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)

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]

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: [4,3^1,1]x[∞]
• A^~₃xI^~₁: [3^[4]]x[∞]

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)]

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}

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

There are 9 forms:

Vertex figures (As 3D Schlegel diagrams)

Operation Coxeter-Dynkin	Euclidean
Coxeter group	[31,1,1,1]

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