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Graphs of regular and uniform polytopes.
5-simplex graph.png
5-simplex (hexateron)
Truncated 5-simplex.png
Truncated 5-simplex
Rectified 5-simplex.png
Rectified 5-simplex
Birectified 5-simplex.png
Birectified 5-simplex
Cross graph 5.png
5-orthoplex, 211
(Pentacross)
Truncated 5-orthoplex.png
Truncated 5-orthoplex
Rectified pentacross.png
Rectified 5-orthoplex
5-cube.svg
5-cube
(Penteract)
Truncated 5-cube.png
Truncated 5-cube
Rectified 5-cube.png
Rectified 5-cube
Demipenteract graph ortho.svg
5-demicube. 121
(Demipenteract)
Truncated 5-demicube.png
Truncated 5-demicube
Rectified 5-demicube.png
Rectified 5-demicube

In geometry, a uniform polyteron (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-Dynkin diagrams.

Contents

Regular 5-polytopes

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 convex and zero nonconvex regular polytopes.

There are exactly three such convex regular polytopes:

  1. {3,3,3,3} - Hexateron (5-simplex)
  2. {4,3,3,3} - Penteract (5-hypercube)
  3. {3,3,3,4} - Pentacross (5-orthoplex)

There are no nonconvex regular 5-polytopes.

Convex uniform 5-polytopes

There are 105 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.

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Reflection families

The hexateron is the regular form in the A5 family. The penteract and pentacross are the regular forms in the B5 family. The bifurcating graph of the D6 family contains the pentacross, as well as a demipenteract which is an alternated penteract.

Fundamental families

# Coxeter group Coxeter-Dynkin diagram
1 A5 [34] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
2 B5 [4,33] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
3 D5 [32,1,1] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png

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

# Coxeter groups Coxeter graph
1 A4 × A1 [3,3,3] × [ ] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
2 B4 × A1 [4,3,3] × [ ] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
3 F4 × A1 [3,4,3] × [ ] CDW dot.pngCDW 3.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
4 H4 × A1 [5,3,3] × [ ] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.png
5 D4 × A1 [31,1,1] × [ ] CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 2.pngCD dot.png

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

Coxeter groups Coxeter graph
I2(p) × I2(q) × A1 [p] × [q] × [ ] CDW dot.pngCDW p.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW q.pngCDW dot.pngCDW 2.pngCDW dot.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 graph
1 A3 × I2(p) [3,3] × [p] CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.png
2 B3 × I2(p) [4,3] × [p] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.png
3. H3 × I2(p) [5,3] × [p] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW dot.pngCDW p.pngCDW dot.png

Enumerating the convex uniform 5-polytopes

  1. Simplex family: A5 [34] - CDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
    • 19 uniform 5-polytopes as permutations of rings in the group diagram, including one regular:
      1. {34} - 5-simplex, hexateron, or hexa-5-tope, CDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
  2. Hypercube/orthoplex family: B5 [4,33] - CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
    • 31 uniform 5-polytopes as permutations of rings in the group diagram, including two regular forms:
      1. {4,33} — 5-cube, penteract, or deca-5-tope. CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
      2. {33,4} — 5-orthoplex, pentacross, or triacontadi-5-tope. CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.png
  3. Demihypercube D5/E5 family: [32,1,1] - CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png
    • 23 uniform 5-polytopes (8 unique) as permutations of rings in the group diagram, including:
      1. 121 demipenteract - CD ring.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD dot.png; also as h{4,33}, CDW hole.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.png
      2. 211 pentacross - CD dot.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.pngCD 3.pngCD ring.png
  4. Prisms and duoprisms:
    1. 56 uniform 5-polytope (46 unique) constructions based on prismatic families: [3,3,3]x[ ], [4,3,3]x[ ], [5,3,3]x[ ], [31,1,1]x[ ].
    2. 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:

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

The A5 [3,3,3,3] family (5-simplex)

There are 19 forms based on all permutations of the Coxeter-Dynkin 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).

