Uniform 5polytope
Unsolved problem in mathematics: Find the complete set of uniform 5polytopes (more unsolved problems in mathematics)

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 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 Coxeter diagrams.
Contents
 1 History of discovery
 2 Regular 5polytopes
 3 Convex uniform 5polytopes
 4 Notes on the Wythoff construction for the uniform 5polytopes
 5 Regular and uniform honeycombs
 6 Notes
 7 References
 8 External links
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 4polytopes) in his publication On the Regular and SemiRegular Figures in Space of n Dimensions.^{[1]}
 Convex uniform polytopes:
 19401988: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and SemiRegular 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
Regular 5polytopes[edit]
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:
 {3,3,3,3}  5simplex
 {4,3,3,3}  5cube
 {3,3,3,4}  5orthoplex
There are no nonconvex regular polytopes in 5 or more dimensions.
Convex uniform 5polytopes[edit]
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]}
Symmetry of uniform 5polytopes in four dimensions[edit]
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.
 Fundamental families^{[2]}
Group symbol 
Order  Coxeter graph 
Bracket notation 
Commutator subgroup 
Coxeter number (h) 
Reflections m=5/2 h^{[3]}  

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 
 Uniform prisms
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 (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 
 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)  
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 
Enumerating the convex uniform 5polytopes[edit]
 Simplex family: A_{5} [3^{4}]
 19 uniform 5polytopes
 Hypercube/Orthoplex family: BC_{5} [4,3^{3}]
 31 uniform 5polytopes
 Demihypercube D_{5}/E_{5} family: [3^{2,1,1}]
 23 uniform 5polytopes (8 unique)
 Prisms and duoprisms:
 56 uniform 5polytope (45 unique) constructions based on prismatic families: [3,3,3]×[ ], [4,3,3]×[ ], [5,3,3]×[ ], [3^{1,1,1}]×[ ].
 One nonWythoffian  The grand antiprism prism is the only known nonWythoffian convex uniform 5polytope, 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 5polytope constructions based on duoprism prismatic families: [p]×[q]×[ ].
 Infinitely many uniform 5polytope constructions based on duoprismatic families: [3,3]×[p], [4,3]×[p], [5,3]×[p].
The A_{5} family[edit]
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)  
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} 
(5) {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}×{ } 
(4) r{3,3,3} 
      (2) {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 
(4) t{3,3,3} 
      (1) {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 
(3) rr{3,3,3} 
    (1) × { }×{3,3} 
(1) 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  (3) 2t{3,3,3} 
      (2) 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  (2) t_{0,3}{3,3,3} 
  (3) {3}×{3} 
(3) × { }×r{3,3} 
(1) 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} 
rr{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} 
#  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)  
13  (0,0,0,1,1,1)  Birectified 5simplex dodecateron (dot) 
12  60  120  90  20  {3}×{3} 
(3) r{3,3,3} 
      (3) r{3,3,3} 
14  (0,0,1,1,2,2)  Bicantellated 5simplex small birhombated dodecateron (sibrid) 
32  180  420  360  90  (2) rr{3,3,3} 
  (8) {3}×{3} 
  (2) 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 
(1) {3,3,3} 
(4) × { }×{3,3} 
(6) {3}×{3} 
(4) × { }×{3,3} 
(1) {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} 
(1) t_{0,1,2,3}{3,3,3} 
(1) × { }×tr{3,3} 
(1) {6}×{6} 
(1) × { }×tr{3,3} 
(1) t_{0,1,2,3}{3,3,3} 
The B_{5} family[edit]
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.
