Uniform 4-honeycomb

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets.

The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of Coxeter groups, symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.

History of discovery

*Regular polytopes: (convex faces)
**1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more dimensions.
*Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
**1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimensions''.
*Convex uniform polytopes:
**1940-1988: The search was expanded systematically by Harold Scott MacDonald Coxeter, H.S.M. Coxeter in his publication ''Regular and Semi-Regular Polytopes I, II, and III''.
**1966: Norman W. Johnson completed his Ph.D. Dissertation under Coxeter, ''The Theory of Uniform Polytopes and Honeycombs'', University of Toronto
* Non-convex uniform polytopes:
**1966: Johnson describes two non-convex uniform antiprisms in 5-space in his dissertation.
**2000-2022: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes, with a current count of 1294 known uniform 5-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 4-polytopes. The list is not proven complete.

Regular 5-polytopes

Regular 5-polytopes can be represented by the Schläfli symbol , with s 4-polytope Facet (mathematics), facets around each Face (geometry), face. There are exactly three such regular polytopes, all convex:

* - 5-simplex
* - 5-cube
* - 5-orthoplex

There are no nonconvex regular polytopes in 5 dimensions or above.

Convex uniform 5-polytopes

There are 104 known convex uniform 5-polytopes, plus a number of infinite families of duoprism prisms, and polygon-polyhedron duoprisms. All except the ''grand antiprism prism'' are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.

Symmetry of uniform 5-polytopes in four dimensions

The 5-simplex is the regular form in the A₅ family. The 5-cube and 5-orthoplex are the regular forms in the B₅ family. The bifurcating graph of the D₅ family contains the 5-orthoplex, as well as a 5-demicube which is an alternation (geometry), alternated 5-cube.

Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,b,a], have an extended symmetry, a,b,b,a, like [3,3,3,3], doubling the symmetry order. Uniform polytopes in these group with symmetric rings contain this extended symmetry.

If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation (geometry), alternation operation can generate a new 5-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.

;Fundamental families

;Uniform prisms

There are 5 finite categorical Uniform polytope, uniform Prismatic polytope, prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. There is one infinite family of 5-polytopes based on prisms of the uniform duoprisms ××.

;Uniform duoprisms

There are 3 categorical Uniform polytope, uniform duoprismatic families of polytopes based on Cartesian products of the uniform polyhedron, uniform polyhedra and regular polygons: ×.

Enumerating the convex uniform 5-polytopes

* Simplex family: A₅ [3⁴]
** 19 uniform 5-polytopes
* Hypercube/Orthoplex family: B₅ [4,3³]
** 31 uniform 5-polytopes
* Demihypercube D₅/E₅ family: [3^2,1,1]
** 23 uniform 5-polytopes (8 unique)
* Polychoral prisms:
** 56 uniform 5-polytope (45 unique) constructions based on prismatic families: [3,3,3]×[ ], [4,3,3]×[ ], [5,3,3]×[ ], [3^1,1,1]×[ ].
** One non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two grand antiprisms connected by polyhedral prisms.

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

In addition there are:
* Infinitely many uniform 5-polytope constructions based on duoprism prismatic families: [''p'']×[''q'']×[ ].
* Infinitely many uniform 5-polytope constructions based on duoprismatic families: [3,3]×[''p''], [4,3]×[''p''], [5,3]×[''p''].

The A₅ family

There are 19 forms based on all permutations of the Coxeter diagrams with one or more rings. (16+4-1 cases)

They are named by Norman Johnson (mathematician), Norman Johnson from the Wythoff construction operations upon regular 5-simplex (hexateron).

The Coxeter group#Finite Coxeter groups, A₅ family has symmetry of order 720 (6 factorial). 7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440.

The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, all in hyperplanes with normal vector (1,1,1,1,1,1).

The B₅ family

The Coxeter group#Finite Coxeter groups, B₅ family has symmetry of order 3840 (5!×2⁵).

This family has 2⁵−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter diagram. Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D₅ family as ... = ..... (There are more alternations that are not listed because they produce only repetitions, as ... = .... and ... = .... These would give a complete duplication of the uniform 5-polytopes numbered 20 through 34 with symmetry broken in half.)

For simplicity it is divided into two subgroups, each with 12 forms, and 7 "middle" forms which equally belong in both.

The 5-cube family of 5-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 5-polytope. All coordinates correspond with uniform 5-polytopes of edge length 2.

