Uniform 5-polytope

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Graphs of regular and uniform polytopes.
5-simplex t0.svg
5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t1.svg
Rectified 5-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t01.svg
Truncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t02.svg
Cantellated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t03.svg
Runcinated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t04.svg
Stericated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t4.svg
5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t34.svg
Truncated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t3.svg
Rectified 5-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t24.svg
Cantellated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t14.svg
Runcinated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-cube t02.svg
Cantellated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t03.svg
Runcinated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t04.svg
Stericated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t0.svg
5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t01.svg
Truncated 5-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t1.svg
Rectified 5-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t0 D5.svg
5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t01 D5.svg
Truncated 5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t02 D5.svg
Cantellated 5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-demicube t03 D5.svg
Runcinated 5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png

In geometry, a uniform polyteron[1][2] (or uniform 5-polytope) is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive 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 Coxeter diagrams.

History of discovery[edit]

  • Regular polytopes: (convex faces)
    • 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions.
  • Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
    • 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular polychora) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.[3]
  • Convex uniform polytopes:
    • 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
  • Nonregular uniform star polytopes: (similar to the nonconvex uniform polyhedra)
    • Ongoing: Thousands of nonconvex uniform polytera are known, but mostly unpublished. The list is presumed not to be complete, and there is no estimate of how long the complete list will be, although 3067 convex and nonconvex uniform polytera are currently known. Participating researchers include Jonathan Bowers, Richard Klitzing and Norman Johnson.[4]

Regular 5-polytopes[edit]

Regular 5-polytopes 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 such regular polytopes, all convex:

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

Convex uniform 5-polytopes[edit]

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.

Reflection families[edit]

Coxeter diagram finite rank5 correspondence.png
Coxeter diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

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 D6 family contains the pentacross, as well as a 5-demicube which is an alternated 5-cube.

Fundamental families

# Coxeter group Coxeter diagram
1 A5 [34] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
2 B5 [4,33] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3 D5 [32,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Uniform prisms There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes:

# Coxeter groups Coxeter diagram
1 A4 × A1 [3,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
2 B4 × A1 [4,3,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
3 F4 × A1 [3,4,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
4 H4 × A1 [5,3,3,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
5 D4 × A1 [31,1,1,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png

There is one infinite family of 5-polytopes based on prisms of the uniform duoprisms {p}×{q}×{ }:

Coxeter groups Coxeter diagram
I2(p) × I2(q) × A1 [p,2,q,2] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.png

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 diagram
1 A3 × I2(p) [3,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
2 B3 × I2(p) [4,3,2,p] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
3. H3 × I2(p) [5,3,2,p] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png

Enumerating the convex uniform 5-polytopes[edit]

  • Simplex family: A5 [34]
    • 19 uniform 5-polytopes
  • Hypercube/Orthoplex family: BC5 [4,33]
    • 31 uniform 5-polytopes
  • Demihypercube D5/E5 family: [32,1,1]
    • 23 uniform 5-polytopes (8 unique)
  • Prisms and duoprisms:
    • 56 uniform 5-polytope (46 unique) constructions based on prismatic families: [3,3,3]×[ ], [4,3,3]×[ ], [5,3,3]×[ ], [31,1,1]×[ ].
    • One non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two grand antiprisms connected by polyhedral prisms.

That brings the tally to: 19+31+8+46+1=105

In addition there are:

  • Infinitely many uniform 5-polytope constructions based on duoprism prismatic families: [p]×[q]×[ ].
  • Infinitely many uniform 5-polytope constructions based on duoprismatic families: [3,3]×[p], [4,3]×[p], [5,3]×[p].

The A5 family[edit]

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

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

The A5 family has symmetry of order 720 (6 factorial).

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).

