Étale homotopy type

Example paracompact regular honeycombs
order-6 cubic honeycomb Template:CDD	order-4 hexagonal tiling honeycomb Template:CDD
square tiling honeycomb Template:CDD	order-4 octahedral honeycomb Template:CDD
triangular tiling honeycomb Template:CDD	order-6 hexagonal tiling honeycomb Template:CDD.

In geometry, a uniform honeycombs in hyperbolic space is a tessellation of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter–Dynkin diagrams for each family. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity, similar to the hyperbolic uniform tilings in 2-dimensions.

Enumeration



These graphs show subgroup relations of paracompact hyperbolic Coxeter groups. Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetry.

This is a complete enumeration of the 150+ unique Wythoffian paracompact uniform honeycombs generated from tetrahedral fundamental domains (rank 4 paracompact coxeter groups). The alternations are listed for completeness, but don't in general generate uniform honeycombs. Single-hole alternations represent a mirror removal operation. If an end-node is removed, another simplex family is generated. If a hole has two branches, a Vinberg polytope is generated, although only Vinberg polytope with mirror symmetry are related to the simplex groups, and their uniform honeycombs have not been systematically explored.

Simplectic hyperbolic paracompact group summary
Type	Coxeter groups	Unique honeycomb count
Linear graphs	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	4×15+6+8+8 = 82
Tridental graphs	Template:CDD \| Template:CDD \| Template:CDD	4+4+0 = 8
Cyclic graphs	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	4×9+5+1+4+1+0 = 47
Loop-n-tail graphs	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	4+4+4+2 = 14

The complete list of nonsimpletic paracompact solutions was published by P. Tumarkin in 2003.^[1] The smallest paracompact form in H³ can be represented by Template:CDD, or [∞,3,3,∞] which can be constructed by a mirror removal of paracompact hyperbolic group [3,4,4] as [3,4,1⁺,4]. The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramids include [4,4,1⁺,4] = [∞,4,4,∞], Template:CDD. Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: [(3,3,4,1⁺,4)] = [(3,∞,3),(3,∞,3)] or Template:CDD, [(3,4,4,1⁺,4)] = [(4,∞,3),(3,∞,4)] or Template:CDD, [(4,4,4,1⁺,4)] = [(4,∞,4),(4,∞,4)] or Template:CDD.

Vinberg polytopes
Dimension	Rank	Graphs
H³	5	Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD ... Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD ...

Linear graphs

Paracompact hyperbolic enumeration
Group	Extended symmetry	Honeycombs		Chiral extended symmetry	Alternation honeycombs
${\bar{R}}_{3}$ [4,4,3] Template:CDD	[4,4,3] Template:CDD	15	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1⁺,4,1⁺,4,3⁺]	(6)	Template:CDD (= Template:CDD) Template:CDD (= Template:CDD) Template:CDD \| Template:CDD Template:CDD \| Template:CDD
${\bar{R}}_{3}$ [4,4,3] Template:CDD	[4,4,3] Template:CDD	15		[4,4,3]⁺	(1)	Template:CDD
${\bar{N}}_{3}$ [4,4,4] Template:CDD	[4,4,4] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[1⁺,4,1⁺,4,1⁺,4,1⁺]	(3)	Template:CDD (= Template:CDD = Template:CDD) Template:CDD \| Template:CDD
	[4,4,4] Template:CDD = Template:CDD	(3)	Template:CDD \| Template:CDD \| Template:CDD	[1⁺,4,1⁺,4,1⁺,4,1⁺]	(3)	Template:CDD (= Template:CDD) Template:CDD \| Template:CDD
	[2⁺[4,4,4]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(4,4⁺,4,2⁺)]]	(2)	Template:CDD \| Template:CDD
	[2⁺[4,4,4]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[4,4,4]]⁺	(1)	Template:CDD
${\bar{V}}_{3}$ [6,3,3] Template:CDD	[6,3,3] Template:CDD	15	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1⁺,6,(3,3)⁺]	(2)	Template:CDD (= Template:CDD) Template:CDD
${\bar{V}}_{3}$ [6,3,3] Template:CDD	[6,3,3] Template:CDD	15		[6,3,3]⁺	(1)	Template:CDD
${\bar{B V}}_{3}$ [6,3,4] Template:CDD	[6,3,4] Template:CDD	15	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1⁺,6,3⁺,4,1⁺]	(6)	Template:CDD (= Template:CDD) Template:CDD (= Template:CDD) Template:CDD \| Template:CDD Template:CDD \| Template:CDD
${\bar{B V}}_{3}$ [6,3,4] Template:CDD	[6,3,4] Template:CDD	15		[6,3,4]⁺	(1)	Template:CDD
${\bar{H V}}_{3}$ [6,3,5] Template:CDD	[6,3,5] Template:CDD	15	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1⁺,6,(3,5)⁺]	(2)	Template:CDD (= Template:CDD) Template:CDD
${\bar{H V}}_{3}$ [6,3,5] Template:CDD	[6,3,5] Template:CDD	15		[6,3,5]⁺	(1)	Template:CDD
${\bar{Y}}_{3}$ [3,6,3] Template:CDD	[3,6,3] Template:CDD	5	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
	[3,6,3] Template:CDD = Template:CDD	(1)	Template:CDD	[2⁺[3⁺,6,3⁺]]	(1)	Template:CDD
	[2⁺[3,6,3]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[3,6,3]]⁺	(1)	Template:CDD
${\bar{Z}}_{3}$ [6,3,6] Template:CDD	[6,3,6] Template:CDD	6	Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD	[1⁺,6,3⁺,6,1⁺]	(2)	Template:CDD (= Template:CDD) Template:CDD
	[2⁺[6,3,6]] Template:CDD = Template:CDD	(1)	Template:CDD	[2⁺[(6,3⁺,6,2⁺)]]	(2)	Template:CDD
	[2⁺[6,3,6]] Template:CDD	2	Template:CDD \| Template:CDD	[2⁺[(6,3⁺,6,2⁺)]]	(2)	Template:CDD
	[2⁺[6,3,6]] Template:CDD	2	Template:CDD \| Template:CDD	[2⁺[6,3,6]]⁺	(1)	Template:CDD