{, class="wikitable"
!rowspan=2, #
!rowspan=2, Base point
!rowspan=2, Name
Coxeter diagram
!colspan=5, Element counts
!rowspan=2, Vertex figure, Vertex
figure
!colspan=6 , Facet counts by location: [4,3,3,3]
, - BGCOLOR="#e0e0f0"
!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
, - BGCOLOR="#f0e0e0"
!20
, , (0,0,0,0,1)√2, , 5-orthoplex
triacontaditeron (tac)
, , 32, , 80, , 80, , 40, , 10
, ,
16-cell, {3,3,4}, , - , , - , , - , , - , ,
5-cell, {3,3,3}, ,
, - BGCOLOR="#f0e0e0"
!21
, , (0,0,0,1,1)√2, , Rectified 5-orthoplex
rectified triacontaditeron (rat)
, , 42, , 240, , 400, , 240, , 40
, ,
Octahedral prism, { }×{3,4}, ,
16-cell, {3,3,4} , , - , , - , , - , ,
Rectified 5-cell, r{3,3,3} , ,
, - BGCOLOR="#f0e0e0"
!22
, , (0,0,0,1,2)√2, , Truncated 5-orthoplex
truncated triacontaditeron (tot)
, , 42, , 240, , 400, , 280, , 80
, ,
(Octah.pyr), ,
16-cell, {3,3,4} , , - , , - , , - , ,
Truncated 5-cell, t{3,3,3}, ,
, - BGCOLOR="#e0f0e0"
!23
, , (0,0,1,1,1)√2, , Birectified 5-cube
penteractitriacontaditeron (nit)
(Birectified 5-orthoplex)
, , 42, , 280, , 640, , 480, , 80
, ,
3-4 duoprism, {4}×{3}, ,
24-cell, r{3,3,4} , , - , , - , , - , ,
Rectified 5-cell, r{3,3,3} , ,
, -BGCOLOR="#f0e0e0"
!24
, , (0,0,1,1,2)√2, , Cantellated 5-orthoplex
small rhombated triacontaditeron (sart)
, , 82, , 640, , 1520, , 1200, , 240
, ,
Prism-wedge, ,
24-cell, r{3,3,4}, ,
Octahedral prism, { }×{3,4} , , - , , - , ,
Cantellated 5-cell, rr{3,3,3} , ,
, - BGCOLOR="#f0e0e0"
!25
, , (0,0,1,2,2)√2, , Bitruncated 5-orthoplex
bitruncated triacontaditeron (bittit)
, , 42, , 280, , 720, , 720, , 240
, , , ,
Truncated 16-cell, t{3,3,4} , , - , , - , , - , ,
Bitruncated 5-cell, 2t{3,3,3} , ,
, - BGCOLOR="#f0e0e0"
!26
, , (0,0,1,2,3)√2, , Cantitruncated 5-orthoplex
great rhombated triacontaditeron (gart)
, , 82, , 640, , 1520, , 1440, , 480
, , , ,
Truncated 16-cell, t{3,3,4}, ,
Octahedral prism, { }×{3,4} , , -, , - , ,
Runcitruncated 5-cell, t_0,1,3{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!27
, , (0,1,1,1,1)√2, , Rectified 5-cube
rectified penteract (rin)
, , 42, , 200, , 400, , 320, , 80
, ,
Tetrahedral prism, {3,3}×{ }, ,
Rectified tesseract, r{4,3,3}, , - , , - , , - , ,
5-cell, {3,3,3} , ,
, -BGCOLOR="#f0e0e0"
!28
, , (0,1,1,1,2)√2, , Runcinated 5-orthoplex
small prismated triacontaditeron (spat)
, , 162, , 1200, , 2160, , 1440, , 320
, , , ,
Rectified tesseract, r{4,3,3} , ,
Cuboctahedral prism, { }×r{3,4} , ,
3-4 duoprism, {3}×{4}, , , ,
Runcinated 5-cell, t_0,3{3,3,3} , ,
, - BGCOLOR="#e0f0e0"
!29
, , (0,1,1,2,2)√2, , Bicantellated 5-cube
small birhombated penteractitriacontaditeron (sibrant)
(Bicantellated 5-orthoplex)
, , 122, , 840, , 2160, , 1920, , 480
, , , ,
Rectified 24-cell, rr{3,3,4}, , - , ,
3-4 duoprism, {4}×{3}, , - , ,
Cantellated 5-cell, rr{3,3,3} , ,
, - BGCOLOR="#f0e0e0"
!30
, , (0,1,1,2,3)√2, , Runcitruncated 5-orthoplex
prismatotruncated triacontaditeron (pattit)
, , 162, , 1440, , 3680, , 3360, , 960
, , , ,
Rectified 24-cell, rr{3,3,4} , ,
Cuboctahedral prism, { }×r{3,4} , ,
4-6 duoprism, {6}×{4}, , - , ,
Runcitruncated 5-cell, t_0,1,3{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!31
, , (0,1,2,2,2)√2, , Bitruncated 5-cube
bitruncated penteract (bittin)
, , 42, , 280, , 720, , 800, , 320
, , , ,
Bitruncated tesseract, 2t{4,3,3}, , - , , - , , - , ,
Truncated 5-cell, t{3,3,3} , ,
, - BGCOLOR="#f0e0e0"
!32
, , (0,1,2,2,3)√2, , Runcicantellated 5-orthoplex
prismatorhombated triacontaditeron (pirt)
, , 162, , 1200, , 2960, , 2880, , 960
, , , ,
Bitruncated tesseract, 2t{4,3,3}, ,
Truncated octahedral prism, { }×t{3,4}, ,
3-4 duoprism, {3}×{4} , , - , ,
Runcitruncated 5-cell, t_0,1,3{3,3,3} , ,
, - BGCOLOR="#e0f0e0"
!33
, , (0,1,2,3,3)√2, , Bicantitruncated 5-cube
great birhombated triacontaditeron (gibrant)
(Bicantitruncated 5-orthoplex)
, , 122, , 840, , 2160, , 2400, , 960
, , , ,
Truncated 24-cell, tr{3,3,4}, , - , ,
3-4 duoprism, {4}×{3}, , - , ,
Cantellated 5-cell, rr{3,3,3} , ,
, - BGCOLOR="#f0e0e0"
!34
, , (0,1,2,3,4)√2, , Runcicantitruncated 5-orthoplex
great prismated triacontaditeron (gippit)
, , 162, , 1440, , 4160, , 4800, , 1920
, , , ,
Truncated 24-cell, tr{3,3,4} , ,
Truncated octahedral prism, { }×t{3,4} , ,
4-6 duoprism, {6}×{4}, , - , ,
Omnitruncated 5-cell, t_0,1,2,3{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!35
, , (1,1,1,1,1), , 5-cube
penteract (pent)
, , 10, , 40, , 80, , 80, , 32
, ,
5-cell, {3,3,3}, ,
Tesseract, {4,3,3}, , - , , - , , - , , - , ,
, - BGCOLOR="#e0f0e0"
!36
, , (1,1,1,1,1)
+ (0,0,0,0,1)√2, , Stericated 5-cube
small cellated penteractitriacontaditeron (scant)
(Stericated 5-orthoplex)
, , 242, , 800, , 1040, , 640, , 160
, ,
Tetr.antiprm, ,
tesseract, {4,3,3}, ,
Tesseract, {4,3}×{ }, ,
3-4 duoprism, {4}×{3}, ,
Tetrahedral prism, { }×{3,3}, ,
5-cell, {3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!37
, , (1,1,1,1,1)
+ (0,0,0,1,1)√2, , Runcinated 5-cube
small prismated penteract (span)
