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

Hyperbolic

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

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
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11
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13
14
15
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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
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3
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5
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7