Tridental graphs

Paracompact hyperbolic enumeration
Group	Extended symmetry	Honeycombs		Chiral extended symmetry	Alternation honeycombs
${\bar{D V}}_{3}$ [6,3^1,1] Template:CDD	[6,3^1,1]	4	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
	[1[6,3^1,1]]=[6,3,4] Template:CDD = Template:CDD	(7)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1[1⁺,6,3^1,1]]⁺	(2)	Template:CDD (= Template:CDD) Template:CDD
	[1[6,3^1,1]]=[6,3,4] Template:CDD = Template:CDD	(7)		[1[6,3^1,1]]⁺=[6,3,4]⁺	(1)	Template:CDD
${\bar{O}}_{3}$ [3,4^1,1] Template:CDD	[3,4^1,1]	4	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[3⁺,4^1,1]⁺	(2)	Template:CDD (= Template:CDD) Template:CDD
	[1[3,4^1,1]]=[3,4,4] Template:CDD = Template:CDD	(7)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1[3⁺,4^1,1]]⁺	(2)	Template:CDD \| Template:CDD
	[1[3,4^1,1]]=[3,4,4] Template:CDD = Template:CDD	(7)		[1[3,4^1,1]]⁺	(1)	Template:CDD
${\bar{M}}_{3}$ [4^1,1,1] Template:CDD	[4^1,1,1]	0	(none)
	[1[4^1,1,1]]=[4,4,4] Template:CDD = Template:CDD	(4)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1[1⁺,4,1⁺,4^1,1]]⁺=[(4,1⁺,4,1⁺,4,2⁺)]	(4)	Template:CDD (= Template:CDD) Template:CDD \| Template:CDD \| Template:CDD
	[3[4^1,1,1]]=[4,4,3] Template:CDD = Template:CDD	(3)	Template:CDD \| Template:CDD \| Template:CDD	[3[1⁺,4^1,1,1]]⁺=[1⁺,4,1⁺,4,3⁺]	(2)	Template:CDD (= Template:CDD) Template:CDD
	[3[4^1,1,1]]=[4,4,3] Template:CDD = Template:CDD	(3)	Template:CDD \| Template:CDD \| Template:CDD	[3[4^1,1,1]]⁺=[4,4,3]⁺	(1)	Template:CDD

