# User:Tomruen/Uniform polyteron verf

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

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

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

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

## Linear Coxeter graphs

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

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

There are 9 forms:

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

## Cyclic Coxeter graphs

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