, , 202, , 1240, , 2160, , 1440, , 320
, , , ,
Runcinated tesseract, t_0,3{4,3,3}, , - , ,
3-4 duoprism, {4}×{3}, ,
Rectified tetrahedral prism, { }×r{3,3}, ,
Rectified 5-cell, r{3,3,3} , ,
, - BGCOLOR="#f0e0e0"
!38
, , (1,1,1,1,1)
+ (0,0,0,1,2)√2, , Steritruncated 5-orthoplex
celliprismated triacontaditeron (cappin)
, , 242, , 1520, , 2880, , 2240, , 640
, , , ,
Runcinated tesseract, t_0,3{4,3,3} , ,
Tesseract, {4,3}×{ } , ,
4-6 duoprism, {6}×{4} , ,
Truncated tetrahedral prism, { }×t{3,3} , ,
Truncated 5-cell, t{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!39
, , (1,1,1,1,1)
+ (0,0,1,1,1)√2, , Cantellated 5-cube
small rhombated penteract (sirn)
, , 122, , 680, , 1520, , 1280, , 320
, ,
Prism-wedge, ,
Cantellated tesseract, rr{4,3,3}, , - , , - , ,
Tetrahedral prism, { }×{3,3}, ,
Rectified 5-cell, r{3,3,3} , ,
, - BGCOLOR="#e0f0e0"
!40
, , (1,1,1,1,1)
+ (0,0,1,1,2)√2, , Stericantellated 5-cube
cellirhombated penteractitriacontaditeron (carnit)
(Stericantellated 5-orthoplex)
, , 242, , 2080, , 4720, , 3840, , 960
, , , ,
Cantellated tesseract, rr{4,3,3}, ,
Rhombicuboctahedral prism, rr{4,3}×{ }, ,
3-4 duoprism, {4}×{3}, ,
Cantellated tetrahedral prism, { }×rr{3,3}, ,
Cantellated 5-cell, rr{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!41
, , (1,1,1,1,1)
+ (0,0,1,2,2)√2, , Runcicantellated 5-cube
prismatorhombated penteract (prin)
, , 202, , 1240, , 2960, , 2880, , 960
, , , ,
Runcitruncated 16-cell, t_0,2,3{4,3,3}, , - , ,
3-4 duoprism, {4}×{3}, ,
Truncated tetrahedral prism, { }×t{3,3}, ,
Bitruncated 5-cell, 2t{3,3,3} , ,
, - BGCOLOR="#f0e0e0"
!42
, , (1,1,1,1,1)
+ (0,0,1,2,3)√2, , Stericantitruncated 5-orthoplex
celligreatorhombated triacontaditeron (cogart)
, , 242, , 2320, , 5920, , 5760, , 1920
, , , ,
Runcitruncated 16-cell, t_0,2,3{4,3,3}, ,
Rhombicuboctahedral prism, rr{4,3}×{ }, ,
4-6 duoprism, {6}×{4}, ,
Truncated octahedral prism, { }×tr{3,3}, ,
Cantitruncated 5-cell, tr{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!43
, , (1,1,1,1,1)
+ (0,1,1,1,1)√2, , Truncated 5-cube
truncated penteract (tan)
, , 42, , 200, , 400, , 400, , 160
, ,
5-cell, Tetrah.pyr, ,
Truncated tesseract, t{4,3,3}, , - , , - , , - , ,
5-cell, {3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!44
, , (1,1,1,1,1)
+ (0,1,1,1,2)√2, , Steritruncated 5-cube
celliprismated triacontaditeron (capt)
, , 242, , 1600, , 2960, , 2240, , 640
, , , ,
Truncated tesseract, t{4,3,3}, ,
Truncated cubic prism, t{4,3}×{ }, ,
3-8 duoprism, {8}×{3}, ,
Tetrahedral prism, { }×{3,3}, ,
Runcinated 5-cell, t_0,3{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!45
, , (1,1,1,1,1)
+ (0,1,1,2,2)√2, , Runcitruncated 5-cube
prismatotruncated penteract (pattin)
, , 202, , 1560, , 3760, , 3360, , 960
, , , ,
Runcitruncated tesseract, t_0,1,3{4,3,3} , , - , ,
3-8 duoprism, {8}×{3}, ,
Octahedral prism, { }×r{3,3} , ,
Cantellated 5-cell, rr{3,3,3} , ,
, - BGCOLOR="#e0f0e0"
!46
, , (1,1,1,1,1)
+ (0,1,1,2,3)√2, , Steriruncitruncated 5-cube
celliprismatotruncated penteractitriacontaditeron (captint)
(Steriruncitruncated 5-orthoplex)
, , 242, , 2160, , 5760, , 5760, , 1920
, , , ,
Runcitruncated tesseract, t_0,1,3{4,3,3}, ,
Truncated cubic prism, t{4,3}×{ }, ,
6-8 duoprism, {8}×{6}, ,
Truncated tetrahedral prism, { }×t{3,3}, ,
Runcitruncated 5-cell, t_0,1,3{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!47
, , (1,1,1,1,1)
+ (0,1,2,2,2)√2, , Cantitruncated 5-cube
great rhombated penteract (girn)
, , 122, , 680, , 1520, , 1600, , 640
, , , ,
Cantitruncated tesseract, tr{4,3,3}, , - , , - , ,
Tetrahedral prism, { }×{3,3}, ,
Truncated 5-cell, t{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!48
, , (1,1,1,1,1)
+ (0,1,2,2,3)√2, , Stericantitruncated 5-cube
celligreatorhombated penteract (cogrin)
, , 242, , 2400, , 6000, , 5760, , 1920
, , , ,
Cantitruncated tesseract, tr{4,3,3}, ,
truncated cuboctahedral prism, tr{4,3}×{ }, ,
3-8 duoprism, {8}×{3}, ,
Cantellated tetrahedral prism, { }×rr{3,3}, ,
Runcitruncated 5-cell, t_0,1,3{3,3,3} , ,
, - BGCOLOR="#e0e0f0"
!49
, , (1,1,1,1,1)
+ (0,1,2,3,3)√2, , Runcicantitruncated 5-cube
great prismated penteract (gippin)
, , 202, , 1560, , 4240, , 4800, , 1920
, , , ,
Omnitruncated tesseract, t_0,1,2,3{4,3,3}, , - , ,
3-8 duoprism, {8}×{3}, ,
Truncated tetrahedral prism, { }×t{3,3}, ,
Cantitruncated 5-cell, tr{3,3,3} , ,
, - BGCOLOR="#e0f0e0"
!50
, , (1,1,1,1,1)
+ (0,1,2,3,4)√2, , Omnitruncated 5-cube
great cellated penteractitriacontaditeron (gacnet)
(omnitruncated 5-orthoplex)
, , 242, , 2640, , 8160, , 9600, , 3840
, ,
5-cell, Irr. {3,3,3}, ,
truncated cuboctahedral prism, tr{4,3}×{ }, ,
truncated cuboctahedral prism, tr{4,3}×{ }, ,
6-8 duoprism, {8}×{6}, ,
Omnitruncated tetrahedral prism, { }×tr{3,3}, ,
Omnitruncated 5-cell, t_0,1,2,3{3,3,3} , ,
, - BGCOLOR="#d0f0f0"
!51
,
, 5-demicube
hemipenteract (hin)
=
, 26
, 120
, 160
, 80
, 16
,
Rectified 5-cell, r{3,3,3}
,
16-cell, h{4,3,3}
, -
, -
, -
, -
, (16)