Cyclic graphs

Paracompact hyperbolic enumeration
Group	Extended symmetry	Honeycombs		Chiral extended symmetry	Alternation honeycombs
${\hat{C R}}_{3}$ [(4,4,4,3)] Template:CDD	[(4,4,4,3)]	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[(4,1⁺,4,1⁺,4,3⁺)]	(2)	Template:CDD (= Template:CDD) Template:CDD
	[2⁺[(4,4,4,3)]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(4,4⁺,4,3⁺)]]	(2)	Template:CDD \| Template:CDD
	[2⁺[(4,4,4,3)]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(4,4,4,3)]]⁺	(1)	Template:CDD
${\hat{R R}}_{3}$ [4^[4]] Template:CDD	[4^[4]]	(none)
	[2⁺[4^[4]]] Template:CDD	1	Template:CDD	[2⁺[(4⁺,4)^[2]]]	(1)	Template:CDD
	[1[4^[4]]]=[4,4^1,1] Template:CDD = Template:CDD	(2)	Template:CDD Template:CDD	[(1⁺,4)^[4]]	(2)	Template:CDD (= Template:CDD) Template:CDD
	[2[4^[4]]]=[4,4,4] Template:CDD = Template:CDD	(1)	Template:CDD	[2⁺[(1⁺,4,4)^[2]]]	(1)	Template:CDD
	[(2⁺,4)[4^[4]]]=[2⁺[4,4,4]] Template:CDD = Template:CDD	(1)	Template:CDD	[(2⁺,4)[4^[4]]]⁺ = [2⁺[4,4,4]]⁺	(1)	Template:CDD
${\hat{A V}}_{3}$ [(6,3,3,3)] Template:CDD	[(6,3,3,3)]	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
${\hat{A V}}_{3}$ [(6,3,3,3)] Template:CDD	[2⁺[(6,3,3,3)]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(6,3,3,3)]]⁺	(1)	Template:CDD
${\hat{B V}}_{3}$ [(3,4,3,6)] Template:CDD	[(3,4,3,6)]	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[(3⁺,4,3⁺,6)]	(1)	Template:CDD
${\hat{B V}}_{3}$ [(3,4,3,6)] Template:CDD	[2⁺[(3,4,3,6)]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(3,4,3,6)]]⁺	(1)	Template:CDD
${\hat{H V}}_{3}$ [(3,5,3,6)] Template:CDD	[(3,5,3,6)]	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
${\hat{H V}}_{3}$ [(3,5,3,6)] Template:CDD	[2⁺[(3,5,3,6)]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(3,5,3,6)]]⁺	(1)	Template:CDD
${\hat{V V}}_{3}$ [(3,6)^[2]] Template:CDD	[(3,6)^[2]]	2	Template:CDD \| Template:CDD
	[2⁺[(3,6)^[2]]] Template:CDD	1	Template:CDD
	[2⁺[(3,6)^[2]]] Template:CDD	1	Template:CDD
	[2⁺[(3,6)^[2]]] Template:CDD = Template:CDD	(1)	Template:CDD	[2⁺[(3⁺,6)^[2]]]	(1)	Template:CDD
	[(2,2)⁺[(3,6)^[2]]] Template:CDD	1	Template:CDD	[(2,2)⁺[(3,6)^[2]]]⁺	(1)	Template:CDD

Paracompact hyperbolic enumeration
Group	Extended symmetry	Honeycombs		Chiral extended symmetry	Alternation honeycombs
${\hat{B R}}_{3}$ [(3,3,4,4)] Template:CDD	[(3,3,4,4)]	4	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
	[1[(4,4,3,3)]]=[3,4^1,1] Template:CDD = Template:CDD	(7)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1[(3,3,4,1⁺,4)]]⁺ = [3⁺,4^1,1]⁺	(2)	Template:CDD (= Template:CDD) Template:CDD
	[1[(4,4,3,3)]]=[3,4^1,1] Template:CDD = Template:CDD	(7)		[1[(3,3,4,4)]]⁺ = [3,4^1,1]⁺	(1)	Template:CDD
${\bar{D P}}_{3}$ [3^{[ ]x[ ]}] Template:CDD	[3^{[ ]x[ ]}]	1	Template:CDD
	[1[3^{[ ]x[ ]}]]=[6,3^1,1] Template:CDD = Template:CDD	(2)	Template:CDD \| Template:CDD
	[1[3^{[ ]x[ ]}]]=[4,3^[3]] Template:CDD = Template:CDD	(2)	Template:CDD \| Template:CDD
	[2[3^{[ ]x[ ]}]]=[6,3,4] Template:CDD = Template:CDD	(3)	Template:CDD \| Template:CDD \| Template:CDD	[2[3^{[ ]x[ ]}]]⁺ =[6,3,4]⁺	(1)	Template:CDD
${\bar{P P}}_{3}$ [3^[3,3]] Template:CDD Template:CDD	[3^[3,3]]	0	(none)
	[1[3^[3,3]]]=[6,3^[3]] Template:CDD = Template:CDD	0	(none)
	[3[3^[3,3]]]=[3,6,3] Template:CDD = Template:CDD	(2)	Template:CDD \| Template:CDD
	[2[3^[3,3]]]=[6,3,6] Template:CDD = Template:CDD	(1)	Template:CDD
	[(3,3)[3^[3,3]]]=[6,3,3] Template:CDD = Template:CDD	(1)	Template:CDD	[(3,3)[3^[3,3]]]⁺ = [6,3,3]⁺	(1)	Template:CDD