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)  
20  (0,0,0,0,1)√2  5orthoplex (tac) 
32  80  80  40  10  {3,3,4} 
{3,3,3} 
       
21  (0,0,0,1,1)√2  Rectified 5orthoplex (rat) 
42  240  400  240  40  { }×{3,4} 
{3,3,4} 
      r{3,3,3} 
22  (0,0,0,1,2)√2  Truncated 5orthoplex (tot) 
42  240  400  280  80  (Octah.pyr) 
t{3,3,3} 
{3,3,3} 
     
23  (0,0,1,1,1)√2  Birectified 5cube (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 (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 (bittit) 
42  280  720  720  240  t{3,3,4}        2t{3,3,3}  
26  (0,0,1,2,3)√2  Cantitruncated 5orthoplex (gart) 
82  640  1520  1440  480  rr{3,3,4}  { }×r{3,4}  {6}×{4} 
  t_{0,1,3}{3,3,3}  
27  (0,1,1,1,1)√2  Rectified 5cube (rin) 
42  200  400  320  80  {3,3}×{ } 
r{4,3,3} 
      {3,3,3} 
28  (0,1,1,1,2)√2  Runcinated 5orthoplex (spat) 
162  1200  2160  1440  320  r{4,3,3}    {3}×{4} 
t_{0,3}{3,3,3}  
29  (0,1,1,2,2)√2  Bicantellated 5cube (sibrant) (Bicantellated 5orthoplex) 
122  840  2160  1920  480  rr{4,3,3} 
  {4}×{3} 
  rr{3,3,3}  
30  (0,1,1,2,3)√2  Runcitruncated 5orthoplex (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 (tan) 
42  280  720  800  320  2t{4,3,3} 
      t{3,3,3}  
32  (0,1,2,2,3)√2  Runcicantellated 5orthoplex (pirt) 
162  1200  2960  2880  960  { }×t{3,4}  2t{3,3,4}  {3}×{4} 
  t_{0,1,3}{3,3,3}  
33  (0,1,2,3,3)√2  Bicantitruncated 5cube (gibrant) (Bicantitruncated 5orthoplex) 
122  840  2160  2400  960  rr{4,3,3} 
  {4}×{3} 
  rr{3,3,3}  
34  (0,1,2,3,4)√2  Runcicantitruncated 5orthoplex (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 (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 (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 (span) 
202  1240  2160  1440  320  t_{0,3}{4,3,3} 
  {4}×{3} 
{ }×r{3,3} 
{3,3,3}  
38  (1,1,1,1,1) + (0,0,0,1,2)√2 
Steritruncated 5orthoplex (cappin) 
242  1520  2880  2240  640  t_{0,3}{3,3,4}  { }×{4,3}      t{3,3,3}  
39  (1,1,1,1,1) + (0,0,1,1,1)√2 
Cantellated 5cube (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 (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 (prin) 
202  1240  2960  2880  960  t_{0,1,3}{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 (cogart) 
242  2320  5920  5760  1920  { }×rr{3,4} 
t_{0,1,3}{3,3,4} 
{6}×{4} 
{ }×t{3,3} 
tr{3,3,3}  
43  (1,1,1,1,1) + (0,1,1,1,1)√2 
Truncated 5cube (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 (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 (pattin) 
202  1560  3760  3360  960  t_{0,1,3}{4,3,3} 
{ }×t{4,3}  {6}×{8} 
{ }×t{3,3}  t_{0,1,3}{3,3,3}]]  
46  (1,1,1,1,1) + (0,1,1,2,3)√2 
Steriruncitruncated 5cube (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 (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 (cogrin) 
242  2400  6000  5760  1920  tr{4,3,3} 
tr{4,3}×{ } 
{8}×{3} 
{ }×t_{0,2}{3,3} 
t_{0,1,3}{3,3,3}  
49  (1,1,1,1,1) + (0,1,2,3,3)√2 
Runcicantitruncated 5cube (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 (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} 
The D_{5} family[edit]
The D_{5} family has symmetry of order 1920 (5! x 2^{4}).
This family has 23 Wythoffian uniform polyhedra, from 3x81 permutations of the D_{5} Coxeter diagram with one or more rings. 15 (2x81) are repeated from the B_{5} family and 8 are unique to this family.
#  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)  
51  = h{4,3,3,3}, 5demicube Hemipenteract (hin) 
26  120  160  80  16  t_{1}{3,3,3} 
{3,3,3}  t_{0}(1_{11})       
52  = h_{2}{4,3,3,3}, cantic 5cube Truncated hemipenteract (thin) 
42  280  640  560  160  
53  = h_{3}{4,3,3,3}, runcic 5cube Small rhombated hemipenteract (sirhin) 
42  360  880  720  160  
54  = h_{4}{4,3,3,3}, steric 5cube Small prismated hemipenteract (siphin) 
82  480  720  400  80  
55  = h_{2,3}{4,3,3,3}, runcicantic 5cube Great rhombated hemipenteract (girhin) 
42  360  1040  1200  480  
56  = h_{2,4}{4,3,3,3}, stericantic 5cube Prismatotruncated hemipenteract (pithin) 
82  720  1840  1680  480  
57  = h_{3,4}{4,3,3,3}, steriruncic 5cube Prismatorhombated hemipenteract (pirhin) 
82  560  1280  1120  320  
58  = h_{2,3,4}{4,3,3,3}, steriruncicantic 5cube Great prismated hemipenteract (giphin) 
82  720  2080  2400  960 
Uniform prismatic forms[edit]
There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4polytopes:
A_{4} × A_{1}[edit]
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 
7  20  30  25  10 
60  = r{3,3,3}×{ } Rectified 5cell prism 
12  50  90  70  20 
61  = t{3,3,3}×{ } Truncated 5cell prism 
12  50  100  100  40 
62  = rr{3,3,3}×{ } Cantellated 5cell prism 
22  120  250  210  60 
63  = t_{0,3}{3,3,3}×{ } Runcinated 5cell prism 
32  130  200  140  40 
64  = 2t{3,3,3}×{ } Bitruncated 5cell prism 
12  60  140  150  60 
65  = tr{3,3,3}×{ } Cantitruncated 5cell prism 
22  120  280  300  120 
66  = t_{0,1,3}{3,3,3}×{ } Runcitruncated 5cell prism 
32  180  390  360  120 
67  = t_{0,1,2,3}{3,3,3}×{ } Omnitruncated 5cell prism 
32  210  540  600  240 
B_{4} × A_{1}[edit]
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!).
