105  grand antiprism prism Gappip 
322  1360  1940  1100  200 
Construction of the reflective 5dimensional uniform polytopes are done through a Wythoff construction process, and represented through a CoxeterDynkin 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's the primary operators available for constructing and naming the uniform 5polytopes.
The last operation, the snub, and more generally the alternation, are the operation 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 Dynkin diagram 
Description 

Parent  t_{0}{p,q,r,s}  Any regular 5polytope  
Rectified  t_{1}{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}  Birectification reduces cells to their duals.  
Truncated  t_{0,1}{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}  In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms. 

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}  Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for polyterons.)  
Omnitruncated  t_{0,1,2,3,4}{p,q,r,s}  All four operators, truncation, cantellation, runcination, and sterication are applied. 
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 4space. ^{[1]}
#  Coxeter group  CoxeterDynkin diagram  

1  A^{~}_{4}  (3 3 3 3 3)  [3^{[5]}]  
2  B^{~}_{4}  [4,3,3,4]  
3  C^{~}_{4}  [4,3,3^{1,1}]  h[4,3,3,4]  
4  D^{~}_{4}  [3^{1,1,1,1}]  q[4,3,3,4]  
5  F^{~}_{4}  [3,4,3,3]  h[4,3,3,4] 
There are three regular honeycomb 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.
The singleringed tessellations are given below, indexed by Olshevsky's listing.
#  CoxeterDynkin andSchläfli symbols 
Name  Facets by location: [4,3,3,4]  

4  3  2  1  0  
[4,3,3] 
[4,3]×[ ] 
[4]×[4] 
[ ]×[3,4] 
[3,3,4] 

1  t_{0}{4,3,3,4} 
Tesseractic honeycomb  {4,3,3}         
87  t_{1}{4,3,3,4} 
Rectified tesseractic honeycomb  t_{1}{4,3,3}        {3,3,4} 
88  t_{2}{4,3,3,4} 
Birectified tesseractic honeycomb (Same as Icositetrachoric honeycomb {3,4,3,3}) 
t_{1}{3,3,4} or {3,4,3} 
      t_{1}{3,3,4} or {3,4,3} 
^{[2]}
#  CoxeterDynkin andSchläfli symbols 
Name  Facets by location: [3^{1,1},3,4]  

4  3  2  1  0  
[3,3,4] 
[3^{1,1,1}] 
[3,3]×[ ] 
[ ]×[3]×[ ] 
[3,3,4] 

104  {3^{1,1},3,4} 
Demitesseractic honeycomb (Same as hexadecachoric honeycomb) 

[88]  t_{1}{3^{1,1},3,4} 
Rectified demitesseractic honeycomb (Same as icositetrachoric honeycomb, {3,4,3,3}) (Also birectified tesseractic honeycomb) 

[87]  t_{2}{3^{1,1},3,4} 
Birectified demitesseractic honeycomb (Same as Rectified tesseractic honeycomb) 

[1]  t_{3}{3^{1,1},3,4} 
Trirectified demitesseractic honeycomb (Same as tesseractic honeycomb) 
^{[3]}
#  CoxeterDynkin andSchläfli symbols 
Name  Facets by location: [3,4,3,3]  

4  3  2  1  0  
[3,4,3] 
[3,4]×[ ] 
[3]×[3] 
[ ]×[3,3] 
[4,3,3] 

104  t_{0}{3,4,3,3} 
Icositetrachoric honeycomb  
[88]  t_{1}{3,4,3,3} 
Rectified icositetrachoric honeycomb (Same as Hexadecachoric honeycomb) 

106  t_{2}{3,4,3,3} 
Birectified icositetrachoric honeycomb  
?  t_{1}{3,3,4,3} 
Rectified hexadecachoric honeycomb  
88  t_{0}{3,3,4,3} 
Hexadecachoric honeycomb 
#  CoxeterDynkin andSchläfli symbols 
Name  Facets by location  

4  3  2  1  0  
[3,3,3] 
[]x[3,3] 
[3]x[3] 
[3,3]x[] 
[3,3,3] 

134  Pentachoricdispentachoric honeycomb 
#  CoxeterDynkin andSchläfli symbols 

?  q[4,3,3,4] 
?  q_{1}[4,3,3,4] 
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in hyperbolic 4space. They generate 5 regular hyperbolic honeycombs and numerous uniform ones.
#  Coxeter group  CoxeterDynkin diagram  Uniform tessellations 

1  (4 3 3 3 3)  25 forms  
2  [5,3,3,3]  31 forms  
3  [5,3,3,4]  31 forms  
4  [5,3,3^{1,1}]  23 forms, most overlapping [5,3,3,4] family  
5  [5,3,3,5]  19 forms 
There are five kinds of convex regular honeycombs and four kinds of starhoneycombs in H^{4} space: ^{[4]}
Honeycomb name  Schläfli Symbol {p,q,r,s} 
CoxeterDynkin 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 pentachoric  {3,3,3,5}  {3,3,3}  {3,3}  {3}  {5}  {3,5}  {3,3,5}  {5,3,3,3}  
Order3 hecatonicosachoric  {5,3,3,3}  {5,3,3}  {5,3}  {5}  {3}  {3,3}  {3,3,3}  {3,3,3,5}  
Order5 tesseractic  {4,3,3,5}  {4,3,3}  {4,3}  {4}  {5}  {3,5}  {3,3,5}  {5,3,3,4}  
Order4 hecatonicosachoric  {5,3,3,4}  {5,3,3}  {5,3}  {5}  {4}  {3,4}  {3,3,4}  {4,3,3,5}  
Order5 hecatonicosachoric  {5,3,3,5}  {5,3,3}  {5,3}  {5}  {5}  {3,5}  {3,3,5}  Selfdual 
There are four regular starhoneycombs in H^{4} space:
Honeycomb name  Schläfli Symbol {p,q,r,s} 
CoxeterDynkin 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 stellated hecatonicosachoric  {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 hexacosichoric  {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 hecatonicosachoric  {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 hecatonicosachoric  {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} 
Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