Loop-n-tail graphs

Symmetry in these graphs can be doubled by adding a mirror: [1[n,3^[3]]] = [n,3,6]. Therefore ring-symmetry graphs are repeated in the linear graph families.

Paracompact hyperbolic enumeration
Group	Extended symmetry	Honeycombs		Chiral extended symmetry	Alternation honeycombs
${\bar{P}}_{3}$ [3,3^[3]] Template:CDD	[3,3^[3]]	4	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
${\bar{P}}_{3}$ [3,3^[3]] Template:CDD	[1[3,3^[3]]]=[3,3,6] Template:CDD = Template:CDD	(7)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1[3,3^[3]]]⁺ = [3,3,6]⁺	(1)	Template:CDD
${\bar{B P}}_{3}$ [4,3^[3]] Template:CDD	[4,3^[3]]	4	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
	[1[4,3^[3]]]=[4,3,6] Template:CDD = Template:CDD	(7)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1⁺,4,(3^[3])⁺]	(2)	Template:CDD (= Template:CDD) Template:CDD
	[1[4,3^[3]]]=[4,3,6] Template:CDD = Template:CDD	(7)		[4,3^[3]]⁺	(1)	Template:CDD
${\bar{H P}}_{3}$ [5,3^[3]] Template:CDD	[5,3^[3]]	4	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
${\bar{H P}}_{3}$ [5,3^[3]] Template:CDD	[1[5,3^[3]]]=[5,3,6] Template:CDD = Template:CDD	(7)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1[5,3^[3]]]⁺ = [5,3,6]⁺	(1)	Template:CDD
${\bar{V P}}_{3}$ [6,3^[3]] Template:CDD	[6,3^[3]]	2	Template:CDD \| Template:CDD
	[6,3^[3]]=(?)	(2)	(Template:CDD = Template:CDD) \| (Template:CDD = Template:CDD)
	[1[6,3^[3]]]=[6,3,6] Template:CDD = Template:CDD	(7)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[3[1⁺,6,3^[3]]]⁺ = [3,6,3]⁺	(1)	Template:CDD (= Template:CDD = Template:CDD )
				[1[1⁺,6,3^[3]]]⁺	(1)	Template:CDD
				[1[6,3^[3]]]⁺ = [6,3,6]⁺	(1)	Template:CDD