#  Coxeter diagram and Schläfli symbols Name 
Element counts  

Facets  Cells  Faces  Edges  Vertices  
[16]  = {4,3,3}×{ } Tesseractic prism (Same as 5cube) 
10  40  80  80  32 
68  = r{4,3,3}×{ } Rectified tesseractic prism 
26  136  272  224  64 
69  = t{4,3,3}×{ } Truncated tesseractic prism 
26  136  304  320  128 
70  = rr{4,3,3}×{ } Cantellated tesseractic prism 
58  360  784  672  192 
71  = t_{0,3}{4,3,3}×{ } Runcinated tesseractic prism 
82  368  608  448  128 
72  = 2t{4,3,3}×{ } Bitruncated tesseractic prism 
26  168  432  480  192 
73  = tr{4,3,3}×{ } Cantitruncated tesseractic prism 
58  360  880  960  384 
74  = t_{0,1,3}{4,3,3}×{ } Runcitruncated tesseractic prism 
82  528  1216  1152  384 
75  = t_{0,1,2,3}{4,3,3}×{ } Omnitruncated tesseractic prism 
82  624  1696  1920  768 
76  = {3,3,4}×{ } 16cell prism 
18  64  88  56  16 
77  = r{3,3,4}×{ } Rectified 16cell prism (Same as 24cell prism) 
26  144  288  216  48 
78  = t{3,3,4}×{ } Truncated 16cell prism 
26  144  312  288  96 
79  = rr{3,3,4}×{ } Cantellated 16cell prism (Same as rectified 24cell prism) 
50  336  768  672  192 
80  = tr{3,3,4}×{ } Cantitruncated 16cell prism (Same as truncated 24cell prism) 
50  336  864  960  384 
81  = t_{0,1,3}{3,3,4}×{ } Runcitruncated 16cell prism 
82  528  1216  1152  384 
82  = sr{3,3,4}×{ } snub 24cell prism 
146  768  1392  960  192 
F_{4} × A_{1}[edit]
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 
26  144  288  216  48 
[79]  = r{3,4,3}×{ } rectified 24cell prism 
50  336  768  672  192 
[80]  = t{3,4,3}×{ } truncated 24cell prism 
50  336  864  960  384 
83  = rr{3,4,3}×{ } cantellated 24cell prism 
146  1008  2304  2016  576 
84  = t_{0,3}{3,4,3}×{ } runcinated 24cell prism 
242  1152  1920  1296  288 
85  = 2t{3,4,3}×{ } bitruncated 24cell prism 
50  432  1248  1440  576 
86  = tr{3,4,3}×{ } cantitruncated 24cell prism 
146  1008  2592  2880  1152 
87  = t_{0,1,3}{3,4,3}×{ } runcitruncated 24cell prism 
242  1584  3648  3456  1152 
88  = t_{0,1,2,3}{3,4,3}×{ } omnitruncated 24cell prism 
242  1872  5088  5760  2304 
[82]  = s{3,4,3}×{ } snub 24cell prism 
146  768  1392  960  192 
H_{4} × A_{1}[edit]
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 
122  960  2640  3000  1200 
90  = r{5,3,3}×{ } Rectified 120cell prism 
722  4560  9840  8400  2400 
91  = t{5,3,3}×{ } Truncated 120cell prism 
722  4560  11040  12000  4800 
92  = rr{5,3,3}×{ } Cantellated 120cell prism 
1922  12960  29040  25200  7200 
93  = t_{0,3}{5,3,3}×{ } Runcinated 120cell prism 
2642  12720  22080  16800  4800 
94  = 2t{5,3,3}×{ } Bitruncated 120cell prism 
722  5760  15840  18000  7200 
95  = tr{5,3,3}×{ } Cantitruncated 120cell prism 
1922  12960  32640  36000  14400 
96  = t_{0,1,3}{5,3,3}×{ } Runcitruncated 120cell prism 
2642  18720  44880  43200  14400 
97  = t_{0,1,2,3}{5,3,3}×{ } Omnitruncated 120cell prism 
2642  22320  62880  72000  28800 
98  = {3,3,5}×{ } 600cell prism 
602  2400  3120  1560  240 
99  = r{3,3,5}×{ } Rectified 600cell prism 
722  5040  10800  7920  1440 
100  = t{3,3,5}×{ } Truncated 600cell prism 
722  5040  11520  10080  2880 
101  = rr{3,3,5}×{ } Cantellated 600cell prism 
1442  11520  28080  25200  7200 
102  = tr{3,3,5}×{ } Cantitruncated 600cell prism 
1442  11520  31680  36000  14400 
103  = t_{0,1,3}{3,3,5}×{ } Runcitruncated 600cell prism 
2642  18720  44880  43200  14400 
Grand antiprism prism[edit]
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 
322  1360  1940  1100  200 
Notes on the Wythoff construction for the uniform 5polytopes[edit]