See symmetry graphs: List of A5 polytopes

# Base point Johnson naming system
Bowers name and (acronym)
Coxeter diagram
k-face element counts Vertex
figure
Facet counts by location: [3,3,3,3]
4 3 2 1 0 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(6)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[3,3,2]
(15)
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[3,2,3]
(20)
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[2,3,3]
(15)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(6)
1 (0,0,0,0,0,1) or (0,1,1,1,1,1) 5-simplex
hexateron (hix)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6 15 20 15 6 5-simplex verf.png
{3,3,3}
(5)
4-simplex t0.svg
{3,3,3}
- - - -
2 (0,0,0,0,1,1) or (0,0,1,1,1,1) Rectified 5-simplex
rectified hexateron (rix)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
12 45 80 60 15 Rectified 5-simplex verf.png
t{3,3}×{ }
(4)
4-simplex t1.svg
r{3,3,3}
- - - (2)
4-simplex t0.svg
{3,3,3}
3 (0,0,0,0,1,2) or (0,1,2,2,2,2) Truncated 5-simplex
truncated hexateron (tix)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
12 45 80 75 30 Truncated 5-simplex verf.png
Tetrah.pyr
(4)
4-simplex t01.svg
t{3,3,3}
- - - (1)
4-simplex t0.svg
{3,3,3}
4 (0,0,0,1,1,1) Birectified 5-simplex
dodecateron (dot)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
12 60 120 90 20 Birectified hexateron verf.png
{3}×{3}
(3)
4-simplex t1.svg
r{3,3,3}
- - - (3)
4-simplex t1.svg
r{3,3,3}
5 (0,0,0,1,1,2) or (0,1,1,2,2,2) Cantellated 5-simplex
small rhombated hexateron (sarx)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
27 135 290 240 60 Cantellated hexateron verf.png
prism-wedge
(3)
4-simplex t02.svg
rr{3,3,3}
- - (1)
1-simplex t0.svg×3-simplex t0.svg
{ }×{3,3}
(1)
4-simplex t1.svg
r{3,3,3}
6 (0,0,0,1,2,2) or (0,0,1,2,2,2) Bitruncated 5-simplex
bitruncated hexateron (bittix)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
12 60 140 150 60 Bitruncated 5-simplex verf.png (3)
4-simplex t12.svg
2t{3,3,3}
- - - (2)
4-simplex t01.svg
t{3,3,3}
7 (0,0,0,1,2,3) or (0,1,2,3,3,3) Cantitruncated 5-simplex
great rhombated hexateron (garx)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
27 135 290 300 120 Canitruncated 5-simplex verf.png 4-simplex t012.svg
tr{3,3,3}
- - 1-simplex t0.svg×3-simplex t0.svg
{ }×{3,3}
4-simplex t01.svg
t{3,3,3}
8 (0,0,1,1,1,2) or (0,1,1,1,2,2) Runcinated 5-simplex
small prismated hexateron (spix)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
47 255 420 270 60 Runcinated 5-simplex verf.png (2)
4-simplex t03.svg
t0,3{3,3,3}
- (3)
3-3 duoprism ortho-skew.png
{3}×{3}
(3)
1-simplex t0.svg×3-simplex t1.svg
{ }×r{3,3}
(1)
4-simplex t1.svg
r{3,3,3}
9 (0,0,1,1,2,2) Bicantellated 5-simplex
small birhombated dodecateron (sibrid)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
32 180 420 360 90 Bicantellated 5-simplex verf.png (2)
4-simplex t02.svg
rr{3,3,3}
- (8)
3-3 duoprism ortho-skew.png
{3}×{3}
- (2)
4-simplex t02.svg
rr{3,3,3}
10 (0,0,1,1,2,3) or (0,1,2,2,3,3) Runcitruncated 5-simplex
prismatotruncated hexateron (pattix)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
47 315 720 630 180 Runcitruncated 5-simplex verf.png 4-simplex t013.svg
t0,1,3{3,3,3}
- 2-simplex t0.svg×2-simplex t01.svg
{6}×{3}
1-simplex t0.svg×3-simplex t1.svg
{ }×r{3,3}
4-simplex t02.svg
rr{3,3,3}
11 (0,0,1,2,2,3) or (0,1,1,2,3,3) Runcicantellated 5-simplex