5-cell, {3,3,3}
, - BGCOLOR="#d0f0f0"
!52
,
, Cantic 5-cube
Truncated hemipenteract (thin)
=
, 42
, 280
, 640
, 560
, 160
,
,
Truncated 16-cell, h₂{4,3,3}
, -
, -
, -
, (16)

Rectified 5-cell, r{3,3,3}
, (16)

Truncated 5-cell, t{3,3,3}
, - BGCOLOR="#d0f0f0"
!53
,
, Runcic 5-cube
Small rhombated hemipenteract (sirhin)
=
, 42
, 360
, 880
, 720
, 160
,
,
Rectified tesseract, h₃{4,3,3}
, -
, -
, -
, (16)

Rectified 5-cell, r{3,3,3}
, (16)

Cantellated 5-cell, rr{3,3,3}
, - BGCOLOR="#d0f0f0"
!54
,
, Steric 5-cube
Small prismated hemipenteract (siphin)
=
, 82
, 480
, 720
, 400
, 80
,
,
16-cell, h{4,3,3}
,
Tetrahedral prism, h{4,3}×{}
, -
, -
, (16)

5-cell, {3,3,3}
, (16)

Runcinated 5-cell, t_0,3{3,3,3}
, - BGCOLOR="#d0f0f0"
!55
,
, Runcicantic 5-cube
Great rhombated hemipenteract (girhin)
=
, 42
, 360
, 1040
, 1200
, 480
,
,
Bitruncated tesseract, h_2,3{4,3,3}
, -
, -
, -
, (16)

Bitruncated 5-cell, 2t{3,3,3}
, (16)

Cantitruncated 5-cell, tr{3,3,3}
, - BGCOLOR="#d0f0f0"
!56
,
, Stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
=
, 82
, 720
, 1840
, 1680
, 480
,
,
Truncated 16-cell, h₂{4,3,3}
,
Truncated tetrahedral prism, h₂{4,3}×{}
, -
, -
, (16)

Cantitruncated 5-cell, rr{3,3,3}
, (16)

Runcitruncated 5-cell, t_0,1,3{3,3,3}
, - BGCOLOR="#d0f0f0"
!57
,
, Steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
=
, 82
, 560
, 1280
, 1120
, 320
,
,
Rectified tesseract, h₃{4,3,3}
,
Tetrahedral prism, h{4,3}×{}
, -
, -
, (16)

Truncated 5-cell, t{3,3,3}
, (16)

Runcitruncated 5-cell, t_0,1,3{3,3,3}
, - BGCOLOR="#d0f0f0"
!58
,
, Steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
=
, 82
, 720
, 2080
, 2400
, 960
,
,
Bitruncated tesseract, h_2,3{4,3,3}
,
Truncated tetrahedral prism, h₂{4,3}×{}
, -
, -
, (16)

Cantitruncated 5-cell, tr{3,3,3}
, (16)

Omnitruncated 5-cell, t_0,1,2,3{3,3,3}
, - BGCOLOR="#d0f0f0"
!Nonuniform
,
, Alternated runcicantitruncated 5-orthoplex
Snub prismatotriacontaditeron (snippit)
Snub hemipenteract (snahin)
=
, 1122
, 6240
, 10880
, 6720
, 960
,
,
Snub 24-cell, sr{3,3,4}
, Truncated octahedral prism#Related polytopes, sr{2,3,4}
, Duoprism#Duoantiprism, sr{3,2,4}
, -
, Runcinated 5-cell#Full snub 5-cell, ht_0,1,2,3{3,3,3}
, (960)

5-cell, Irr. {3,3,3}
, - BGCOLOR="#d0f0f0"
!Nonuniform
,
, Edge-snub 5-orthoplex
Pyritosnub penteract (pysnan)

, 1202
, 7920
, 15360
, 10560
, 1920
,
, sr₃{3,3,4}
, Truncated_cuboctahedral_prism#Related_polytopes, sr₃{2,3,4}
, sr₃{3,2,4}
,
Icosahedral prism, s{3,3}×{ }
, Runcinated 5-cell#Full snub 5-cell, ht_0,1,2,3{3,3,3}
, (960)

Tetrahedral prism, Irr. {3,3}×{ }
, - BGCOLOR="#d0f0f0"
!Nonuniform
,
, Snub 5-cube
Snub penteract (snan)

, 2162
, 12240
, 21600
, 13440
, 960
,
, Runcinated_tesseracts#Full_snub_tesseract, ht_0,1,2,3{3,3,4}
, Truncated_cuboctahedral_prism#Related_polytopes, ht_0,1,2,3{2,3,4}
, Duoprism#Duoantiprism, ht_0,1,2,3{3,2,4}
, Truncated octahedral prism#Related polytopes, ht_0,1,2,3{3,3,2}
, Runcinated 5-cell#Full snub 5-cell, ht_0,1,2,3{3,3,3}
, (1920)

5-cell, Irr. {3,3,3}

The D₅ family

The Coxeter group#Finite Coxeter groups, D₅ family has symmetry of order 1920 (5! x 2⁴).

This family has 23 Wythoffian uniform polytopes, from ''3×8-1'' permutations of the D₅ Coxeter diagram with one or more rings. 15 (2×8-1) are repeated from the B₅ family and 8 are unique to this family, though even those 8 duplicate the alternations from the B₅ family.

In the 15 repeats, both of the nodes terminating the length-1 branches are ringed, so the two kinds of element are identical and the symmetry doubles: the relations are ... = .... and ... = ..., creating a complete duplication of the uniform 5-polytopes 20 through 34 above. The 8 new forms have one such node ringed and one not, with the relation ... = ... duplicating uniform 5-polytopes 51 through 58 above.

{, class="wikitable"
!rowspan=2, #
!rowspan=2, Coxeter diagram
Schläfli symbol symbols
Johnson and Bowers names
!colspan=5, Element counts
!rowspan=2, Vertex figure, Vertex
figure
!colspan=6 , 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}, 5-demicube
Hemipenteract (hin)
, 26
, 120
, 160
, 80
, 16
,
rectified 5-cell, r{3,3,3}
,
5-cell, {3,3,3}
,
16-cell, h{4,3,3}
, -
, -
, -
,
, -
![52]
, =
h₂{4,3,3,3}, cantic 5-cube
Truncated hemipenteract (thin)
, 42
, 280
, 640
, 560
, 160
,
,
Truncated 5-cell, t{3,3,3}
,
Truncated 16-cell, h₂{4,3,3}
, -
, -
,
Rectified 5-cell, r{3,3,3}
,
, -
![53]
, =
h₃{4,3,3,3}, runcic 5-cube
Small rhombated hemipenteract (sirhin)
, 42
, 360
, 880
, 720
, 160
,
,
Cantellated 5-cell, rr{3,3,3}
,
Rectified tesseract, h₃{4,3,3}
, -
, -
,
Rectified 5-cell, r{3,3,3}
,
, -
![54]
, =
h₄{4,3,3,3}, steric 5-cube
Small prismated hemipenteract (siphin)
, 82
, 480
, 720
, 400
, 80
,
,
Runcinated 5-cell, t_0,3{3,3,3}
,
16-cell, h{4,3,3}
,
Tetrahedral prism, h{4,3}×{}
, -
,
5-cell, {3,3,3}
,
, -
![55]
, =
h_2,3{4,3,3,3}, runcicantic 5-cube
Great rhombated hemipenteract (girhin)
, 42
, 360
, 1040
, 1200
, 480
,
,
Bitruncated 5-cell, 2t{3,3,3}
,
Bitruncated tesseract, h_2,3{4,3,3}
, -
, -
,
Cantitruncated 5-cell, tr{3,3,3}
,
, -
![56]
, =
h_2,4{4,3,3,3}, stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
, 82
, 720
, 1840
, 1680
, 480
,
,
Runcitruncated 5-cell, t_0,1,3{3,3,3}
,
Truncated 16-cell, h₂{4,3,3}
,
Truncated tetrahedral prism, h₂{4,3}×{}
, -
,
Cantitruncated 5-cell, rr{3,3,3}
,
, -
![57]
, =
h_3,4{4,3,3,3}, steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
, 82
, 560
, 1280
, 1120
, 320
,
,
Runcitruncated 5-cell, t_0,1,3{3,3,3}
,
Rectified tesseract, h₃{4,3,3}
,
Tetrahedral prism, h{4,3}×{}
, -
,
Truncated 5-cell, t{3,3,3}
,
, -
![58]
, =
h_2,3,4{4,3,3,3}, steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
, 82
, 720
, 2080
, 2400
, 960
,
,
Omnitruncated 5-cell, t_0,1,2,3{3,3,3}
,
Bitruncated tesseract, h_2,3{4,3,3}
,
Truncated tetrahedral prism, h₂{4,3}×{}
, -
,
Cantitruncated 5-cell, tr{3,3,3}
,
, - bgcolor="#D0F0F0"
! Nonuniform
, =
ht_0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplex
Snub hemipenteract (snahin)
, 1122
, 6240
, 10880
, 6720
, 960
,
, Runcinated 5-cell#Full snub 5-cell, ht_0,1,2,3{3,3,3}
,
Snub 24-cell, sr{3,3,4}
, Truncated octahedral prism#Related polytopes, sr{2,3,4}
, Duoprism#Duoantiprism, sr{3,2,4}
, Runcinated 5-cell#Full snub 5-cell, ht_0,1,2,3{3,3,3}
, (960)