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 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 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  
Runcisteric  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 
Regular and uniform honeycombs[edit]
There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4space.^{[4]}^{[5]}
#  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:
 tesseractic honeycomb, with symbols {4,3,3,4}, = . There are 19 uniform honeycombs in this family.
 24cell honeycomb, with symbols {3,4,3,3}, . There are 31 reflective uniform honeycombs in this family, and one alternated form.
 Truncated 24cell honeycomb with symbols t{3,4,3,3},
 Snub 24cell honeycomb, with symbols s{3,4,3,3}, and constructed by four snub 24cell, one 16cell, and five 5cells at each vertex.
 16cell honeycomb, with symbols {3,3,4,3},
Other families that generate uniform honeycombs:
 There are 23 uniquely ringed forms, 8 new ones in the 16cell honeycomb family. With symbols h{4,3^{2},4} it is geometrically identical to the 16cell honeycomb, =
 There are 7 uniquely ringed forms from the , family, all new, including:
 There are 9 uniquely ringed forms in the : [3^{1,1,1,1}] family, two new ones, including the quarter tesseractic honeycomb, = , and the bitruncated tesseractic honeycomb, = .
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] 
Compact regular tessellations of hyperbolic 4space[edit]
There are five kinds of convex regular honeycombs and four kinds of starhoneycombs in H^{4} space:^{[6]}
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  {3,3,3,5}  {3,3,3}  {3,3}  {3}  {5}  {3,5}  {3,3,5}  {5,3,3,3}  
Order3 120cell  {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 120cell  {5,3,3,4}  {5,3,3}  {5,3}  {5}  {4}  {3,4}  {3,3,4}  {4,3,3,5}  
Order5 120cell  {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} 
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} 
Regular and uniform hyperbolic honeycombs[edit]
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. There are also 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.
= [(3,3,3,3,4)]: 
= [5,3,3^{1,1}]: 
= [3,3,3,5]: = [4,3,3,5]: 
= [3,3^{[4]}]: = [4,3^{[4]}]: 
= [4,/3\,3,4]: 
= [3,4,3,4]: 
Notes[edit]
 ^ T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 ^ Regular and semiregular polytopes III, p.315 Three finite groups of 5dimensions
 ^ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
 ^ Regular polytopes, p.297. Table IV, Fundamental regions for irreducible groups generated by reflections.
 ^ Regular and Semiregular polytopes, II, pp.298302 Fourdimensional honeycombs
 ^ Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213
References[edit]
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 (3 regular and one semiregular 4polytope)
 A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
 H.S.M. Coxeter:
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 (p. 297 Fundamental regions for irreducible groups generated by reflections, Spherical and Euclidean)
 H.S.M. Coxeter, The Beauty of Geometry: Twelve Essays (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591] (p. 287 5D Euclidean groups, p. 298 Fourdimensionsal honeycombs)
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
 James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990) (Page 141, 6.9 List of hyperbolic Coxeter groups, figure 2) [2]
External links[edit]
 Klitzing, Richard. "5D uniform polytopes (polytera)".
Fundamental convex regular and uniform honeycombs in dimensions 29
 

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 