prismatorhombated hexateron (pirx)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
47 255 570 540 180 Runcicantellated 5-simplex verf.png 4-simplex t03.svg
t0,1,3{3,3,3}
- 3-3 duoprism ortho-skew.png
{3}×{3}
1-simplex t0.svg×4-simplex t01.svg
{ }×t{3,3}
4-simplex t12.svg
2t{3,3,3}
12 (0,0,1,2,3,3) Bicantitruncated 5-simplex
great birhombated dodecateron (gibrid)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
32 180 420 450 180 Bicanitruncated 5-simplex verf.png 4-simplex t012.svg
tr{3,3,3}
- 3-3 duoprism ortho-skew.png
{3}×{3}
- 4-simplex t012.svg
tr{3,3,3}
13 (0,0,1,2,3,4) or (0,1,2,3,4,4) Runcicantitruncated 5-simplex
great prismated hexateron (gippix)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
47 315 810 900 360 Runcicantitruncated 5-simplex verf.png
Irr.5-cell
4-simplex t0123.svg
t0,1,2,3{3,3,3}
- 2-simplex t0.svg×2-simplex t01.svg
{3}×{6}
1-simplex t0.svg×4-simplex t01.svg
{ }×t{3,3}
4-simplex t02.svg
rr{3,3,3}
14 (0,1,1,1,1,2) Stericated 5-simplex
small cellated dodecateron (scad)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
62 180 210 120 30 Stericated hexateron verf.png
Irr.16-cell
(1)
4-simplex t0.svg
{3,3,3}
(4)
1-simplex t0.svg×3-simplex t0.svg
{ }×{3,3}
(6)
3-3 duoprism ortho-skew.png
{3}×{3}
(4)
1-simplex t0.svg×3-simplex t0.svg
{ }×{3,3}
(1)
4-simplex t0.svg
{3,3,3}
15 (0,1,1,1,2,3) or (0,1,2,2,2,3) Steritruncated 5-simplex
celliprismated hexateron (cappix)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
62 330 570 420 120 Steritruncated 5-simplex verf.png 4-simplex t01.svg
t{3,3,3}
1-simplex t0.svg×4-simplex t01.svg
{ }×t{3,3}
2-simplex t0.svg×2-simplex t01.svg
{3}×{6}
1-simplex t0.svg×3-simplex t0.svg
{ }×{3,3}
4-simplex t03.svg
t0,3{3,3,3}
16 (0,1,1,2,2,3) Stericantellated 5-simplex
small cellirhombated dodecateron (card)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
62 420 900 720 180 Stericantellated 5-simplex verf.png 4-simplex t02.svg
rr{3,3,3}
1-simplex t0.svg×3-simplex t02.svg
{ }×rr{3,3}
3-3 duoprism ortho-skew.png
{3}×{3}
1-simplex t0.svg×3-simplex t02.svg
{ }×rr{3,3}
4-simplex t02.svg
rr{3,3,3}
17 (0,1,1,2,3,4) or (0,1,2,3,3,4) Stericantitruncated 5-simplex
celligreatorhombated hexateron (cograx)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
62 480 1140 1080 360 Stericanitruncated 5-simplex verf.png 4-simplex t012.svg
tr{3,3,3}
1-simplex t0.svg×3-simplex t012.svg
{ }×tr{3,3}
2-simplex t0.svg×2-simplex t01.svg
{3}×{6}
1-simplex t0.svg×3-simplex t02.svg
{ }×rr{3,3}
4-simplex t013.svg
t0,1,3{3,3,3}
18 (0,1,2,2,3,4) Steriruncitruncated 5-simplex
celliprismatotruncated dodecateron (captid)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
62 450 1110 1080 360 Steriruncitruncated 5-simplex verf.png 4-simplex t013.svg
t0,1,3{3,3,3}
1-simplex t0.svg×4-simplex t01.svg
{ }×t{3,3}
6-6 duoprism ortho-3.png
{6}×{6}
1-simplex t0.svg×4-simplex t01.svg
{ }×t{3,3}
4-simplex t013.svg
t0,1,3{3,3,3}
19 (0,1,2,3,4,5) Omnitruncated 5-simplex
great cellated dodecateron (gocad)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
62 540 1560 1800 720 Omnitruncated 5-simplex verf.png
Irr. {3,3,3}
(1)
4-simplex t0123.svg
t0,1,2,3{3,3,3}
(1)
1-simplex t0.svg×3-simplex t012.svg
{ }×tr{3,3}
(1)
6-6 duoprism ortho-3.png
{6}×{6}
(1)
1-simplex t0.svg×3-simplex t012.svg
{ }×tr{3,3}
(1)
4-simplex t0123.svg
t0,1,2,3{3,3,3}