# Coxeter-Dynkin diagram
Schläfli symbol
Name
Graph Vertex
figure
Facet counts by location: [3,3,3,3] Element counts
4 3 2 1 0
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.png
[3,3,3]
(6)
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.png
[3,3]×[ ]
(15)
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.png
[3]×[3]
(20)
CDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
[ ]×[3,3]
(15)
CDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
[3,3,3]
(6)
4-faces Cells Faces Edges Vertices
1 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0{3,3,3,3}
Hexateron (hix)
5-simplex graph.png 5-simplex verf.png
{3,3,3}
(5)
Schlegel wireframe 5-cell.png
{3,3,3}
- - - - 6 15 20 15 6
2 CDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t1{3,3,3,3}
Rectified hexateron (rix)
Rectified 5-simplex.png Rectified 5-simplex verf.png
{3,3}x{ }
(4)
Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
- - - (2)
Schlegel wireframe 5-cell.png
{3,3,3}
12 45 80 60 15
3 CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t2{3,3,3,3}
Birectified hexateron (dot)
Birectified 5-simplex.png Birectified hexateron verf.png
{3}x{3}
(3)
Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
- - - (3)
Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
12 60 120 90 20
4 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0,1{3,3,3,3}
Truncated hexateron (tix)
Truncated 5-simplex.png Truncated 5-simplex verf.png
Tetrah.pyr
(4)
Schlegel half-solid truncated pentachoron.png
t0,1{3,3,3}
- - - (1)
Schlegel wireframe 5-cell.png
{3,3,3}
12 45 80 75 30
5 CDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t1,2{3,3,3,3}
Bitruncated hexateron (bittix)
Bitruncated 5-simplex verf.png (3)
Schlegel half-solid bitruncated 5-cell.png
t1,2{3,3,3}
- - - (2)
Schlegel half-solid truncated pentachoron.png
t0,1{3,3,3}
12 60 140 150 60
6 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0,2{3,3,3,3}
Cantellated hexateron (sarx)
Cantellated Hexateron HP.svg Cantellated hexateron verf.png
prism-wedge
(3)
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
- - (1)
Tetrahedral prism.png
{ }×{3,3}
(1)
Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
27 135 290 240 60
7 CDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t1,3{3,3,3,3}
Bicantellated hexateron (sibrid)
Bicantellated 5-simplex verf.png (2)
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
- (8)
3-3 duoprism.png
{3}×{3}
- (2)
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
32 180 420 360 90
8 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t0,3{3,3,3,3}
Runcinated hexateron (spix)
Runcinated 5-simplex verf.png (2)
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
- (3)
3-3 duoprism.png
{3}×{3}
(3)
Octahedral prism.png
{ }×t1{3,3}
(1)
Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
47 255 420 270 60
9 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0,4{3,3,3,3}
Stericated hexateron (scad)
Stericated hexateron ortho.svg Stericated hexateron verf.png
Irr.16-cell
(1)
Schlegel wireframe 5-cell.png
{3,3,3}
(4)
Tetrahedral prism.png
{ }×{3,3}
(6)
3-3 duoprism.png
{3}×{3}
(4)
Tetrahedral prism.png
{ }×{3,3}
(1)
Schlegel wireframe 5-cell.png
{3,3,3}
62 180 210 120 30
10 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0,1,2{3,3,3,3}
Cantitruncated hexateron (garx)
Canitruncated 5-simplex verf.png Schlegel half-solid cantitruncated 5-cell.png
t0,1,2{3,3,3}
- - Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid truncated pentachoron.png
t0,1{3,3,3}
27 135 290 300 120
11 CDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t1,2,3{3,3,3,3}
Bicantitruncated hexateron (gibrid)
Bicanitruncated 5-simplex verf.png Schlegel half-solid cantitruncated 5-cell.png
t0,1,2{3,3,3}
- 3-3 duoprism.png
{3}×{3}
- Schlegel half-solid cantitruncated 5-cell.png
t0,1,2{3,3,3}
32 180 420 450 180
12 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t0,1,3{3,3,3,3}
Runcitruncated hexateron (pattix)
Runcitruncated 5-simplex verf.png Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
- 3-6 duoprism.png
{6}×{3}
Octahedral prism.png
{ }×t1{3,3}
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
47 315 720 630 180
13 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t0,2,3{3,3,3,3}
Runcicantellated hexateron (pirx)
Runcicantellated 5-simplex verf.png Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
- 3-3 duoprism.png
{3}×{3}
Truncated tetrahedral prism.png
{ }×t0,1{3,3}
Schlegel half-solid bitruncated 5-cell.png
t1,2{3,3,3}
47 255 570 540 180
14 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0,1,4{3,3,3,3}
Steritruncated hexateron (cappix)
Steritruncated 5-simplex verf.png Schlegel half-solid truncated pentachoron.png
t0,1{3,3,3}
Truncated tetrahedral prism.png
{ }×t0,1{3,3}
3-6 duoprism.png
{3}×{6}
Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
62 330 570 420 120
15 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0,2,4{3,3,3,3}
Stericantellated hexateron (card)
Stericantellated 5-simplex verf.png Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
Cuboctahedral prism.png
{ }×t0,2{3,3}
3-3 duoprism.png
{3}×{3}
Cuboctahedral prism.png
{ }×t0,2{3,3}
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
62 420 900 720 180
16 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t0,1,2,3{3,3,3,3}
Runcicantitruncated hexateron (gippix)
Runcicantitruncated 5-simplex verf.png
Irr.5-cell
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
- 3-6 duoprism.png
{3}×{6}
Truncated tetrahedral prism.png
{ }×t0,1{3,3}
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
47 315 810 900 360
17 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0,1,2,4{3,3,3,3}
Stericantitruncated hexateron (cograx)
Stericanitruncated 5-simplex verf.png Schlegel half-solid cantitruncated 5-cell.png
t0,1,2{3,3,3}
Truncated octahedral prism.png
{ }×t0,1,2{3,3}
3-6 duoprism.png
{3}×{6}
Cuboctahedral prism.png
{ }×t0,2{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
62 480 1140 1080 360
18 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.png
t0,1,3,4{3,3,3,3}
Steriruncitruncated hexateron (captid)
Steriruncitruncated 5-simplex verf.png Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
Truncated tetrahedral prism.png
{ }×t0,1{3,3}
6-6 duoprism.png
{6}×{6}
Truncated tetrahedral prism.png
{ }×t0,1,3{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
62 450 1110 1080 360
19 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.png
t0,1,2,3,4{3,3,3,3}
Omnitruncated hexateron (gocad)
Omnitruncated 5-simplex verf.png
Irr. {3,3,3}
(1)
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
(1)
Truncated octahedral prism.png
{ }×t0,1,2{3,3}
(1)
6-6 duoprism.png
{6}×{6}
(1)
Truncated octahedral prism.png
{ }×t0,1,2{3,3}
(1)
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
62 540 1560 1800 720

The B5 [4,3,3,3] family (penteract/pentacross)

This family has 31 Wythoffian uniform polyhedra, from 25-1 permutations of the Coxeter-Dynkin diagram with one or more rings.

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

The B5 family has symmetry of order 3840 (2^5*5!).

The penteract subfamily

There are 20 forms here, 7 shared with the pentacross family. Four are shared with the demipenteract family.

