User:Tomruen/Uniform polyteron verf

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Vertex figures (as Schlegel diagrams) for uniform polyterons, uniform honeycombs (Euclidean and hyperbolic). (Excluding prismatic forms, and nonwythoffian forms)

Tables are expanded for finite and infinite forms (spherical/Euclidean/hyperbolic) for completeness, not that I expect ever to include all of the hyperbolic forms! (Compare to 4-polytopes: Talk:Vertex figure/polychoron)

Spherical[edit]

There are three fundamental affine Coxeter groups that generate regular and uniform tessellations on the 3-sphere:

# Coxeter group Coxeter graph
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

In addition there are prismatic groups:

Uniform prismatic forms:

# Coxeter groups Coxeter graph
1 A4 × A1 [3,3,3] × [ ] 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] × [ ] 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] × [ ] 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] × [ ] 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] × [ ] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png

Uniform duoprism prismatic forms:

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

Uniform duoprismatic forms:

# Coxeter groups Coxeter graph
1 A3 × I2(p) [3,3] × [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] × [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] × [p] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png

Euclidean[edit]

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 4-space:

# Coxeter group Coxeter-Dynkin diagram
1 A~4 [(3,3,3,3,3)] CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
2 B~4 [4,3,3,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
3 C~4 [4,3,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
4 D~4 [31,1,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
5 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

In addition there are prismatic groups:

Duoprismatic forms

  • B~2xB~2: [4,4]x[4,4] = [4,3,3,4] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png = CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png (Same as tesseractic honeycomb family)
  • B~2xH~2: [4,4]x[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
  • H~2xH~2: [6,3]x[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
  • A~2xB~2: [3[3]]]x[4,4] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png (Same forms as [6,3]x[4,4])
  • A~2xH~2: [3[3]]]x[6,3] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png (Same forms as [6,3]x[6,3])
  • A~2xA~2: [3[3]]]x[3[3]] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel split1.pngCDel branch.png (Same forms as [6,3]x[6,3])

Prismatic forms

  • B~3xI~1: [4,3,4]x[∞] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
  • D~3xI~1: [4,31,1]x[∞] CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
  • A~3xI~1: [3[4]]x[∞] CDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png

Hyperbolic[edit]

1 [5,3,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
2 [5,3,3,4] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
3 [5,3,3,5] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
4 [5,3,31,1] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
5 [(4,3,3,3,3)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

Linear Coxeter graphs[edit]

There are 31 truncation forms for each group, or 19 subgrouped as half-families as given below (with 7 overlapped).