The B5 family[edit]

The B5 family has symmetry of order 3840 (5!×25).

This family has 25−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter diagram.

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 polytera 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 polyteron. All coordinates correspond with uniform polytera of edge length 2.

See symmetry graph: List of B5 polytopes

# Base point Name
Coxeter diagram
Element counts Vertex
figure
Facet counts by location: [4,3,3,3]
4 3 2 1 0 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[4,3,3]
(10)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[4,3,2]
(40)
CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[4,2,3]
(80)
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[2,3,3]
(80)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(32)
1 (0,0,0,0,1)√2 5-orthoplex (tac)
(Quadrirectified 5-cube)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
32 80 80 40 10 Pentacross verf.png
{3,3,4}
Schlegel wireframe 5-cell.png
{3,3,3}
- - - -
2 (0,0,0,1,1)√2 Rectified 5-orthoplex (rat)
(Trirectified 5-cube)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
42 240 400 240 40 Rectified pentacross verf.png
{ }×{3,4}
Schlegel wireframe 16-cell.png

{3,3,4}
- - - Schlegel half-solid rectified 5-cell.png
r{3,3,3}
3 (0,0,0,1,2)√2 Truncated 5-orthoplex (tot)
(Quadritruncated 5-cube)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
42 240 400 280 80 Truncated pentacross.png
(Octah.pyr)
Schlegel half-solid truncated pentachoron.png
t{3,3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
- - -
4 (0,0,1,1,1)√2 Birectified 5-cube (nit)
(Birectified 5-orthoplex)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
42 280 640 480 80 Birectified penteract verf.png
{4}×{3}
Schlegel half-solid rectified 16-cell.png
r{3,3,4}
- - - Schlegel half-solid rectified 5-cell.png
r{3,3,3}
5 (0,0,1,1,2)√2 Cantellated 5-orthoplex (sart)
(Tricantellated 5-cube)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
82 640 1520 1200 240 Cantellated pentacross verf.png
Prism-wedge
r{3,3,4} { }×{3,4} - - Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
6 (0,0,1,2,2)√2 Bitruncated 5-orthoplex (bittit)
(tritruncated 5-cube)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
42 280 720 720 240 Bitruncated pentacross verf.png t{3,3,4} - - - Schlegel half-solid bitruncated 5-cell.png
2t{3,3,3}
7 (0,0,1,2,3)√2 Cantitruncated 5-orthoplex (gart)
(tricantitruncated 5-orthoplex)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
82 640 1520 1440 480 Canitruncated 5-orthoplex verf.png rr{3,3,4} { }×r{3,4} 6-4 duoprism.png
{6}×{4}
- Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
8 (0,1,1,1,1)√2 Rectified 5-cube (rit)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
42 200 400 320 80 Rectified 5-cube verf.png
{3,3}×{ }
Schlegel half-solid rectified 8-cell.png
r{4,3,3}
- - - Schlegel wireframe 5-cell.png
{3,3,3}
9 (0,1,1,1,2)√2 Runcinated 5-orthoplex (spat)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
162 1200 2160 1440 320 Runcinated pentacross verf.png r{4,3,3} - 3-4 duoprism.png
{3}×{4}
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
10 (0,1,1,2,2)√2 Bicantellated 5-cube (sibrant)
(Bicantellated 5-orthoplex)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
122 840 2160 1920 480 Bicantellated penteract verf.png Schlegel half-solid cantellated 8-cell.png
rr{4,3,3}
- 3-4 duoprism.png
{4}×{3}
- Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
11 (0,1,1,2,3)√2 Runcitruncated 5-orthoplex (pattit)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
162 1440 3680 3360 960 Runcitruncated 5-orthoplex verf.png rr{3,3,4} { }×r{3,4} 6-4 duoprism.png
{6}×{4}
- Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
12 (0,1,2,2,2)√2 Bitruncated 5-cube (pattin)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
42 280 720 800 320 Bitruncated penteract verf.png Schlegel half-solid bitruncated 8-cell.png
2t{4,3,3}
- - - Schlegel half-solid truncated pentachoron.png
t{3,3,3}
13 (0,1,2,2,3)√2 Runcicantellated 5-orthoplex (pirt)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
162 1200 2960 2880 960 Runcicantellated 5-orthoplex verf.png { }×t{3,4} 2t{3,3,4} 3-4 duoprism.png
{3}×{4}
- Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
14 (0,1,2,3,3)√2 Bicantitruncated 5-cube (gibrant)
(Bicantitruncated 5-orthoplex)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
122 840 2160 2400 960 Bicantellated penteract verf.png Schlegel half-solid cantellated 8-cell.png
rr{4,3,3}
- 3-4 duoprism.png
{4}×{3}
- Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
15 (0,1,2,3,4)√2 Runcicantitruncated 5-orthoplex (gippit)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
162 1440 4160 4800 1920 Runcicantitruncated 5-orthoplex verf.png tr{3,3,4} { }×t{3,4} 6-4 duoprism.png
{6}×{4}
- Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
16 (1,1,1,1,1) 5-cube (pent)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10 40 80 80 32 5-cube verf.png
{3,3,3}
Schlegel wireframe 8-cell.png
{4,3,3}
- - - -
17 (1,1,1,1,1)
+ (0,0,0,0,1)√2