# Coxeter-Dynkin
andSchläfli
symbols
Name
Graph Vertex
figure
Facet counts by location: [4,3,3,3] Element counts
4 3 2 1 0
CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.png
[4,3,3]
(10)
CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.png
[4,3]×[ ]
(40)
CDW dot.pngCDW 4.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.png
[4]×[3]
(80)
CDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
[ ]×[3,3]
(80)
CDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
[3,3,3]
(32)
Facets Cells Faces Edges Vertices
20 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0{4,3,3,3}
Penteract
Pent
5-cube.svg 5-cube verf.png
{3,3,3}
Hypercube.svg
{4,3,3}
- - - - 10 40 80 80 32
21 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t1{4,3,3,3}
Rectified penteract
Rin
Rectified 5-cube.png Rectified 5-cube verf.png
{3,3}x{ }
Schlegel half-solid rectified 8-cell.png
t1{4,3,3}
- - - Schlegel wireframe 5-cell.png
{3,3,3}
42 200 400 320 80
22 CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t2{4,3,3,3}
Birectified penteract
Nit
Rectified 5-demicube.png Birectified penteract verf.png
{4}×{3}
Schlegel half-solid rectified 8-cell.png
t1{4,3,3}
- - - Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
42 280 640 480 80
23 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0,1{4,3,3,3}
Truncated penteract
Tan
Truncated 5-cube.png Truncated 5-cube verf.png
Tetrah.pyr
Schlegel half-solid truncated tesseract.png
t0,1{4,3,3}
- - - Schlegel wireframe 5-cell.png
{3,3,3}
42 200 400 400 160
24 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t1,2{4,3,3,3}
Bitruncated penteract
Bittin
Bitruncated penteract verf.png Schlegel half-solid bitruncated 8-cell.png
t1,2{4,3,3}
- - - Schlegel half-solid truncated pentachoron.png
t0,1{3,3,3}
42 280 720 800 320
25 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0,2{4,3,3,3}
Cantellated penteract
Sirn
Cantellated 5-cube vertf.png
Prism-wedge
Schlegel half-solid cantellated 8-cell.png
t0,2{4,3,3}
- - Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
122 680 1520 1280 320
26 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t1,3{4,3,3,3}
Bicantellated penteract
Sibrant
Bicantellated penteract verf.png Schlegel half-solid cantellated 8-cell.png
t0,2{4,3,3}
- 3-4 duoprism.png
{4}×{3}
- Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
122 840 2160 1920 480
27 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t0,3{4,3,3,3}
Runcinated penteract
Span
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
{ }×t1{3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
202 1240 2160 1440 320
28 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0,4{4,3,3,3}
Stericated penteract
Scant
Stericated penteract verf.png
Tetr.antiprm
Hypercube.svg
{4,3,3}
Hypercube.svg
{4,3}×{ }
3-4 duoprism.png
{4}×{3}
Tetrahedral prism.png
{ }×{3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
242 800 1040 640 160
29 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0,1,2{4,3,3,3}
Cantitruncated penteract
Girn
Schlegel half-solid cantitruncated 8-cell.png
t0,1,2{4,3,3}
- - Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid truncated pentachoron.png
t0,1{3,3,3}
122 680 1520 1600 640
30 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t1,2,3{4,3,3,3}
Bicantitruncated penteract
Gibrant
Schlegel half-solid cantitruncated 8-cell.png
t0,1,2{4,3,3}
- 3-4 duoprism.png
{4}×{3}
- Schlegel half-solid cantitruncated 5-cell.png
t0,1,2{3,3,3}
122 840 2160 2400 960
31 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t0,1,3{4,3,3,3}
Runcitruncated penteract
Pattin
Schlegel half-solid runcitruncated 8-cell.png
t0,1,3{4,3,3}
- {8}×{3} Octahedral prism.png
{ }×t1{3,3}
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
202 1560 3760 3360 960
32 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t0,2,3{4,3,3,3}
Runcicantellated penteract
Prin
Schlegel half-solid runcitruncated 8-cell.png
t0,1,3{4,3,3}
- 3-4 duoprism.png
{4}×{3}
Truncated tetrahedral prism.png
{ }×t0,1{3,3}
Schlegel half-solid bitruncated 5-cell.png
t1,2{3,3,3}
202 1240 2960 2880 960
33 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0,1,4{4,3,3,3}
Steritruncated penteract
Capt
Schlegel half-solid truncated tesseract.png
t0,1{4,3,3}
Truncated cubic prism.png
t0,1{4,3}×{ }
{8}×{3} Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
242 1600 2960 2240 640
34 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0,2,4{4,3,3,3}
Stericantellated penteract
Carnit
Schlegel half-solid cantellated 8-cell.png
t0,2{4,3,3}
Rhombicuboctahedral prism.png
t0,2{4,3}×{ }
3-4 duoprism.png
{4}×{3}
Cuboctahedral prism.png
{ }×t0,2{3,3}
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
242 2080 4720 3840 960
35 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t0,1,2,3{4,3,3,3}
Runcicantitruncated penteract
Gippin
Schlegel half-solid omnitruncated 8-cell.png
t0,1,2,3{4,3,3}
- {8}×{3} Truncated tetrahedral prism.png
{ }×t0,1{3,3}
Schlegel half-solid cantitruncated 5-cell.png
t0,1,2{3,3,3}
202 1560 4240 4800 1920
36 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0,1,2,4{4,3,3,3}
Stericantitruncated penteract
Cogrin
Schlegel half-solid cantitruncated 8-cell.png
t0,1,2{4,3,3}
Truncated cuboctahedral prism.png
t0,1,2{4,3}×{ }
{8}×{3} Cuboctahedral prism.png
{ }×t0,2{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
242 2400 6000 5760 1920
37 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.png
t0,1,3,4{4,3,3,3}
Steriruncitruncated penteract
Captint
Schlegel half-solid runcitruncated 8-cell.png
t0,1,3{4,3,3}
Truncated cubic prism.png
t0,1{4,3}×{ }
{8}×{6} Truncated tetrahedral prism.png
{ }×t0,1{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
242 2160 5760 5760 1920
38 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.png
t0,1,2,3,4{4,3,3,3}
Omnitruncated penteract
Gacnet
Omnitruncated 5-cube.svg Omnitruncated 5-cube verf.png
Irr. {3,3,3}
Schlegel half-solid omnitruncated 8-cell.png
t0,1,2{4,3}×{ }
Truncated cuboctahedral prism.png
t0,1,2{4,3}×{ }
{8}×{6} Truncated octahedral prism.png
{ }×t0,1,2{3,3}
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
242 2640 8160 9600 3840
[51] CDW hole.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
h0{4,3,3,3}
Demipenteract
Hin
Demipenteract graph ortho.svg 5-demicube verf.png
t1{3,3,3}
Schlegel wireframe 5-cell.png
(16) {3,3,3}
- - - Schlegel wireframe 16-cell.png
{3,3,4}
26 120 160 80 16

Pentacross subfamily

There are 19 forms, 12 new ones. 7 are shared from the penteract family, and 10 shared with the demipenteract family.

