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:

In addition there are prismatic groups:

Uniform prismatic forms:

Uniform duoprism prismatic forms:

Uniform duoprismatic forms:

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

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[&infin;]
 * D~3xI~1: [4,31,1]x[&infin;]
 * A~3xI~1: [3[4]]x[&infin;]

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

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

Trifurcating Coxeter graphs
There are 9 forms:

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