Stericated 5-cube (scant)
(Stericated 5-orthoplex)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
242 800 1040 640 160 Stericated penteract verf.png
Tetr.antiprm
Schlegel wireframe 8-cell.png
{4,3,3}
Schlegel wireframe 8-cell.png
{4,3}×{ }
3-4 duoprism.png
{4}×{3}
Tetrahedral prism.png
{ }×{3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
18 (1,1,1,1,1)
+ (0,0,0,1,1)√2
Runcinated 5-cube (span)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
202 1240 2160 1440 320 Runcinated penteract verf.png Schlegel half-solid runcinated 8-cell.png
t0,3{4,3,3}
- 3-4 duoprism.png
{4}×{3}
Octahedral prism.png
{ }×r{3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
19 (1,1,1,1,1)
+ (0,0,0,1,2)√2
Steritruncated 5-orthoplex (cappin)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
242 1520 2880 2240 640 Steritruncated 5-orthoplex verf.png t0,3{3,3,4} { }×{4,3} - - Schlegel half-solid truncated pentachoron.png
t{3,3,3}
20 (1,1,1,1,1)
+ (0,0,1,1,1)√2
Cantellated 5-cube (sirn)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
122 680 1520 1280 320 Cantellated 5-cube vertf.png
Prism-wedge
Schlegel half-solid cantellated 8-cell.png
rr{4,3,3}
- - Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
21 (1,1,1,1,1)
+ (0,0,1,1,2)√2
Stericantellated 5-cube (carnit)
(Stericantellated 5-orthoplex)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
242 2080 4720 3840 960 Stericantellated 5-orthoplex verf.png Schlegel half-solid cantellated 8-cell.png
rr{4,3,3}
Rhombicuboctahedral prism.png
rr{4,3}×{ }
3-4 duoprism.png
{4}×{3}
Cuboctahedral prism.png
{ }×rr{3,3}
Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
22 (1,1,1,1,1)
+ (0,0,1,2,2)√2
Runcicantellated 5-cube (prin)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
202 1240 2960 2880 960 Runcicantellated 5-cube verf.png Schlegel half-solid runcitruncated 8-cell.png
t0,1,3{4,3,3}
- 3-4 duoprism.png
{4}×{3}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid bitruncated 5-cell.png
2t{3,3,3}
23 (1,1,1,1,1)
+ (0,0,1,2,3)√2
Stericantitruncated 5-orthoplex (cogart)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
242 2320 5920 5760 1920 Stericanitruncated 5-orthoplex verf.png Truncated tetrahedral prism.png
{ }×rr{3,4}
Runcitruncated 16-cell.png
t0,1,3{3,3,4}
6-4 duoprism.png
{6}×{4}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
24 (1,1,1,1,1)
+ (0,1,1,1,1)√2
Truncated 5-cube (tan)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
42 200 400 400 160 Truncated 5-cube verf.png
Tetrah.pyr
Schlegel half-solid truncated tesseract.png
t{4,3,3}
- - - Schlegel wireframe 5-cell.png
{3,3,3}
25 (1,1,1,1,1)
+ (0,1,1,1,2)√2
Steritruncated 5-cube (cappin)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
242 1600 2960 2240 640 Steritruncated 5-cube verf.png Schlegel half-solid truncated tesseract.png
t{4,3,3}
Truncated cubic prism.png
t{4,3}×{ }
8-3 duoprism.png
{8}×{3}
Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
26 (1,1,1,1,1)
+ (0,1,1,2,2)√2
Runcitruncated 5-cube (pattin)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
202 1560 3760 3360 960 Runcitruncated 5-cube verf.png Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{4,3,3}
{ }×t{4,3} 6-8 duoprism.png
{6}×{8}
{ }×t{3,3} t0,1,3{3,3,3}]]
27 (1,1,1,1,1)
+ (0,1,1,2,3)√2
Steriruncitruncated 5-cube (captint)
(Steriruncitruncated 5-orthoplex)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
242 2160 5760 5760 1920 Steriruncitruncated 5-cube verf.png Schlegel half-solid runcitruncated 8-cell.png
t0,1,3{4,3,3}
Truncated cubic prism.png
t{4,3}×{ }
8-6 duoprism.png
{8}×{6}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
28 (1,1,1,1,1)
+ (0,1,2,2,2)√2
Cantitruncated 5-cube (girn)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
122 680 1520 1600 640 Canitruncated 5-cube verf.png Schlegel half-solid cantitruncated 8-cell.png
tr{4,3,3}
- - Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid truncated pentachoron.png
t{3,3,3}
29 (1,1,1,1,1)
+ (0,1,2,2,3)√2
Stericantitruncated 5-cube (cogrin)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
242 2400 6000 5760 1920 Stericanitruncated 5-cube verf.png Schlegel half-solid cantitruncated 8-cell.png
tr{4,3,3}
Truncated cuboctahedral prism.png
tr{4,3}×{ }
8-3 duoprism.png
{8}×{3}
Cuboctahedral prism.png
{ }×t0,2{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
30 (1,1,1,1,1)
+ (0,1,2,3,3)√2
Runcicantitruncated 5-cube (gippin)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
202 1560 4240 4800 1920 Runcicantitruncated 5-cube verf.png Schlegel half-solid omnitruncated 8-cell.png
t0,1,2,3{4,3,3}
- 8-3 duoprism.png
{8}×{3}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
31 (1,1,1,1,1)
+ (0,1,2,3,4)√2
Omnitruncated 5-cube (gacnet)
(omnitruncated 5-orthoplex)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
242 2640 8160 9600 3840 Omnitruncated 5-cube verf.png
Irr. {3,3,3}
Schlegel half-solid omnitruncated 8-cell.png
tr{4,3}×{ }
Truncated cuboctahedral prism.png
tr{4,3}×{ }
8-6 duoprism.png
{8}×{6}
Truncated octahedral prism.png
{ }×tr{3,3}
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}