# Coxeter-Dynkin
andSchläfli
symbols
Name
Graph Vertex
figure
Facet counts by location: [4,3,3,3] Element counts
4 3 2 1 0
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.png
[3,3,3]
(32)
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.png
[3,3]×[ ]
(80)
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 4.pngCDW dot.png
[3]×[4]
(80)
CDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
[ ]×[3,4]
(40)
CDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
[3,3,4]
(10)
Facets Cells Faces Edges Vertices
39 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t0{3,3,3,4}
Pentacross
Tac
Cross graph 5.png Pentacross verf.png
{3,3,4}
Schlegel wireframe 5-cell.png
{3,3,3}
- - - - 32 80 80 40 10
40 CDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t1{3,3,3,4}
Rectified pentacross
Rat
Rectified pentacross.png Rectified pentacross verf.png
{ }×{3,4}
Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
- - - Schlegel wireframe 16-cell.png

{3,3,4}
42 240 400 240 40
[22] CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t2{3,3,3,4}
Birectified pentacross
Nit
Rectified 5-demicube.png Birectified penteract verf.png
{4}×{3}
Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
- - - Schlegel half-solid rectified 16-cell.png
t1{3,3,4}
42 280 640 480 80
41 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t0,1{3,3,3,4}
Truncated pentacross
Tot
Truncated 5-orthoplex.png Truncated pentacross.png
(Octah.pyr)
Schlegel half-solid truncated pentachoron.png
t0,1{3,3,3}
- - - {3,3,3} 42 240 400 280 80
42 CDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t1,2{3,3,3,4}
Bitruncated pentacross
Bittit
Bitruncated pentacross verf.png Schlegel half-solid bitruncated 5-cell.png
t1,2{3,3,3}
- - - t0,1{3,3,4} 42 280 720 720 240
43 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t0,2{3,3,3,4}
Cantellated pentacross
Sart
Cantellated pentacross verf.png
Prism-wedge
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
- - { }×{3,4} t1{3,3,4} 82 640 1520 1200 240
[26] CDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW dot.png
t1,3{3,3,3,4}
Bicantellated pentacross
Sibrant
Bicantellated penteract verf.png Schlegel half-solid cantellated 5-cell.png
t1,3{3,3,3}
- {3}×{4} - t0,2{3,3,4} 122 840 2160 1920 480
44 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW dot.png
t0,3{3,3,3,4}
Runcinated pentacross
Spat
Runcinated pentacross verf.png Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
- {3}×{4} t1{4,3,3} 162 1200 2160 1440 320
[28] CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW ring.png
t0,4{3,3,3,4}
Stericated pentacross
Scant
Stericated penteract verf.png
Tetr.antiprm
Schlegel wireframe 5-cell.png
{3,3,3}
- - - {4,3,3} 242 800 1040 640 160
45 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t0,1,2{3,3,3,4}
Cantitruncated pentacross
Gart
Schlegel half-solid cantitruncated 5-cell.png
t0,1,2{3,3,3}
- - - t0,1{3,3,4} 82 640 1520 1440 480
[30] CDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW dot.png
t1,2,3{3,3,3,4}
Bicantitruncated pentacross
Gibrant
Schlegel half-solid cantitruncated 5-cell.png
t1,2,3{3,3,3}
- {3}×{4} - t0,1,2{3,3,4} 122 840 2160 2400 960
46 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW dot.png
t0,1,3{3,3,3,4}
Runcitruncated pentacross
Pattit
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
- {6}×{4} { }×t1{3,4} t0,2{3,3,4} 162 1440 3680 3360 960
47 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW dot.png
t0,2,3{3,3,3,4}
Runcicantellated pentacross
Pirt
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
- {3}×{4} { }×t0,1{3,4} t1,2{3,3,4} 162 1200 2960 2880 960
48 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW ring.png
t0,1,4{3,3,3,4}
Steritruncated pentacross
Cappin
Schlegel half-solid truncated pentachoron.png
t0,1{3,3,3}
- - { }×{4,3} t0,3{3,3,4} 242 1520 2880 2240 640
[34] CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW ring.png
t0,2,4{3,3,3,4}
Stericantellated pentacross
Carnit
Schlegel half-solid cantellated 5-cell.png
t0,2{3,3,3}
{ }×t0,2{3,3} {3}×{4} { }×t0,2{3,4} t0,2{4,3,3} 242 2080 4720 3840 960
49 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW dot.png
t0,1,2,3{3,3,3,4}
Runcicantitruncated pentacross
Gippit
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
- {6}×{4} { }×t0,1{3,4} t0,1,2{3,3,4} 162 1440 4160 4800 1920
50 CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW ring.png
t0,1,2,4{3,3,3,4}
Stericantitruncated pentacross
Cogart
Schlegel half-solid cantitruncated 5-cell.png
t0,1,2{3,3,3}
{ }×t0,1,{3,3} {6}×{4} { }×t0,2{3,4} t0,1,3{3,3,4} 242 2320 5920 5760 1920
[37] CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW ring.png
t0,1,3,4{3,3,3,4}
Steriruncitruncated pentacross
Captint
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
{ }×t0,1{3,3} {6}×{8} { }×t0,1{4,3} t0,1,3{4,3,3} 242 2160 5760 5760 1920
[38] CDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW ring.png
t0,1,2,3,4{3,3,3,4}
Omnitruncated pentacross
Gacnet
Omnitruncated 5-cube.svg Omnitruncated 5-cube verf.png
Irr. {3,3,3}
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
{ }×t0,1,2{3,3} {6}×{8} { }×t0,1,2{3,4} t0,1,2,3{3,3,4} 242 2640 8160 9600 3840

The D5 [31,2,1] family (demipenteract)

There are 23 forms. 16 are repeated from the [4,3,3,3] family and 7 are new ones.

The D5 family has symmetry of order 1920 (2^4*5!).

