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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

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

# Coxeter group Coxeter graph
1 A5 [34]
2 B5 [4,33]
3 D5 [32,1,1]

In addition there are prismatic groups:

Uniform prismatic forms:

# Coxeter groups Coxeter graph
1 A4 × A1 [3,3,3] × [ ]
2 B4 × A1 [4,3,3] × [ ]
3 F4 × A1 [3,4,3] × [ ]
4 H4 × A1 [5,3,3] × [ ]
5 D4 × A1 [31,1,1] × [ ]

Uniform duoprism prismatic forms:

Coxeter groups Coxeter graph
I2(p) × I2(q) × A1 [p] × [q] × [ ]

Uniform duoprismatic forms:

# Coxeter groups Coxeter graph
1 A3 × I2(p) [3,3] × [p]
2 B3 × I2(p) [4,3] × [p]
3. H3 × I2(p) [5,3] × [p]

Euclidean

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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)]
2 B~4 [4,3,3,4]
3 C~4 [4,3,31,1]
4 D~4 [31,1,1,1]
5 F~4 [3,4,3,3]

In addition there are prismatic groups:

Duoprismatic forms

  • B~2xB~2: [4,4]x[4,4] = [4,3,3,4] = (Same as tesseractic honeycomb family)
  • B~2xH~2: [4,4]x[6,3]
  • H~2xH~2: [6,3]x[6,3]
  • A~2xB~2: [3[3]]]x[4,4] (Same forms as [6,3]x[4,4])
  • A~2xH~2: [3[3]]]x[6,3] (Same forms as [6,3]x[6,3])
  • A~2xA~2: [3[3]]]x[3[3]] (Same forms as [6,3]x[6,3])

Prismatic forms

  • B~3xI~1: [4,3,4]x[∞]
  • D~3xI~1: [4,31,1]x[∞]
  • A~3xI~1: [3[4]]x[∞]

Hyperbolic

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1 [5,3,3,3]
2 [5,3,3,4]
3 [5,3,3,5]
4 [5,3,31,1]
5 [(4,3,3,3,3)]

Linear Coxeter graphs

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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]
5-cube
[4,3,3,3]
5-orthoplex
[3,3,3,4]
[4,3,3,4]
[3,4,3,3]
[3,3,4,3]
[3,3,3,5]
[5,3,3,3]
[4,3,3,5]
[5,3,3,4]
[5,3,3,5]
1 Regular
{q,r,s}:(p)
{3,3,3}:(3)

{3,3,3}:(4)

{3,3,4}:(3)

{3,3,4}:(4)

{4,3,3}:(3)

{3,4,3}:(3)

{3,3,5}:(3)

{3,3,3}:(5)

{3,3,5}:(4)

{3,3,4}:(5)

{3,3,5}:(5)
2 Rectified

{r,s}-prism
3 Birectified

p-s duoprism

3-3 duoprism

3-4 duoprism

3-4 duoprism

4-4 duoprism

3-3 duoprism

3-3 duoprism
4 Truncated

{r,s}-pyramid
5 Bitruncated
6 Cantellated

s-prism-wedge
7 Bicantellated
8 Runcinated
9 Stericated

{q,r}-{r,q} antiprism
10 Cantitruncated
11 Bicantitruncated
12 Runcitruncated

wedge-pyramid
13 Steritruncated
14 Runcicantellated
15 Stericantellated
16 Runcicantitruncated
17 Stericantitruncated
18 Steriruncitruncated
19 Omnitruncated

Irr. 5-simplex
20 Alternated regular
t1{3,3,p}
t1{3,3,3}

t1{3,3,4}

Bifurcating Coxeter graphs

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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
[s,3,31,1]
[3,3,31,1]
[4,3,31,1]
[5,3,31,1]
1 t1{3,3,s}
t1{3,3,3}
t1{3,3,4} t1{3,3,5}
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

Trifurcating Coxeter graphs

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There are 9 forms:

Vertex figures (As 3D Schlegel diagrams)
Operation
Coxeter-Dynkin
Euclidean
Coxeter group [31,1,1,1]

Cyclic Coxeter graphs

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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)]
[(3,3,3,3,3)]
[(4,3,3,3,3)]
1
2
3
4
5
6
7