5-cell, Irr. {3,3,3}

Uniform prismatic forms

There are 5 finite categorical Uniform polytope, uniform Prismatic polytope, prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. For simplicity, most alternations are not shown.

A₄ × A₁

This prismatic family has Uniform 4-polytope#The A4 .5B3.2C3.2C3.5D family - .285-cell.29, 9 forms:

The Coxeter group#Finite Coxeter groups, A₁ x A₄ family has symmetry of order 240 (2*5!).

{, class="wikitable"
!rowspan=2, #
!rowspan=2, Coxeter diagram
and Schläfli symbol, Schläfli
symbols
Name
!colspan=5 rowspan=1, Element counts
, -
! Facets, , Cells, , Faces, , Edges, , Vertices
, -
, 59
, = {3,3,3}×{ }
5-cell prism (penp)
, 7, , 20, , 30, , 25, , 10
, -
, 60
, = r{3,3,3}×{ }
Rectified 5-cell prism (rappip)
, 12, , 50, , 90, , 70, , 20
, -
, 61
, = t{3,3,3}×{ }
Truncated 5-cell prism (tippip)
, 12, , 50, , 100, , 100, , 40
, -
, 62
, = rr{3,3,3}×{ }
Cantellated 5-cell prism (srippip)
, 22, , 120, , 250, , 210, , 60
, - BGCOLOR="#e0f0e0"
, 63
, = t_0,3{3,3,3}×{ }
Runcinated 5-cell prism (spiddip)
, 32, , 130, , 200, , 140, , 40
, - BGCOLOR="#e0f0e0"
, 64
, = 2t{3,3,3}×{ }
Bitruncated 5-cell prism (decap)
, 12, , 60, , 140, , 150, , 60
, -
, 65
, = tr{3,3,3}×{ }
Cantitruncated 5-cell prism (grippip)
, 22, , 120, , 280, , 300, , 120
, -
, 66
, = t_0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism (prippip)
, 32, , 180, , 390, , 360, , 120
, - BGCOLOR="#e0f0e0"
, 67
, = t_0,1,2,3{3,3,3}×{ }
Omnitruncated 5-cell prism (gippiddip)
, 32, , 210, , 540, , 600, , 240

B₄ × A₁

This prismatic family has Uniform 4-polytope#The B.2FC4 .5B4.2C3.2C3.5D family - .28tesseract.2F16-cell.29, 16 forms. (Three are shared with [3,4,3]×[ ] family)

The Coxeter group#Finite Coxeter groups, A₁×B₄ family has symmetry of order 768 (2⁵4!).

The last three snubs can be realised with equal-length edges, but turn out nonuniform anyway because some of their 4-faces are not uniform 4-polytopes.

{, class="wikitable"
!rowspan=2, #
!rowspan=2, Coxeter diagram
and Schläfli symbol, Schläfli
symbols
Name
!colspan=5 rowspan=1, Element counts
, -
! Facets, , Cells, , Faces, , Edges, , Vertices
, - BGCOLOR="#f0e0e0"
, [16], , = {4,3,3}×{ }
Tesseractic prism (pent)
(Same as 5-cube)
, 10, , 40, , 80, , 80, , 32
, - BGCOLOR="#f0e0e0"
, 68, , = r{4,3,3}×{ }
Rectified tesseractic prism (rittip)
, 26, , 136, , 272, , 224, , 64
, - BGCOLOR="#f0e0e0"
, 69, , = t{4,3,3}×{ }
Truncated tesseractic prism (tattip)
, 26, , 136, , 304, , 320, , 128
, - BGCOLOR="#f0e0e0"
, 70, , = rr{4,3,3}×{ }
Cantellated tesseractic prism (srittip)
, 58, , 360, , 784, , 672, , 192
, - BGCOLOR="#e0f0e0"
, 71, , = t_0,3{4,3,3}×{ }
Runcinated tesseractic prism (sidpithip)
, 82, , 368, , 608, , 448, , 128
, - BGCOLOR="#e0f0e0"
, 72, , = 2t{4,3,3}×{ }
Bitruncated tesseractic prism (tahp)
, 26, , 168, , 432, , 480, , 192
, - BGCOLOR="#f0e0e0"
, 73, , = tr{4,3,3}×{ }
Cantitruncated tesseractic prism (grittip)
, 58, , 360, , 880, , 960, , 384
, - BGCOLOR="#f0e0e0"
, 74, , = t_0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism (prohp)
, 82, , 528, , 1216, , 1152, , 384
, - BGCOLOR="#e0f0e0"
, 75, , = t_0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism (gidpithip)
, 82, , 624, , 1696, , 1920, , 768
, - BGCOLOR="#e0e0f0"
, 76, , = {3,3,4}×{ }
16-cell prism (hexip)
, 18, , 64, , 88, , 56, , 16
, - BGCOLOR="#e0e0f0"
, 77, , = r{3,3,4}×{ }
Rectified 16-cell prism (icope)
(Same as 24-cell prism)
, 26, , 144, , 288, , 216, , 48
, - BGCOLOR="#e0e0f0"
, 78, , = t{3,3,4}×{ }
Truncated 16-cell prism (thexip)
, 26, , 144, , 312, , 288, , 96
, - BGCOLOR="#e0e0f0"
, 79, , = rr{3,3,4}×{ }
Cantellated 16-cell prism (ricope)
(Same as rectified 24-cell prism)
, 50, , 336, , 768, , 672, , 192
, - BGCOLOR="#e0e0f0"
, 80, , = tr{3,3,4}×{ }
Cantitruncated 16-cell prism (ticope)
(Same as truncated 24-cell prism)
, 50, , 336, , 864, , 960, , 384
, - BGCOLOR="#e0e0f0"
, 81, , = t_0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism (prittip)
, 82, , 528, , 1216, , 1152, , 384
, - BGCOLOR="#a0e0f0"
, 82, , = sr{3,3,4}×{ }
snub 24-cell prism (sadip)
, 146, , 768, , 1392, , 960, , 192
, - BGCOLOR="#a0e0f0"
, Nonuniform, ,
rectified tesseractic alterprism (rita)
, 50, , 288, , 464, , 288, , 64
, - BGCOLOR="#a0e0f0"
, Nonuniform, ,
truncated 16-cell alterprism (thexa)
, 26, , 168, , 384, , 336, , 96
, - BGCOLOR="#a0e0f0"
, Nonuniform, ,
bitruncated tesseractic alterprism (taha)
, 50, , 288, , 624, , 576, , 192

F₄ × A₁

This prismatic family has Uniform 4-polytope#The F4 .5B3.2C4.2C3.5D family - .2824-cell.29, 10 forms.