The D5 family[edit]

The D5 family has symmetry of order 1920 (5! x 24).

This family has 23 Wythoffian uniform polyhedra, from 3x8-1 permutations of the D5 Coxeter diagram with one or more rings. 15 (2x8-1) are repeated from the B5 family and 8 are unique to this family.

See symmetry graphs: List of D5 polytopes

# Coxeter diagram
Schläfli symbol symbols
Johnson and Bowers names
Element counts Vertex
figure
Facets by location: CD B5 nodes.png [31,2,1]
4 3 2 1 0 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(16)
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
[31,1,1]
(10)
CDel nodes.pngCDel split2.pngCDel node.pngCDel 2.pngCDel node.png
[3,3]×[ ]
(40)
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[ ]×[3]×[ ]
(80)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(16)
51 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
h{4,3,3,3}, 5-demicube
Hemipenteract (hin)
26 120 160 80 16 Demipenteract verf.png
t1{3,3,3}
{3,3,3} t0(111) - - -
52 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
h2{4,3,3,3}, cantic 5-cube
Truncated hemipenteract (thin)
42 280 640 560 160 Truncated 5-demicube verf.png
53 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
h3{4,3,3,3}, runcic 5-cube
Small rhombated hemipenteract (sirhin)
42 360 880 720 160
54 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
h4{4,3,3,3}, steric 5-cube
Small prismated hemipenteract (siphin)
82 480 720 400 80
55 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
h2,3{4,3,3,3}, runcicantic 5-cube
Great rhombated hemipenteract (girhin)
42 360 1040 1200 480
56 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
h2,4{4,3,3,3}, stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
82 720 1840 1680 480
57 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
h3,4{4,3,3,3}, steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
82 560 1280 1120 320
58 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
h2,3,4{4,3,3,3}, steriruncicantic 5-cube
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 4-polytopes:

A4 × A1[edit]

This prismatic family has 9 forms:

The A1 x A4 family has symmetry of order 240 (2*5!).

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
59 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{3,3,3}×{ }
5-cell prism
7 20 30 25 10
60 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{3,3,3}×{ }
Rectified 5-cell prism
12 50 90 70 20
61 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{3,3,3}×{ }
Truncated 5-cell prism
12 50 100 100 40
62 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
rr{3,3,3}×{ }
Cantellated 5-cell prism
22 120 250 210 60
63 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,3{3,3,3}×{ }
Runcinated 5-cell prism
32 130 200 140 40
64 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
2t{3,3,3}×{ }
Bitruncated 5-cell prism
12 60 140 150 60
65 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
tr{3,3,3}×{ }
Cantitruncated 5-cell prism
22 120 280 300 120
66 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism
32 180 390 360 120
67 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2,3{3,3,3}×{ }
Omnitruncated 5-cell prism
32 210 540 600 240

B4 × A1[edit]

This prismatic family has 16 forms. (Three are shared with [3,4,3]×[ ] family)

The A1 x B4 family has symmetry of order 768 (2*2^4*4!).

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
[16] CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{4,3,3}×{ }
Tesseractic prism
(Same as 5-cube)
10 40 80 80 32
68 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{4,3,3}×{ }
Rectified tesseractic prism
26 136 272 224 64
69 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{4,3,3}×{ }
Truncated tesseractic prism
26 136 304 320 128
70 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
rr{4,3,3}×{ }
Cantellated tesseractic prism
58 360 784 672 192
71 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,3{4,3,3}×{ }
Runcinated tesseractic prism
82 368 608 448 128
72 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
2t{4,3,3}×{ }
Bitruncated tesseractic prism
26 168 432 480 192
73 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
tr{4,3,3}×{ }
Cantitruncated tesseractic prism
58 360 880 960 384
74 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism
82 528 1216 1152 384
75 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism
82 624 1696 1920 768
76 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{3,3,4}×{ }
16-cell prism
18 64 88 56 16
77 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{3,3,4}×{ }
Rectified 16-cell prism
(Same as 24-cell prism)
26 144 288 216 48
78 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{3,3,4}×{ }
Truncated 16-cell prism
26 144 312 288 96
79 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
rr{3,3,4}×{ }
Cantellated 16-cell prism
(Same as rectified 24-cell prism)
50 336 768 672 192
80 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{3,3,4}×{ }
Cantitruncated 16-cell prism
(Same as truncated 24-cell prism)
50 336 864 960 384
81 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism
82 528 1216 1152 384
82 CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
sr{3,3,4}×{ }
snub 24-cell prism
146 768 1392 960 192

F4 × A1[edit]

This prismatic family has 10 forms.

The 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.

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
[77] CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{3,4,3}×{ }
24-cell prism
26 144 288 216 48
[79] CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{3,4,3}×{ }
rectified 24-cell prism
50 336 768 672 192
[80] CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{3,4,3}×{ }
truncated 24-cell prism
50 336 864 960 384
83 CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
rr{3,4,3}×{ }
cantellated 24-cell prism
146 1008 2304 2016 576
84 CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,3{3,4,3}×{ }
runcinated 24-cell prism
242 1152 1920 1296 288
85 CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
2t{3,4,3}×{ }
bitruncated 24-cell prism
50 432 1248 1440 576
86 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
tr{3,4,3}×{ }
cantitruncated 24-cell prism
146 1008 2592 2880 1152
87 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,3{3,4,3}×{ }
runcitruncated 24-cell prism
242 1584 3648 3456 1152
88 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2,3{3,4,3}×{ }
omnitruncated 24-cell prism
242 1872 5088 5760 2304
[82] CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
s{3,4,3}×{ }
snub 24-cell prism
146 768 1392 960 192

H4 × A1[edit]

This prismatic family has 15 forms:

The A1 x H4 family has symmetry of order 28800 (2*14400).