# Coxeter-Dynkin
and Schläfli symbols
Name
Graph Vertex
figure
Facets by location: CD B5 nodes.png [31,2,1] Element counts
4 3 2 1 0
CD dot.pngCD 3b.pngCD downbranch-0dash.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png
[3,3,3]
(16)
CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 2c.pngCD dash.png
[31,1,1]
(10)
CD dot.pngCD 3b.pngCD downbranch-00.pngCD 2c.pngCD dash.pngCD 2c.pngCD dot.png
[3,3]×[ ]
(40)
CD dot.pngCD 2c.pngCD downbranch-dash0.pngCD 2c.pngCD dot.pngCD 3b.pngCD dot.png
[ ]×[3]×[ ]
(80)
CD dash.pngCD 2c.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png
[3,3,3]
(16)
Facets Cells Faces Edges Vertices
51 CD ring.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png (121)
Demipenteract
Hin
Demipenteract graph ortho.svg Demipenteract verf.png
t1{3,3,3}
{3,3,3} t0(111) - - - 26 120 160 80 16
[22] CD dot.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png t1(121)
Rectified demipenteract
(Same as birectified penteract)
Nit
Rectified 5-demicube.png Birectified penteract verf.png
{ }×{ }×{3}
t1{3,3,3} t1(111) - - t1{3,3,3} 42 280 640 480 80
[40] CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD ring.pngCD 3b.pngCD dot.png t2(121)
Birectified demipenteract
(Same as rectified pentacross)
Rat
Rectified pentacross.png Rectified pentacross verf.png
{ }×t1{3,3}
t1{3,3,3} t0(111) - - t1{3,3,3} 42 240 400 240 40
[39] CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png t3(121)
Trirectified demipenteract
(Same as pentacross)
Tac
Cross graph 5.png Pentacross verf.png
(111)
{3,3,3} - - - {3,3,3} 32 80 80 40 10
52 CD ring.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png t0,1(121)
Truncated demipenteract
Thin
Truncated 5-demicube.png 42 280 640 560 160
53 CD ring.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD ring.pngCD 3b.pngCD dot.png t0,2(121)
Cantellated demipenteract
Sirhin
42 360 880 720 160
54 CD ring.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png t0,3(121)
Runcinated demipenteract
Siphin
82 480 720 400 80
[21] CD ring.pngCD 3b.pngCD downbranch-01.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png t0,4(121)
Stericated demipenteract
(Same as rectified penteract)
Rin
Rectified 5-cube.png Rectified 5-cube verf.png
{3,3}x{ }
42 200 400 320 80
[42] CD dot.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD ring.pngCD 3b.pngCD dot.png t1,2(121)
Bitruncated demipenteract
(Same as bitruncated pentacross)
Bittit
Bitruncated pentacross verf.png 42 280 720 720 240
[43] CD dot.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png t1,3(121)
Bicantellated demipenteract
(Same as cantellated pentacross)
Sart
Cantellated pentacross verf.png
Prism-wedge
82 640 1520 1200 240
[41] CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD ring.pngCD 3b.pngCD ring.png t2,3(121)
Tritruncated demipenteract
(Same as truncated pentacross)
Tot
Truncated 5-orthoplex.png Truncated pentacross.png
(Octah.pyr)
42 240 400 280 80
[24] CD ring.pngCD 3b.pngCD downbranch-11.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png t0,1,4(121)
Steritruncated demipenteract
(Same as bitruncated penteract)
Bittin
Bitruncated penteract verf.png 42 280 720 800 320
55 CD ring.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD ring.pngCD 3b.pngCD dot.png t0,1,2(121)
Cantitruncated demipenteract
Girhin
42 360 1040 1200 480
56 CD ring.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png t0,1,3(121)
Runcitruncated demipenteract
Pithin
82 720 1840 1680 480
[26] CD ring.pngCD 3b.pngCD downbranch-01.pngCD 3b.pngCD ring.pngCD 3b.pngCD dot.png t0,2,4(121)
Stericantellated demipenteract
(Same as bicantellated penteract)
Sibrant
Bicantellated penteract verf.png 122 840 2160 1920 480
[44] CD ring.pngCD 3b.pngCD downbranch-01.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png t0,3,4(121)
Steriruncinated demipenteract
(Same as runcinated pentacross)
Spat
Runcinated pentacross verf.png 162 1200 2160 1440 320
57 CD ring.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD ring.pngCD 3b.pngCD ring.png t0,2,3(121)
Runcicantellated demipenteract
Pirhin
82 560 1280 1120 320
[45] CD dot.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD ring.pngCD 3b.pngCD ring.png t1,2,3(121)
Bicantitruncated demipenteract
(Same as cantitruncated pentacross)
Gart
82 640 1520 1440 480
[30] CD ring.pngCD 3b.pngCD downbranch-11.pngCD 3b.pngCD ring.pngCD 3b.pngCD dot.png t0,1,2,4(121)
Stericantitruncated demipenteract
(Same as bicantitruncated pentacross)
Gibrant
122 840 2160 2400 960
[46] CD ring.pngCD 3b.pngCD downbranch-11.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png t0,1,3,4(121)
Steriruncitruncated demipenteract
(Same as runcicantellated pentacross)
Pirt
162 1440 3680 3360 960
58 CD ring.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD ring.pngCD 3b.pngCD ring.png t0,1,2,3(121)
Runcicantitruncated demipenteract
Giphin
82 720 2080 2400 960
[47] CD ring.pngCD 3b.pngCD downbranch-01.pngCD 3b.pngCD ring.pngCD 3b.pngCD ring.png t0,2,3,4(121)
Steriruncicantellated demipenteract
(Same as runcitruncated pentacross)
Pattit
162 1200 2960 2880 960
[49] CD ring.pngCD 3b.pngCD downbranch-11.pngCD 3b.pngCD ring.pngCD 3b.pngCD ring.png t0,1,2,3,4(121)
Omnitruncated demipenteract
(Same as runcicantitruncated pentacross)
Gippit
Omnitruncated 5-demicube verf.png
Irr. {3,3,3}
162 1440 4160 4800 1920

Uniform prismatic forms

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

[3,3,3] × [ ]

This prismatic family has 9 forms:

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

# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
59 CDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
{3,3,3}x{ }
5-cell prism
7 20 30 25 10
60 CDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1{3,3,3}x{ }
Rectified 5-cell prism
12 50 90 70 20
61 CDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,1{3,3,3}x{ }
Truncated 5-cell prism
12 50 100 100 40
62 CDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,2{3,3,3}x{ }
Cantellated 5-cell prism
22 120 250 210 60
63 CDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,3{3,3,3}x{ }
Runcinated 5-cell prism
32 130 200 140 40
64 CDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1,2{3,3,3}x{ }
Bitruncated 5-cell prism
12 60 140 150 60
65 CDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,1,2{3,3,3}x{ }
Cantitruncated 5-cell prism
22 120 280 300 120
66 CDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,3{3,3,3}x{ }
Runcitruncated 5-cell prism
32 180 390 360 120
67 CDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,2,3{3,3,3}x{ }
Omnitruncated 5-cell prism
32 210 540 600 240

[4,3,3] × [ ]

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!).

