User:Tomruen/Convex uniform tetracomb

Regular and uniform honeycombs
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 4-space.

There are three regular honeycomb of Euclidean 4-space:
 * 1) tesseractic honeycomb, with symbols {4,3,3,4}, [[Image:CDW ring.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] = [[Image:CD ring.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]]. There are 19 uniform honeycombs in this family.
 * 2) 24-cell honeycomb, with symbols {3,4,3,3}, [[Image:CDW ring.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]]. There are 31 uniform honeycombs in this family.
 * 3) 16-cell honeycomb, with symbols {3,3,4,3}, [[Image:CDW ring.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]]

Other families that generate uniform honeycombs:
 * There are 23 uniform honeycombs, 4 unique in the 16-cell honeycomb family. With symbols h{4,32,4} it is geometrically identical to the 16-cell honeycomb, [[Image:CDW hole.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] = [[Image:CD dot.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD ring.png]]
 * There are 7 uniform honeycombs from the A~4, [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD righttriangleopen 000.png]] family, all unique.
 * There are 9 uniform honeycombs in the D~4: [31,1,1,1] [[File:CDT dot.png]][[File:CDT 3a.png]][[File:CDT branch000.png]][[File:CDT 3a.png]][[File:CDT dot.png]] family, all repeated in other families, including the 16-cell honeycomb.

Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.

The single-ringed tessellations are given below, indexed by Olshevsky's listing.

Duoprismatic forms
 * B~2xB~2: [4,4]x[4,4] = [4,3,3,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (Same as tesseractic honeycomb family)
 * B~2xH~2: [4,4]x[6,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * H~2xH~2: [6,3]x[6,3] [[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * A~2xB~2: [&Delta;]x[4,4] [[Image:CD righttriangle-000.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (Same forms as [6,3]x[4,4])
 * A~2xH~2: [&Delta;]x[6,3] [[Image:CD righttriangle-000.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (Same forms as [6,3]x[6,3])
 * A~2xA~2: [&Delta;]x[&Delta;] [[Image:CD righttriangle-000.png]][[Image:CDW 2.png]][[Image:CD righttriangle-000.png]] (Same forms as [6,3]x[6,3])

Prismatic forms
 * B~3xI~1: [4,3,4]x[&infin;] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * D~3xI~1: [4,31,1]x[&infin;] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD infin.png]][[Image:CD dot.png]]
 * A~3xI~1: [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD infin.png]][[Image:CD dot.png]]

Noncompact prismatic forms
 * A3xI~1: [3,3]x[&infin;] - [[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * B3xI~1: [4,3]x[&infin;] - [[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * H3xI~1: [5,3]x[&infin;] - [[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * I~1xI~1xI2r: [&infin;] x [&infin;] x [r] = [4,4]x[r] - [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW r.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW r.png]][[Image:CDW dot.png]]

Non-Wythoffian forms
The non-Wythoffian forms are built as stacked composites of these prismatic noncompact groups:
 * I2pxI~1xA1: [p]x[&infin;]x[ ] - [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW 2.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]] (Prism column)
 * D~3xA1: [4,31,1]x[ ] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 2.png]][[Image:CD dot.png]] (Prism slab)
 * A~3xA1: [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]][[Image:CD 2.png]][[Image:CD dot.png]] (Prism slab)
 * A~2xI2p: [&Delta;]x[p] [[Image:CD righttriangle-000.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD p.png]][[Image:CD dot.png]] (Prism slab)
 * B~2xI2p: [4,4]x[p] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]] (Prism slab)
 * H~2xI2p: [6,3]x[p] [[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]] (Prism slab)

C~4 [31,1,3,4] family
There are 23 honeycombs in this family, all listed below.

F~4 [3,4,3,3] family
There are 32 honeycombs in this family, 31 reflective forms and one snub. They are named as truncated forms from the regular 16-cell honeycomb and 24-cell honeycomb. These 31 forms are listed by the regular generators in two groups of 19, with 7 shared between.

From the regular 24-cell honeycomb, 19 forms are:

From the regular 16-cell honeycomb, 19 forms are:

A~4 [3[5]] family
There are 7 honeycombs in this family, all unique to this family, all given below.

D~4 [31,1,1,1] family
There are 9 honeycombs in this family, all repeated, with all 9 forms given below.

Duoprismatic forms
Coxeter groups:


 * B~2xB~2: [4,4]x[4,4]
 * B~2xH~2: [4,4]x[6,3]
 * H~2xH~2: [6,3]x[6,3]

[4,4]×[4,4]
There are 15 reflective combinatoric forms, but only 3 unique ones.

[4,4]x[6,3]
There are 35 reflective combinatoric forms.

[6,3]x[6,3]
There are 28 reflective combinatoric forms.