# Coxeter- diagram
and Schläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
89 CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{5,3,3}×{ }
120-cell prism
122 960 2640 3000 1200
90 CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{5,3,3}×{ }
Rectified 120-cell prism
722 4560 9840 8400 2400
91 CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{5,3,3}×{ }
Truncated 120-cell prism
722 4560 11040 12000 4800
92 CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
rr{5,3,3}×{ }
Cantellated 120-cell prism
1922 12960 29040 25200 7200
93 CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,3{5,3,3}×{ }
Runcinated 120-cell prism
2642 12720 22080 16800 4800
94 CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
2t{5,3,3}×{ }
Bitruncated 120-cell prism
722 5760 15840 18000 7200
95 CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
tr{5,3,3}×{ }
Cantitruncated 120-cell prism
1922 12960 32640 36000 14400
96 CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,3{5,3,3}×{ }
Runcitruncated 120-cell prism
2642 18720 44880 43200 14400
97 CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,2,3{5,3,3}×{ }
Omnitruncated 120-cell prism
2642 22320 62880 72000 28800
98 CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{3,3,5}×{ }
600-cell prism
602 2400 3120 1560 240
99 CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{3,3,5}×{ }
Rectified 600-cell prism
722 5040 10800 7920 1440
100 CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{3,3,5}×{ }
Truncated 600-cell prism
722 5040 11520 10080 2880
101 CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
rr{3,3,5}×{ }
Cantellated 600-cell prism
1442 11520 28080 25200 7200
102 CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{3,3,5}×{ }
Cantitruncated 600-cell prism
1442 11520 31680 36000 14400
103 CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t0,1,3{3,3,5}×{ }
Runcitruncated 600-cell prism
2642 18720 44880 43200 14400

Grand antiprism prism[edit]

The grand antiprism prism is the only known convex non-Wythoffian uniform polyteron. 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 Grand antiprism.png, 20 pentagonal antiprism prisms Pentagonal antiprismatic prism.png, and 300 tetrahedral prisms Tetrahedral prism.png).

# 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 5-polytopes[edit]

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.

Operation Extended
Schläfli symbol
Coxeter diagram Description
Parent t0{p,q,r,s} {p,q,r,s} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png Any regular 5-polytope
Rectified t1{p,q,r,s} r{p,q,r,s} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual.
Birectified t2{p,q,r,s} 2r{p,q,r,s} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png Birectification reduces faces to points, cells to their duals.
Trirectified t3{p,q,r,s} 3r{p,q,r,s} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png Trirectification reduces cells to points. (Dual rectification)
Quadrirectified t4{p,q,r,s} 4r{p,q,r,s} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.png Quadrirectification reduces 4-faces to points. (Dual)
Truncated t0,1{p,q,r,s} t{p,q,r,s} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png 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.
Cube truncation sequence.svg
Cantellated t0,2{p,q,r,s} rr{p,q,r,s} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png 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.
Cube cantellation sequence.svg
Runcinated t0,3{p,q,r,s} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png Runcination reduces cells and creates new cells at the vertices and edges.
Stericated t0,4{p,q,r,s} 2r2r{p,q,r,s} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.png Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for 5-polytopes.)
Omnitruncated t0,1,2,3,4{p,q,r,s} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node 1.png All four operators, truncation, cantellation, runcination, and sterication are applied.
Half h{2p,3,q,r} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Alternation, same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
Cantic h2{2p,3,q,r} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png
Runcic h3{2p,3,q,r} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png
Runcicantic h2,3{2p,3,q,r} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png
Steric h4{2p,3,q,r} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png
Runcisteric h3,4{2p,3,q,r} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png
Stericantic h2,4{2p,3,q,r} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png
Steriruncicantic h2,3,4{2p,3,q,r} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png
Snub s{p,2q,r,s} CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png Alternated truncation
Snub rectified sr{p,q,2r,s} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel 2x.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png Alternated truncated rectification
ht0,1,2,3{p,q,r,s} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node h.pngCDel 2x.pngCDel s.pngCDel node.png Alternated runcicantitruncation
Full snub ht0,1,2,3,4{p,q,r,s} CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel node h.pngCDel s.pngCDel node h.png Alternated omnitruncation

Regular and uniform honeycombs[edit]

Coxeter diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.[5]

Fundamental groups
# Coxeter group Coxeter diagram Forms
1 {\tilde{A}}_4 [3[5]] [(3,3,3,3,3)] CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png 7
2 {\tilde{C}}_4 [4,3,3,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 19
3 {\tilde{B}}_4 [4,3,31,1] [4,3,3,4,1+] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 23 (8 new)
4 {\tilde{D}}_4 [31,1,1,1] [1+,4,3,3,4,1+] CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png 9 (0 new)
5 {\tilde{F}}_4 [3,4,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 31 (21 new)

There are three regular honeycombs of Euclidean 4-space:

Other families that generate uniform honeycombs:

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

Prismatic groups
# Coxeter group Coxeter diagram
1 {\tilde{C}}_3×{\tilde{I}}_1 [4,3,4,2,∞] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
2 {\tilde{B}}_3×{\tilde{I}}_1 [4,31,1,2,∞] CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel 4a.pngCDel nodea.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
3 {\tilde{A}}_3×{\tilde{I}}_1 [3[4],2,∞] CDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
4 {\tilde{C}}_2×{\tilde{I}}_1x{\tilde{I}}_1 [4,4,2,∞,2,∞] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
5 {\tilde{H}}_2×{\tilde{I}}_1x{\tilde{I}}_1 [6,3,2,∞,2,∞] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
6 {\tilde{A}}_2×{\tilde{I}}_1x{\tilde{I}}_1 [3[3],2,∞,2,∞] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
7 {\tilde{I}}_1×{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1 [∞,2,∞,2,∞,2,∞] CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
8 {\tilde{A}}_2x{\tilde{A}}_2 [3[3],2,3[3]] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel split1.pngCDel branch.png
9 {\tilde{A}}_2×{\tilde{B}}_2 [3[3],2,4,4] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
10 {\tilde{A}}_2×{\tilde{G}}_2 [3[3],2,6,3] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
11 {\tilde{B}}_2×{\tilde{B}}_2 [4,4,2,4,4] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
12 {\tilde{B}}_2×{\tilde{G}}_2 [4,4,2,6,3] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
13 {\tilde{G}}_2×{\tilde{G}}_2 [6,3,2,6,3] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

Regular tessellations of hyperbolic 4-space[edit]

There are five kinds of convex regular honeycombs and four kinds of star-honeycombs in H4 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
Order-5 5-cell {3,3,3,5} CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png {3,3,3} {3,3} {3} {5} {3,5} {3,3,5} {5,3,3,3}
Order-3 120-cell {5,3,3,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png {5,3,3} {5,3} {5} {3} {3,3} {3,3,3} {3,3,3,5}
Order-5 tesseractic {4,3,3,5} CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png {4,3,3} {4,3} {4} {5} {3,5} {3,3,5} {5,3,3,4}
Order-4 120-cell {5,3,3,4} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {5,3,3} {5,3} {5} {4} {3,4} {3,3,4} {4,3,3,5}
Order-5 120-cell {5,3,3,5} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png {5,3,3} {5,3} {5} {5} {3,5} {3,3,5} Self-dual

There are four regular star-honeycombs in H4 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
Order-3 small stellated 120-cell {5/2,5,3,3} CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png {5/2,5,3} {5/2,5} {5} {5} {3,3} {5,3,3} {3,3,5,5/2}
Order-5/2 600-cell {3,3,5,5/2} CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png {3,3,5} {3,3} {3} {5/2} {5,5/2} {3,5,5/2} {5/2,5,3,3}
Order-5 icosahedral 120-cell {3,5,5/2,5} CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.png {3,5,5/2} {3,5} {3} {5} {5/2,5} {5,5/2,5} {5,5/2,5,3}
Order-3 great 120-cell {5,5/2,5,3} CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node 1.png {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 4-space as permutations of rings of the Coxeter diagrams. There are also 9 noncompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Noncompact groups generate honeycombs with infinite facets or vertex figures.

Compact hyperbolic groups

{\widehat{AF}}_4 = [(3,3,3,3,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

{\bar{DH}}_4 = [5,3,31,1]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

{\bar{H}}_4 = [3,3,3,5]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png

{\bar{BH}}_4 = [4,3,3,5]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{K}}_4 = [5,3,3,5]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png

Paracompact hyperbolic groups

{\bar{P}}_4 = [3,3[4]]: CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{BP}}_4 = [4,3[4]]: CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{FR}}_4 = [(3,3,4,3,4)]: CDel branch.pngCdel 4-4.pngCDel nodes.pngCDel split2.pngCDel node.png
{\bar{DP}}_4 = [3[3]×[]]: CDel node.pngCDel split1.pngCDel branchbranch.pngCDel split2.pngCDel node.png

{\bar{N}}_4 = [4,/3\,3,4]: CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{O}}_4 = [3,4,31,1]: CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{S}}_4 = [4,32,1]: CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{M}}_4 = [4,31,1,1]: CDel nodes.pngCDel split2-43.pngCDel node.pngCDel split1.pngCDel nodes.png

{\bar{R}}_4 = [3,4,3,4]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png

Notes[edit]

  1. ^ A proposed name polyteron (plural: polytera) has been advocated, from the Greek root poly- meaning "many", a shortened tetra- meaning "four", and suffix -on. "Four" refers to the dimension of the 5-polytope facets.
  2. ^ http://www.steelpillow.com/polyhedra/ditela.html
  3. ^ T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  4. ^ Uniform Polytera and Other Five Dimensional Shapes
  5. ^ George Olshevsky (2006), Uniform Panoploid Tetracombs, manuscript. Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs.
  6. ^ 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 Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 (3 regular and one semiregular 4-polytope)
  • 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, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (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)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • Richard Klitzing, 5D, uniform polytopes (polytera)
  • 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]