Tesseractic prism family {4,3,3}x{ }
# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
68 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
{4,3,3}x{ }
Tesseractic prism
10 40 80 80 32
69 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1{4,3,3}x{ }
Rectified tesseractic prism
26 136 272 224 64
70 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,1{4,3,3}x{ }
Truncated tesseractic prism
26 136 304 320 128
71 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,2{4,3,3}x{ }
Cantellated tesseractic prism
58 360 784 672 192
72 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,3{4,3,3}x{ }
Runcinated tesseractic prism
82 368 608 448 128
73 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1,2{4,3,3}x{ }
Bitruncated tesseractic prism
26 168 432 480 192
74 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,1,2{4,3,3}x{ }
Cantitruncated tesseractic prism
58 360 880 960 384
75 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,3{4,3,3}x{ }
Runcitruncated tesseractic prism
82 528 1216 1152 384
76 CDW ring.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,2,3{4,3,3}x{ }
Omnitruncated tesseractic prism
82 624 1696 1920 768
16-cell prism family [3,3,4]x[ ]
# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
77 CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
{3,3,4}x{ }
16-cell prism
18 64 88 56 16
78 CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1{3,3,4}x{ }
Rectified 16-cell prism
(Same as 24-cell prism)
26 144 288 216 48
79 CDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1{3,3,4}x{ }
Truncated 16-cell prism
26 144 312 288 96
80 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,2{3,3,4}x{ }
Cantellated 16-cell prism
(Same as rectified 24-cell prism)
50 336 768 672 192
[72] CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,3{4,3,3}x{ }
Runcinated 16-cell prism
(Same as Runcinated tesseractic prism)
82 368 608 448 128
[73] CDW dot.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1,2{4,3,3}x{ }
Bitruncated 16-cell prism
(Same as bitruncated tesseractic prism)
26 168 432 480 192
81 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,2{3,3,4}x{ }
Cantitruncated 16-cell prism
(Same as truncated 24-cell prism)
50 336 864 960 384
82 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,3{3,3,4}x{ }
Runcitruncated 16-cell prism
82 528 1216 1152 384
[76] CDW ring.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,2,3{3,3,4}x{ }
Omnitruncated 16-cell prism
(Same as omnitruncated tesseractic prism)
82 624 1696 1920 768
83 CDW dot.pngCDW 4.pngCDW hole.pngCDW 3.pngCDW hole.pngCDW 3.pngCDW hole.pngCDW 2.pngCDW ring.png
h0,1,2{3,3,4}x{ }
Snub 24-cell prism
146 768 1392 960 192

[3,4,3] × [ ]

This prismatic family has 10 forms.

The A1 x F4 family has symmetry of order 2304 (2*1152).

# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
[79] CDW ring.pngCDW 3.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
{3,4,3}x{ }
24-cell prism
26 144 288 216 48
[80] CDW dot.pngCDW 3.pngCDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1{3,4,3}x{ }
rectified 24-cell prism
50 336 768 672 192
[81] CDW ring.pngCDW 3.pngCDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,1{3,4,3}x{ }
truncated 24-cell prism
50 336 864 960 384
84 CDW ring.pngCDW 3.pngCDW dot.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,2{3,4,3}x{ }
cantellated 24-cell prism
146 1008 2304 2016 576
85 CDW ring.pngCDW 3.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,3{3,4,3}x{ }
runcinated 24-cell prism
242 1152 1920 1296 288
86 CDW dot.pngCDW 3.pngCDW ring.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1,2{3,4,3}x{ }
bitruncated 24-cell prism
50 432 1248 1440 576
87 CDW ring.pngCDW 3.pngCDW ring.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,1,2{3,4,3}x{ }
cantitruncated 24-cell prism
146 1008 2592 2880 1152
88 CDW ring.pngCDW 3.pngCDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,3{3,4,3}x{ }
runcitruncated 24-cell prism
242 1584 3648 3456 1152
89 CDW ring.pngCDW 3.pngCDW ring.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,2,3{3,4,3}x{ }
omnitruncated 24-cell prism
242 1872 5088 5760 2304
[83] CDW hole.pngCDW 3.pngCDW hole.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
h0,1{3,4,3}x{ }
snub 24-cell prism
146 768 1392 960 192

[5,3,3] × [ ]

This prismatic family has 15 forms:

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

Dodecaplex prism family {5,3,3}x{ }
# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
90 CDW ring.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
{5,3,3}x{ }
120-cell prism
122 960 2640 3000 1200
91 CDW dot.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1{5,3,3}x{ }
Rectified 120-cell prism
722 4560 9840 8400 2400
92 CDW ring.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,1{5,3,3}x{ }
Truncated 120-cell prism
722 4560 11040 12000 4800
93 CDW ring.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,2{5,3,3}x{ }
Cantellated 120-cell prism
1922 12960 29040 25200 7200
94 CDW ring.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,3{5,3,3}x{ }
Runcinated 120-cell prism
2642 12720 22080 16800 4800
95 CDW dot.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1,2{5,3,3}x{ }
Bitruncated 120-cell prism
722 5760 15840 18000 7200
96 CDW ring.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t0,1,2{5,3,3}x{ }
Cantitruncated 120-cell prism
1922 12960 32640 36000 14400
97 CDW ring.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,3{5,3,3}x{ }
Runcitruncated 120-cell prism
2642 18720 44880 43200 14400
98 CDW ring.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,2,3{5,3,3}x{ }
Omnitruncated 120-cell prism
2642 22320 62880 72000 28800
Tetraplex prism family [3,3,5]x[ ]
# Coxeter-Dynkin
andSchläfli
symbols
Name
Element counts
Facets Cells Faces Edges Vertices
99 CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
{3,3,5}x{ }
600-cell prism
602 2400 3120 1560 240
100 CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1{3,3,5}x{ }
Rectified 600-cell prism
722 5040 10800 7920 1440
101 CDW dot.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1{3,3,5}x{ }
Truncated 600-cell prism
722 5040 11520 10080 2880
102 CDW dot.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,2{3,3,5}x{ }
Cantellated 600-cell prism
1442 11520 28080 25200 7200
[94] CDW ring.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,3{3,3,5}x{ }
Runcinated 600-cell prism
(Same as runcinated 120-cell prism)
2642 12720 22080 16800 4800
[95] CDW dot.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 2.pngCDW ring.png
t1,2{3,3,5}x{ }
Bitruncated 600-cell prism
(Same as bitruncated 120-cell prism)
722 5760 15840 18000 7200
103 CDW dot.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,2{3,3,5}x{ }
Cantitruncated 600-cell prism
1442 11520 31680 36000 14400
104 CDW ring.pngCDW 5.pngCDW dot.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,3{3,3,5}x{ }
Runcitruncated 600-cell prism
2642 18720 44880 43200 14400
[98] CDW ring.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 3.pngCDW ring.pngCDW 2.pngCDW ring.png
t0,1,2,3{3,3,5}x{ }
Omnitruncated 600-cell prism
(Same as omnitruncated 120-cell prism)
2642 22320 62880 72000 28800