The Coxeter group#Finite Coxeter groups, A₁ x F₄ family has symmetry of order 2304 (2*1152). Three polytopes 85, 86 and 89 (green background) have double symmetry 3,4,3],2], order 4608. The last one, snub 24-cell prism, (blue background) has [3⁺,4,3,2] symmetry, order 1152.

{, class="wikitable"
!rowspan=2, #
!rowspan=2, Coxeter diagram
and Schläfli symbol, Schläfli
symbols
Name
!colspan=5 rowspan=1, Element counts
, -
! Facets, , Cells, , Faces, , Edges, , Vertices
, -
, [77], , = {3,4,3}×{ }
24-cell prism (icope)
, 26, , 144, , 288, , 216, , 48
, -
, [79], , = r{3,4,3}×{ }
rectified 24-cell prism (ricope)
, 50, , 336, , 768, , 672, , 192
, -
, [80], , = t{3,4,3}×{ }
truncated 24-cell prism (ticope)
, 50, , 336, , 864, , 960, , 384
, -
, 83, , = rr{3,4,3}×{ }
cantellated 24-cell prism (sricope)
, 146, , 1008, , 2304, , 2016, , 576
, - BGCOLOR="#b0f0b0"
, 84, , = t_0,3{3,4,3}×{ }
runcinated 24-cell prism (spiccup)
, 242, , 1152, , 1920, , 1296, , 288
, - BGCOLOR="#b0f0b0"
, 85, , = 2t{3,4,3}×{ }
bitruncated 24-cell prism (contip)
, 50, , 432, , 1248, , 1440, , 576
, -
, 86, , = tr{3,4,3}×{ }
cantitruncated 24-cell prism (gricope)
, 146, , 1008, , 2592, , 2880, , 1152
, -
, 87, , = t_0,1,3{3,4,3}×{ }
runcitruncated 24-cell prism (pricope)
, 242, , 1584, , 3648, , 3456, , 1152
, - BGCOLOR="#b0f0b0"
, 88, , = t_0,1,2,3{3,4,3}×{ }
omnitruncated 24-cell prism (gippiccup)
, 242, , 1872, , 5088, , 5760, , 2304
, - BGCOLOR="#b0e0f0"
, [82], , = s{3,4,3}×{ }
snub 24-cell prism (sadip)
, 146, , 768, , 1392, , 960, , 192

H₄ × A₁

This prismatic family has Uniform 4-polytope#The H4 .5B5.2C3.2C3.5D family .E2.80.94 .28120-cell.2F600-cell.29, 15 forms:

The Coxeter group#Finite Coxeter groups, A₁ x H₄ family has symmetry of order 28800 (2*14400).

{, class="wikitable"
!rowspan=2, #
!rowspan=2, Coxeter diagram
and Schläfli symbol, Schläfli
symbols
Name
!colspan=5 rowspan=1, Element counts
, -
! Facets, , Cells, , Faces, , Edges, , Vertices
, - BGCOLOR="#f0e0e0"
, 89, , = {5,3,3}×{ }
120-cell prism (hipe)
, 122, , 960, , 2640, , 3000, , 1200
, - BGCOLOR="#f0e0e0"
, 90, , = r{5,3,3}×{ }
Rectified 120-cell prism (rahipe)
, 722, , 4560, , 9840, , 8400, , 2400
, - BGCOLOR="#f0e0e0"
, 91, , = t{5,3,3}×{ }
Truncated 120-cell prism (thipe)
, 722, , 4560, , 11040, , 12000, , 4800
, - BGCOLOR="#f0e0e0"
, 92, , = rr{5,3,3}×{ }
Cantellated 120-cell prism (srahip)
, 1922, , 12960, , 29040, , 25200, , 7200
, - BGCOLOR="#e0f0e0"
, 93, , = t_0,3{5,3,3}×{ }
Runcinated 120-cell prism (sidpixhip)
, 2642, , 12720, , 22080, , 16800, , 4800
, - BGCOLOR="#e0f0e0"
, 94, , = 2t{5,3,3}×{ }
Bitruncated 120-cell prism (xhip)
, 722, , 5760, , 15840, , 18000, , 7200
, - BGCOLOR="#f0e0e0"
, 95, , = tr{5,3,3}×{ }
Cantitruncated 120-cell prism (grahip)
, 1922, , 12960, , 32640, , 36000, , 14400
, - BGCOLOR="#f0e0e0"
, 96, , = t_0,1,3{5,3,3}×{ }
Runcitruncated 120-cell prism (prixip)
, 2642, , 18720, , 44880, , 43200, , 14400
, - BGCOLOR="#e0f0e0"
, 97, , = t_0,1,2,3{5,3,3}×{ }
Omnitruncated 120-cell prism (gidpixhip)
, 2642, , 22320, , 62880, , 72000, , 28800
, - BGCOLOR="#e0e0f0"
, 98, , = {3,3,5}×{ }
600-cell prism (exip)
, 602, , 2400, , 3120, , 1560, , 240
, - BGCOLOR="#e0e0f0"
, 99, , = r{3,3,5}×{ }
Rectified 600-cell prism (roxip)
, 722, , 5040, , 10800, , 7920, , 1440
, - BGCOLOR="#e0e0f0"
, 100, , = t{3,3,5}×{ }
Truncated 600-cell prism (texip)
, 722, , 5040, , 11520, , 10080, , 2880
, - BGCOLOR="#e0e0f0"
, 101, , = rr{3,3,5}×{ }
Cantellated 600-cell prism (srixip)
, 1442, , 11520, , 28080, , 25200, , 7200
, - BGCOLOR="#e0e0f0"
, 102, , = tr{3,3,5}×{ }
Cantitruncated 600-cell prism (grixip)
, 1442, , 11520, , 31680, , 36000, , 14400
, - BGCOLOR="#e0e0f0"
, 103, , = t_0,1,3{3,3,5}×{ }
Runcitruncated 600-cell prism (prahip)
, 2642, , 18720, , 44880, , 43200, , 14400