Summary chart: File:Uniform polyteron vertex figure chart.png

Vertex figures (As 3D Schlegel diagrams)
# Operation
Coxeter-Dynkin
General
{p,q,r,s}
Spherical Euclidean Hyperbolic
5-simplex
[3,3,3,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube
[4,3,3,3]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-orthoplex
[3,3,3,4]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[4,3,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[3,4,3,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,4,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3,5]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,3,3]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[4,3,3,5]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,3,4]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[5,3,3,5]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
1 Regular
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png
{q,r,s}:(p) 5-simplex verf.png
{3,3,3}:(3)
5-cube verf.png
{3,3,3}:(4)
Pentacross verf.png
{3,3,4}:(3)
Schlegel wireframe 16-cell.png
{3,3,4}:(4)
Schlegel wireframe 8-cell.png
{4,3,3}:(3)
Schlegel wireframe 24-cell.png
{3,4,3}:(3)
Schlegel wireframe 600-cell vertex-centered.png
{3,3,5}:(3)
Schlegel wireframe 5-cell.png
{3,3,3}:(5)
Schlegel wireframe 600-cell vertex-centered.png
{3,3,5}:(4)
Schlegel wireframe 16-cell.png
{3,3,4}:(5)
Schlegel wireframe 600-cell vertex-centered.png
{3,3,5}:(5)
2 Rectified
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png
Rectified polyteron verf.png
{r,s}-prism
Rectified 5-simplex verf.png Rectified 5-cube verf.png Rectified pentacross verf.png Rectified 24-cell honeycomb verf.png
3 Birectified
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png
Birectified polyteron verf.png
p-s duoprism
Birectified hexateron verf.png
3-3 duoprism
Birectified penteract verf.png
3-4 duoprism
Birectified penteract verf.png
3-4 duoprism
Birectified tesseractic honeycomb verf.png
4-4 duoprism
Birectified demitesseractic honeycomb verf.png
3-3 duoprism
Birectified demitesseractic honeycomb verf.png
3-3 duoprism
4 Truncated
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png
Truncated polyteron verf.png
{r,s}-pyramid
Truncated 5-simplex verf.png Truncated 5-cube verf.png Truncated pentacross.png Truncated 24-cell honeycomb verf.png
5 Bitruncated
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png
Bitruncated polyteron verf.png Bitruncated 5-simplex verf.png Bitruncated penteract verf.png Bitruncated pentacross verf.png Bitruncated tesseractic honeycomb verf.png Bitruncated icositetrachoric honeycomb verf.png Bitruncated demitesseractic honeycomb verf.png
6 Cantellated
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png
Cantellated polyteron-order4 verf.png
s-prism-wedge
Cantellated hexateron verf.png Cantellated 5-cube vertf.png Cantellated pentacross verf.png Rectified 24-cell honeycomb F4b verf.png
7 Bicantellated
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png
Bicantellated polyteron verf.png Bicantellated 5-simplex verf.png Bicantellated penteract verf.png Bicantellated penteract verf.png Bicantellated tesseractic honeycomb verf.png Bicantellated icositetrachoric honeycomb verf.png Bicantellated icositetrachoric honeycomb verf.png
8 Runcinated
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png
Runcinated polyteron verf.png Runcinated 5-simplex verf.png Runcinated penteract verf.png Runcinated pentacross verf.png Runcinated tesseractic honeycomb verf.png
9 Stericated
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.png
Stericated polyteron verf.png
{q,r}-{r,q} antiprism
Stericated hexateron verf.png Stericated penteract verf.png Stericated penteract verf.png Stericated tesseractic honeycomb verf.png Stericated demitesseractic honeycomb verf.png Stericated demitesseractic honeycomb verf.png
10 Cantitruncated
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.png
Canitruncated polyteron verf.png Canitruncated 5-simplex verf.png Canitruncated 5-cube verf.png Canitruncated 5-orthoplex verf.png
11 Bicantitruncated
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png
Bicanitruncated polyteron verf.png Bicanitruncated 5-simplex verf.png Bicanitruncated 5-cube verf.png Bicantitruncated 5-orthoplex verf.png
12 Runcitruncated
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png
Runcitruncated polyteron verf.png
wedge-pyramid
Runcitruncated 5-simplex verf.png Runcitruncated 5-cube verf.png Runcitruncated 5-orthoplex verf.png
13 Steritruncated
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.png
Steritruncated polyteron verf.png Steritruncated 5-simplex verf.png Steritruncated 5-cube verf.png Steritruncated 5-orthoplex verf.png
14 Runcicantellated
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png
Runcicantellated polyteron verf.png Runcicantellated 5-simplex verf.png Runcicantellated 5-cube verf.png Runcicantellated 5-orthoplex verf.png
15 Stericantellated
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.png
Stericantellated polyteron verf.png Stericantellated 5-simplex verf.png Stericantellated 5-cube verf.png Stericantellated 5-orthoplex verf.png
16 Runcicantitruncated
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.png
Runcicantitruncated polyteron verf.png Runcicantitruncated 5-simplex verf.png Runcicantitruncated 5-cube verf.png Runcicantitruncated 5-orthoplex verf.png
17 Stericantitruncated
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.png
Stericanitruncated polyteron verf.png Stericanitruncated 5-simplex verf.png Stericanitruncated 5-cube verf.png Stericanitruncated 5-orthoplex verf.png
18 Steriruncitruncated
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node 1.png
Steriruncitruncated polyteron verf.png Steriruncitruncated 5-simplex verf.png Steriruncitruncated 5-cube verf.png Steriruncitruncated 5-orthoplex verf.png
19 Omnitruncated
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node 1.png
Omnitruncated polyteron verf.png
Irr. 5-simplex
Omnitruncated 5-simplex verf.png Omnitruncated 5-cube verf.png Omnitruncated 5-cube verf.png
20 Alternated regular
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
t1{3,3,p} Schlegel half-solid rectified 5-cell.png
t1{3,3,3}
Schlegel half-solid rectified 16-cell.png
t1{3,3,4}

Bifurcating Coxeter graphs[edit]

There are 23 forms from each family, with 15 repeated from the linear [4,3,3,s] families above.

Vertex figures (As 3D Schlegel diagrams)
# Operation
Coxeter-Dynkin
Linear equiv General Spherical Euclidean Hyperbolic
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png [s,3,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png
[3,3,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[4,3,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[5,3,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
1 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png t1{3,3,s} Demipenteract verf.png
t1{3,3,3}
t1{3,3,4} t1{3,3,5}
2 CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png Birectified penteract verf.png
3 CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png Rectified pentacross verf.png
4 CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png Pentacross verf.png
5 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png Truncated bifurcated polyteron verf.png Truncated 5-demicube verf.png
6 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png Cantellated bifurcated polyteron verf.png
7 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png Runcinated bifurcated polyteron verf.png
8 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png Rectified 5-cube verf.png
9 CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png Bitruncated pentacross verf.png
10 CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png Cantellated pentacross verf.png
11 CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png Truncated pentacross.png
12 CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node.png Bitruncated penteract verf.png
13 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png Cantitruncated bifurcated polyteron verf.png
14 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png
15 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png
16 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png
17 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png
18 CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png
19 CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node.png
20 CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel s.pngCDel node 1.png
21 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png
22 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png
23 CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel s.pngCDel node 1.png Omnitruncated 5-demicube verf.png

Trifurcating Coxeter graphs[edit]

There are 9 forms:

Vertex figures (As 3D Schlegel diagrams)
Operation
Coxeter-Dynkin
Euclidean
Coxeter group [31,1,1,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png

Cyclic Coxeter graphs[edit]

There are 7 forms in the first cycle family, and 19 forms in the second cyclic family:

# General Euclidean Hyperbolic
[(p,3,3,3,3)]
CDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
[(3,3,3,3,3)]
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
[(4,3,3,3,3)]
CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
1 CDel branch 01.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
2 CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
3 CDel branch 01.pngCDel 3ab.pngCDel nodes 10.pngCDel split2.pngCDel node.png
4 CDel branch 11.pngCDel 3ab.pngCDel nodes 10.pngCDel split2.pngCDel node.png
5 CDel branch 11.pngCDel 3ab.pngCDel nodes 01.pngCDel split2.pngCDel node.png
6 CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png
7 CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png Omnitruncated 4-simplex honeycomb verf.png