Grand antiprism prism

The grand antiprism prism is the only known convex nonwythoffian uniform polyteron. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (300 tetrahedrons, 20 pentagonal antiprisms, 700 triangular prisms, 20 pentagonal prisms), 322 hypercells (2 grand antiprisms 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
105 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-Dynkin 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's 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-
Dynkin
diagram
Description
Parent t0{p,q,r,s} CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW dot.pngCDW r.pngCDW dot.pngCDW s.pngCDW dot.png Any regular 5-polytope
Rectified t1{p,q,r,s} CDW dot.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.pngCDW r.pngCDW dot.pngCDW s.pngCDW dot.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} CDW dot.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.pngCDW r.pngCDW dot.pngCDW s.pngCDW dot.png Birectification reduces cells to their duals.
Truncated t0,1{p,q,r,s} CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW dot.pngCDW r.pngCDW dot.pngCDW s.pngCDW dot.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} CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW ring.pngCDW r.pngCDW dot.pngCDW s.pngCDW dot.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} CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW dot.pngCDW r.pngCDW ring.pngCDW s.pngCDW dot.png Runcination reduces cells and creates new cells at the vertices and edges.
Stericated t0,4{p,q,r,s} CDW ring.pngCDW p.pngCDW dot.pngCDW q.pngCDW dot.pngCDW r.pngCDW dot.pngCDW s.pngCDW ring.png Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for polyterons.)
Omnitruncated t0,1,2,3,4{p,q,r,s} CDW ring.pngCDW p.pngCDW ring.pngCDW q.pngCDW ring.pngCDW r.pngCDW ring.pngCDW s.pngCDW ring.png All four operators, truncation, cantellation, runcination, and sterication are applied.

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 4-space. [1]

# Coxeter group Coxeter-Dynkin diagram
1 A~4 (3 3 3 3 3) [3[5]] CD downbranch-00.pngCD downbranch-33.pngCD righttriangleopen 000.png
2 B~4 [4,3,3,4] CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
3 C~4 [4,3,31,1] h[4,3,3,4] CD dot.pngCD 4.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png
4 D~4 [31,1,1,1] q[4,3,3,4] CDT dot.pngCDT 3a.pngCDT branch000.pngCDT 3a.pngCDT dot.png
5 F~4 [3,4,3,3] h[4,3,3,4] CDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png

There are three regular honeycomb of Euclidean 4-space:

  1. tesseractic honeycomb, with symbols {4,3,3,4}, CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png = CD ring.pngCD 4.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png. There are 19 uniform honeycombs in this family.
  2. Icositetrachoric honeycomb, with symbols {3,4,3,3}, CDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png. There are 31 uniform honeycombs in this family.
  3. Hexadecachoric honeycomb, with symbols {3,3,4,3}, CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.png

Other families that generate uniform honeycombs:

  • There are 23 uniform honeycombs, 4 unique in the demitesseractic honeycomb family. With symbols h{4,32,4} it is geometrically identical to the hexadecachoric honeycomb, CDW hole.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png = CD dot.pngCD 4.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD ring.png
  • There are 7 uniform honeycombs from the A~4, CD downbranch-00.pngCD downbranch-33.pngCD righttriangleopen 000.png family, all unique.
  • There are 7 uniform honeycombs in the D~4: [31,1,1,1] CDT dot.pngCDT 3a.pngCDT branch000.pngCDT 3a.pngCDT dot.png family, all repeated in other families, including the demitesseractic honeycomb.

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

The single-ringed tessellations are given below, indexed by Olshevsky's listing.

B~4 [4,3,3,4] family (Tesseractic honeycombs)

# Coxeter-Dynkin
andSchläfli
symbols
Name Facets by location: [4,3,3,4]
4 3 2 1 0
CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.png
[4,3,3]
CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.png
[4,3]×[ ]
CDW dot.pngCDW 4.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 4.pngCDW dot.png
[4]×[4]
CDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
[ ]×[3,4]
CDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
[3,3,4]
1 CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t0{4,3,3,4}
Tesseractic honeycomb {4,3,3} - - - -
87 CDW dot.pngCDW 4.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t1{4,3,3,4}
Rectified tesseractic honeycomb t1{4,3,3} - - - {3,3,4}
88 CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
t2{4,3,3,4}
Birectified tesseractic honeycomb
(Same as Icositetrachoric honeycomb {3,4,3,3})
t1{3,3,4}
or {3,4,3}
- - - t1{3,3,4}
or {3,4,3}

C~4 [31,1,3,4] family (Demitesseractic honeycombs)

[2]