Duoprism prisms

Uniform duoprism prisms, {''p''}×{''q''}×{ }, form an infinite class for all integers ''p'',''q''>2. {4}×{4}×{ } makes a lower symmetry form of the 5-cube.
{, class="wikitable"
, -
!rowspan=2, Coxeter diagram
!rowspan=2, Names
!colspan=6, Element counts
, -
! 4-faces
! Cells
! Faces
! Edges
! Vertices
, - align=center
, , , {''p''}×{''q''}×{ }, , ''p''+''q''+2, , 3''pq''+3''p''+3''q'', , 4''pq''+2''p''+2''q'', , 5''pq'', , 2''pq''
, - align=center
, , , {''p''}²×{ }, , 2(''p''+1), , 3''p''(''p''+1), , 4''p''(''p''+1), , 5''p''², , 2''p''²
, - align=center
, , , {3}²×{ }, , 8, , 36, , 48, , 45, , 18
, - align=center
, , , {4}²×{ } = 5-cube, , 10, , 40, , 80, , 80, , 32

Grand antiprism prism

The grand antiprism prism is the only known convex non-Wythoffian uniform 5-polytope. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (600 tetrahedron, tetrahedra, 40 pentagonal antiprisms, 700 triangular prisms, 20 pentagonal prisms), and 322 hypercells (2 grand antiprisms , 20 pentagonal antiprism prisms , and 300 tetrahedral prisms ).

{, class="wikitable"
!rowspan=2, #
!rowspan=2, Name
!colspan=5, 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 5-polytopes

Construction of the reflective 5-dimensional 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 5-polytopes 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 5-polytopes.

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.

{, class="wikitable"
!Operation
!width=200 colspan=2, Extended
Schläfli symbol
!width=80, Coxeter diagram
!Description
, - align=center
! Parent
, t₀{p,q,r,s}
, {p,q,r,s}
,
, Any regular 5-polytope
, - align=center
! Rectification (geometry), Rectified
, t₁{p,q,r,s}, , r{p,q,r,s}
,
, align=left, The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual.
, - align=center
! Rectification (geometry), Birectified
, t₂{p,q,r,s}, , 2r{p,q,r,s}
,
, align=left, Birectification reduces faces to points, Cell (geometry), cells to their Dual polytope, duals.
, - align=center
! Rectification (geometry), Trirectified
, t₃{p,q,r,s}, , 3r{p,q,r,s}
,
, align=left, Trirectification reduces cells to points. (Dual rectification)
, - align=center
! Rectification (geometry), Quadrirectified
, t₄{p,q,r,s}, , 4r{p,q,r,s}
,
, align=left, Quadrirectification reduces 4-faces to points. (Dual)
, - align=center
!Truncation (geometry), Truncated
, t_0,1{p,q,r,s}, , t{p,q,r,s}
,
, align=left, 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 5-polytope. The 5-polytope has its original faces doubled in sides, and contains the faces of the dual.

, - align=center
! Cantellation (geometry), Cantellated
, t_0,2{p,q,r,s}, , rr{p,q,r,s}
,
, align=left, In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place.

, - align=center
! Runcination (geometry), Runcinated
, colspan=2, t_0,3{p,q,r,s}
,
, align=left, Runcination reduces cells and creates new cells at the vertices and edges.
, - align=center
! Sterication (geometry), Stericated
, t_0,4{p,q,r,s}, , 2r2r{p,q,r,s}
,
, align=left, Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as Expansion (geometry), expansion operation for 5-polytopes.)
, - align=center
!Omnitruncation (geometry), Omnitruncated
, colspan=2, t_0,1,2,3,4{p,q,r,s}
,
, align=left, All four operators, truncation, cantellation, runcination, and sterication are applied.

, - align=center
!Half
, colspan=2, h{2p,3,q,r}
,
, align=left, Alternation (geometry), Alternation, same as
, - align=center
!Cantic
, colspan=2, h₂{2p,3,q,r}
,
, align=left, Same as
, - align=center
!Runcic
, colspan=2, h₃{2p,3,q,r}
,
, align=left, Same as
, - align=center
!Runcicantic
, colspan=2, h_2,3{2p,3,q,r}
,
, align=left, Same as
, - align=center
!Steric
, colspan=2, h₄{2p,3,q,r}
,
, align=left, Same as
, - align=center
!Steriruncic
, colspan=2, h_3,4{2p,3,q,r}
,
, align=left, Same as
, - align=center
!Stericantic
, colspan=2, h_2,4{2p,3,q,r}
,
, align=left, Same as
, - align=center
!Steriruncicantic
, colspan=2, h_2,3,4{2p,3,q,r}
,
, align=left, Same as
, - align=center
!Snub
, colspan=2, s{p,2q,r,s}
,
, align=left, Alternated truncation
, - align=center
!Snub rectified
, colspan=2, sr{p,q,2r,s}
,
, align=left, Alternated truncated rectification
, - align=center
!
, colspan=2, ht_0,1,2,3{p,q,r,s}
,
, align=left, Alternated runcicantitruncation
, - align=center
!Full snub
, colspan=2, ht_0,1,2,3,4{p,q,r,s}
,
, align=left, Alternated omnitruncation

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.

{, class=wikitable
, + Fundamental groups
, -
!#
!colspan=3, Coxeter group
!Coxeter diagram
!Forms
, - align=center
, 1, , ${\tilde{A_4$, , [3^[5]], , [(3,3,3,3,3)], , , , 7
, - align=center
, 2, , ${\tilde{C_4$, , [4,3,3,4], , , , , , 19
, - align=center
, 3, , ${\tilde{B_4$, , [4,3,3^1,1], , [4,3,3,4,1⁺], , = , , 23 (8 new)
, - align=center
, 4, , ${\tilde{D_4$, , [3^1,1,1,1], , [1⁺,4,3,3,4,1⁺], , = , , 9 (0 new)
, - align=center
, 5, , ${\tilde{F_4$, , [3,4,3,3], , , , , , 31 (21 new)

There are three List of regular polytopes#Higher dimensions 3, regular honeycombs of Euclidean 4-space:
*tesseractic honeycomb, with symbols {4,3,3,4}, = . There are 19 uniform honeycombs in this family.
* 24-cell honeycomb, with symbols {3,4,3,3}, . There are 31 reflective uniform honeycombs in this family, and one alternated form.
** Truncated 24-cell honeycomb with symbols t{3,4,3,3},
** Snub 24-cell honeycomb, with symbols s{3,4,3,3}, and constructed by four snub 24-cell, one 16-cell, and five 5-cells at each vertex.
* 16-cell honeycomb, with symbols {3,3,4,3},

Other families that generate uniform honeycombs:
* There are 23 uniquely ringed forms, 8 new ones in the 16-cell honeycomb family. With symbols h{4,3²,4} it is geometrically identical to the 16-cell honeycomb, =
* There are 7 uniquely ringed forms from the ${\tilde{A_4$, family, all new, including:
** 4-simplex honeycomb
** Truncated 4-simplex honeycomb
** Omnitruncated 4-simplex honeycomb
* There are 9 uniquely ringed forms in the ${\tilde{D_4$: [3^1,1,1,1] family, two new ones, including the quarter tesseractic honeycomb, = , and the bitruncated tesseractic honeycomb, = .

Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.

{, class=wikitable
, + Prismatic groups
, -
!#
!colspan=2, Coxeter group
!Coxeter diagram
, -
, 1, , ${\tilde{C_3$×${\tilde{I_1$, , [4,3,4,2,∞], ,
, -
, 2, , ${\tilde{B_3$×${\tilde{I_1$, , [4,3^1,1,2,∞], ,
, -
, 3, , ${\tilde{A_3$×${\tilde{I_1$, , [3^[4],2,∞], ,
, -
, 4, , ${\tilde{C_2$×${\tilde{I_1$x${\tilde{I_1$, , [4,4,2,∞,2,∞], ,
, -
, 5, , ${\tilde{H_2$×${\tilde{I_1$x${\tilde{I_1$, , [6,3,2,∞,2,∞], ,
, -
, 6, , ${\tilde{A_2$×${\tilde{I_1$x${\tilde{I_1$, , [3^[3],2,∞,2,∞], ,
, -
, 7, , ${\tilde{I_1$×${\tilde{I_1$x${\tilde{I_1$x${\tilde{I_1$, , [∞,2,∞,2,∞,2,∞], ,
, -
, 8, , ${\tilde{A_2$x${\tilde{A_2$, , [3^[3],2,3^[3]], ,
, -
, 9, , ${\tilde{A_2$×${\tilde{B_2$, , [3^[3],2,4,4], ,
, -
, 10, , ${\tilde{A_2$×${\tilde{G_2$, , [3^[3],2,6,3], ,
, -
, 11, , ${\tilde{B_2$×${\tilde{B_2$, , [4,4,2,4,4], ,
, -
, 12, , ${\tilde{B_2$×${\tilde{G_2$, , [4,4,2,6,3], ,
, -
, 13, , ${\tilde{G_2$×${\tilde{G_2$, , [6,3,2,6,3], ,

Regular and uniform hyperbolic honeycombs

;Hyperbolic compact groups

There are 5 Coxeter-Dynkin diagram#Compact, compact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.

{, class="wikitable"
, valign=top align=
${\widehat{AF_4$ = [(3,3,3,3,4)]:
, valign=top align=
${\bar{DH_4$ = [5,3,3^1,1]:
, valign=top align=${\bar{H_4$ = [3,3,3,5]:

${\bar{BH_4$ = [4,3,3,5]:

${\bar{K_4$ = [5,3,3,5]:

There are 5 regular compact convex hyperbolic honeycombs in H⁴ space:Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213
{, class="wikitable"
, + Compact regular convex hyperbolic honeycombs
, -
!Honeycomb name
!Schläfli symbol, 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, Vertex
figure
{q,r,s}
!dual polyhedron, Dual
, - BGCOLOR="#ffe0e0" align=center
, Order-5 5-cell honeycomb, Order-5 5-cell (pente), , {3,3,3,5}, , , , {3,3,3}, , {3,3}, , {3}, , {5}, , {3,5}, , {3,3,5}, , {5,3,3,3}
, - BGCOLOR="#e0e0ff" align=center
, Order-3 120-cell honeycomb, Order-3 120-cell (hitte), , {5,3,3,3}, , , , {5,3,3}, , {5,3}, , {5}, , {3}, , {3,3}, , {3,3,3}, , {3,3,3,5}
, - BGCOLOR="#ffe0e0" align=center
, Order-5 tesseractic honeycomb, Order-5 tesseractic (pitest), , {4,3,3,5}, , , , {4,3,3}, , {4,3}, , {4}, , {5}, , {3,5}, , {3,3,5}, , {5,3,3,4}
, - BGCOLOR="#e0e0ff" align=center
, Order-4 120-cell honeycomb, Order-4 120-cell (shitte), , {5,3,3,4}, , , , {5,3,3}, , {5,3}, , {5}, , {4}, , {3,4}, , {3,3,4}, , {4,3,3,5}
, - BGCOLOR="#e0ffe0" align=center
, Order-5 120-cell honeycomb, Order-5 120-cell (phitte), , {5,3,3,5}, , , , {5,3,3}, , {5,3}, , {5}, , {5}, , {3,5}, , {3,3,5}, , Self-dual

There are also 4 regular compact hyperbolic star-honeycombs in H⁴ space:
{, class="wikitable"
, + Compact regular hyperbolic star-honeycombs
, -
!Honeycomb name
!Schläfli symbol, 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, Vertex
figure
{q,r,s}
!dual polyhedron, Dual
, - BGCOLOR="#ffe0e0" align=center
, Small stellated 120-cell honeycomb, Order-3 small stellated 120-cell, , {5/2,5,3,3}, , , , {5/2,5,3}, , {5/2,5}, , {5}, , {5}, , {3,3}, , {5,3,3}, , {3,3,5,5/2}
, - BGCOLOR="#e0e0ff" align=center
, Pentagrammic-order 600-cell honeycomb, Order-5/2 600-cell, , {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}
, - BGCOLOR="#ffe0e0" align=center
, Order-5 icosahedral 120-cell honeycomb, Order-5 icosahedral 120-cell, , {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}
, - BGCOLOR="#e0e0ff" align=center
, Great 120-cell honeycomb, Order-3 great 120-cell, , {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}

;Hyperbolic paracompact groups

There are 9 Coxeter-Dynkin diagram#Rank 4 to 10, paracompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Paracompact groups generate honeycombs with infinite Facet (geometry), facets or vertex figures.

{, class=wikitable
, align=
${\bar{P_4$ = [3,3^[4]]:

${\bar{BP_4$ = [4,3^[4]]:

${\bar{FR_4$ = [(3,3,4,3,4)]:

${\bar{DP_4$ = [3^[3]×[]]:

, align=
${\bar{N_4$ = [4,/3\,3,4]:

${\bar{O_4$ = [3,4,3^1,1]:

${\bar{S_4$ = [4,3^2,1]:

${\bar{M_4$ = [4,3^1,1,1]:

, align=
${\bar{R_4$ = [3,4,3,4]:

Notes

References

* Thorold Gosset, T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900 (3 regular and one semiregular 4-polytope)
* Alicia Boole Stott, 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
* Harold Scott MacDonald Coxeter, H.S.M. Coxeter:
** 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)
** Coxeter, 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, Wiley-Interscience Publication, 1995,

** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] (p. 287 5D Euclidean groups, p. 298 Four-dimensionsal honeycombs)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* Norman Johnson (mathematician), 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

External links

* – includes nonconvex forms as well as the duplicate constructions from the B₅ and D₅ families

{{Honeycombs

5-polytopes

eo:5-hiperpluredro