# Coxeter-Dynkin
andSchläfli
symbols
CD B5 nodes.png
Name Facets by location: [31,1,3,4]
4 3 2 1 0
CD dot.pngCD 3b.pngCD downbranch-0dash.pngCD 3b.pngCD dot.pngCD 4.pngCD dot.png
[3,3,4]
CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 2c.pngCD dash.png
[31,1,1]
CD dot.pngCD 3b.pngCD downbranch-00.pngCD 2c.pngCD dash.pngCD 2c.pngCD dot.png
[3,3]×[ ]
CD dot.pngCD 2c.pngCD downbranch-dash0.pngCD 2c.pngCD dot.pngCD 4.pngCD dot.png
[ ]×[3]×[ ]
CD dash.pngCD 2c.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 4.pngCD dot.png
[3,3,4]
104 CD ring.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 4.pngCD dot.png
{31,1,3,4}
Demitesseractic honeycomb
(Same as hexadecachoric honeycomb)
[88] CD dot.pngCD 3b.pngCD downbranch-10.pngCD 3b.pngCD dot.pngCD 4.pngCD dot.png
t1{31,1,3,4}
Rectified demitesseractic honeycomb
(Same as icositetrachoric honeycomb, {3,4,3,3})
(Also birectified tesseractic honeycomb)
[87] CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD ring.pngCD 4.pngCD dot.png
t2{31,1,3,4}
Birectified demitesseractic honeycomb
(Same as Rectified tesseractic honeycomb)
[1] CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 4.pngCD ring.png
t3{31,1,3,4}
Trirectified demitesseractic honeycomb
(Same as tesseractic honeycomb)

F~4 (V5) [3,4,3,3] family (Icositetrachoric-hexadecachoric honeycombs)

[3]

# Coxeter-Dynkin
andSchläfli
symbols
Name Facets by location: [3,4,3,3]
4 3 2 1 0
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.png
[3,4,3]
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.png
[3,4]×[ ]
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.png
[3]×[3]
CDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
[ ]×[3,3]
CDW dash.pngCDW 2c.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
[4,3,3]
104 CDW ring.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t0{3,4,3,3}
Icositetrachoric honeycomb
[88] CDW dot.pngCDW 3b.pngCDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
t1{3,4,3,3}
Rectified icositetrachoric honeycomb
(Same as Hexadecachoric honeycomb)
106 CDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW ring.pngCDW 3.pngCDW dot.pngCDW 3b.pngCDW dot.png
t2{3,4,3,3}
Birectified icositetrachoric honeycomb
? CDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW ring.pngCDW 3b.pngCDW dot.png
t1{3,3,4,3}
Rectified hexadecachoric honeycomb
88 CDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png
t0{3,3,4,3}
Hexadecachoric honeycomb

A~4 (P5) (3 3 3 3) family (Simplectic honeycombs)

# Coxeter-Dynkin
andSchläfli
symbols
Name Facets by location
4 3 2 1 0
CDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
[3,3,3]
CDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
[]x[3,3]
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.pngCDW 3b.pngCDW dot.png
[3]x[3]
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.pngCDW 2c.pngCDW dot.png
[3,3]x[]
CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 2c.pngCDW dash.png
[3,3,3]
134 CD downbranch-01.pngCD downbranch-33.pngCD righttriangleopen 000.png Pentachoric-dispentachoric honeycomb

D~4 (Q5) [31,1,1,1] or q[4,3,3,4] family

# Coxeter-Dynkin
andSchläfli
symbols
? CDT dot.pngCDT 3a.pngCDT branch000.pngCDT 3a.pngCDT ring.png
q[4,3,3,4]
? CDT dot.pngCDT 3a.pngCDT branch010.pngCDT 3a.pngCDT dot.png
q1[4,3,3,4]

Regular and uniform hyperbolic honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in hyperbolic 4-space. They generate 5 regular hyperbolic honeycombs and numerous uniform ones.

# Coxeter group Coxeter-Dynkin diagram Uniform tessellations
1 (4 3 3 3 3) CD downbranch-00-left-4.pngCD downbranch-33.pngCD righttriangleopen 000.png 25 forms
2 [5,3,3,3] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png 31 forms
3 [5,3,3,4] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png 31 forms
4 [5,3,31,1] CD dot.pngCD 5.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png 23 forms, most overlapping [5,3,3,4] family
5 [5,3,3,5] CDW dot.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 5.pngCDW dot.png 19 forms

Regular tessellations of hyperbolic 4-space

There are five kinds of convex regular honeycombs and four kinds of star-honeycombs in H4 space: [4]

Honeycomb name Schläfli
Symbol
{p,q,r,s}
Coxeter-Dynkin
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 pentachoric {3,3,3,5} CDW dot.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW ring.png {3,3,3} {3,3} {3} {5} {3,5} {3,3,5} {5,3,3,3}
Order-3 hecatonicosachoric {5,3,3,3} CDW ring.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png {5,3,3} {5,3} {5} {3} {3,3} {3,3,3} {3,3,3,5}
Order-5 tesseractic {4,3,3,5} CDW dot.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW ring.png {4,3,3} {4,3} {4} {5} {3,5} {3,3,5} {5,3,3,4}
Order-4 hecatonicosachoric {5,3,3,4} CDW ring.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png {5,3,3} {5,3} {5} {4} {3,4} {3,3,4} {4,3,3,5}
Order-5 hecatonicosachoric {5,3,3,5} CDW ring.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 5.pngCDW dot.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-Dynkin
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 stellated hecatonicosachoric {5/2,5,3,3} CD ring.pngCD 5-2.pngCD dot.pngCD 5.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png {5/2,5,3} {5/2,5} {5} {5} {3,3} {5,3,3} {3,3,5,5/2}
Order-5/2 hexacosichoric {3,3,5,5/2} CD dot.pngCD 5-2.pngCD dot.pngCD 5.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png {3,3,5} {3,3} {3} {5/2} {5,5/2} {3,5,5/2} {5/2,5,3,3}
Order-5 icosahedral hecatonicosachoric {3,5,5/2,5} CD ring.pngCD 3b.pngCD dot.pngCD 5.pngCD dot.pngCD 5-2.pngCD dot.pngCD 5.pngCD dot.png {3,5,5/2} {3,5} {3} {5} {5/2,5} {5,5/2,5} {5,5/2,5,3}
Order-3 great hecatonicosachoric {5,5/2,5,3} CD dot.pngCD 3b.pngCD dot.pngCD 5.pngCD dot.pngCD 5-2.pngCD dot.pngCD 5.pngCD ring.png {5,5/2,5} {5,5/2} {5} {3} {5,3} {5/2,5,3} {3,5,5/2,5}

See also

Notes

  1. ^ George Olshevsky, (2006, Uniform Panoploid Tetracombs, Manuscript (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  2. ^ Olshevsky section V
  3. ^ Olshevsky section VI
  4. ^ Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213

References

  • 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, editied 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 quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons [2]
  • 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) [3]

External links


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