[6,3,3] family

#	Honeycomb name Coxeter diagram: Template:CDD Schläfli symbol	Cells by location (and count around each vertex)					Vertex figure	Picture
#		1 Template:CDD	2 Template:CDD	3 Template:CDD	4 Template:CDD	Alt	Vertex figure	Picture
1	Hexagonal tiling Regular Template:CDD {6,3,3}	-	-	-	(4) (6.6.6)		Template:CDD Tetrahedron
2	Rectified hexagonal tiling Template:CDD t₁{6,3,3}	(2) (3.3.3)	-	-	(3) (3.6.3.6)		Template:CDD Triangular prism
3	Rectified order-6 tetrahedral Template:CDD t₁{3,3,6}	(6) (3.3.3.3)	-	-	(2) (3.3.3.3.3.3)		Template:CDD Hexagonal prism
4	Order-6 tetrahedral Regular Template:CDD {3,3,6}	(∞) (3.3.3)	-	-	-		Template:CDD Triangular tiling
5	Truncated hexagonal tiling Template:CDD t_0,1{6,3,3}	(1) (3.3.3)	-	-	(3) (3.12.12)		Triangular pyramid
6	Cantellated hexagonal tiling Template:CDD t_0,2{6,3,3}	(1) 3.3.3.3	(2) (4.4.3)	-	(2) (3.4.6.4)
7	Runcinated hexagonal tiling Template:CDD t_0,3{6,3,3}	(1) (3.3.3)	(3) (4.4.3)	(3) (4.4.6)	(1) (6.6.6)
8	Cantellated order-6 tetrahedral Template:CDD t_0,2{3,3,6}	(1) (3.4.3.4)	-	(2) (4.4.6)	(2) (3.6.3.6)
9	Bitruncated hexagonal tiling Template:CDD t_1,2{6,3,3}	(2) (3.6.6)	-	-	(2) File:Uniform tiling 63-t12.png (6.6.6)		File:Bitruncated order-3 hexagonal tiling honeycomb verf.png
10	Truncated order-6 tetrahedral Template:CDD t_0,1{3,3,6}	(6) File:Uniform polyhedron-33-t12.png (3.6.6)	-	-	(1) (3.3.3.3.3.3)		File:Truncated order-6 tetrahedral honeycomb verf.png
11	Cantitruncated hexagonal tiling Template:CDD t_0,1,2{6,3,3}	(1) (3.6.6)	(1) (4.4.3)	-	(2) File:Uniform tiling 63-t012.png (4.6.12)		File:Cantitruncated order-3 hexagonal tiling honeycomb verf.png
12	Runcitruncated hexagonal tiling Template:CDD t_0,1,3{6,3,3}	(1) (3.4.3.4)	(2) (4.4.3)	(1) File:Dodecagonal prism.png (4.4.12)	(1) (3.12.12)		File:Runcitruncated order-3 hexagonal tiling honeycomb verf.png
13	Runcitruncated order-6 tetrahedral Template:CDD t_0,1,3{3,3,6}	(1) File:Uniform polyhedron-33-t12.png (3.6.6)	(1) (4.4.6)	(2) (4.4.6)	(1) (3.4.6.4)		File:Runcitruncated order-6 tetrahedral honeycomb verf.png
14	Cantitruncated order-6 tetrahedral Template:CDD t_0,1,2{3,3,6}	(2) File:Uniform polyhedron-33-t012.png (4.6.6)	-	(1) (4.4.6)	(1) File:Uniform tiling 63-t12.png (6.6.6)		File:Cantitruncated order-6 tetrahedral honeycomb verf.png
15	Omnitruncated hexagonal tiling Template:CDD t_0,1,2,3{6,3,3}	(1) File:Uniform polyhedron-33-t012.png (4.6.6)	(1) (4.4.6)	(1) File:Dodecagonal prism.png (4.4.12)	(1) File:Uniform tiling 63-t012.png (4.6.12)		File:Omnitruncated order-3 hexagonal tiling honeycomb verf.png
[?]	Alternated hexagonal tiling Template:CDD = Template:CDD h{6,3,3}				File:Uniform tiling 333-t1.png	Error creating thumbnail: (3.3.3)
Nonuniform	Alternated cantitruncated order-6 tetrahedral Template:CDD sr{3,3,6}	File:Uniform polyhedron-33-s012.png			File:Uniform tiling 63-h12.png	Error creating thumbnail: Irr. (3.3.3)
Nonuniform	Full snub hexagonal tiling Template:CDD ht_0,1,2,3{6,3,3}	File:Uniform polyhedron-33-s012.png			File:Uniform tiling 63-snub.png	Error creating thumbnail: Irr. (3.3.3)

[6,3,4] family

There are 15 forms, generated by ring permutations of the Coxeter group: [6,3,4] or Template:CDD

#	Name of honeycomb Coxeter diagram Schläfli symbol	Cells by location and count per vertex					Vertex figure	Picture
#	Name of honeycomb Coxeter diagram Schläfli symbol	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
16	(Regular) order-4 hexagonal tiling Template:CDD {6,3,4}	-	-	-	(8) Template:CDD (6.6.6)		Error creating thumbnail: Template:CDD (3.3.3.3)	Error creating thumbnail:
17	Rectified order-4 hexagonal tiling Template:CDD t₁{6,3,4}	(2) Template:CDD Error creating thumbnail: (3.3.3.3)	-	-	(4) Template:CDD (3.6.3.6)		Error creating thumbnail: Template:CDD (4.4.4)
18	Rectified order-6 cubic Template:CDD t₁{4,3,6}	(6) Template:CDD Error creating thumbnail: (3.4.3.4)	-	-	(2) Template:CDD (3.3.3.3.3.3)		Template:CDD (6.4.4)
19	(Regular) order-6 cubic Template:CDD {4,3,6}	(20) Template:CDD Error creating thumbnail: (4.4.4)	-	-	-		Template:CDD (3.3.3.3.3.3)	Error creating thumbnail:
20	Truncated order-4 hexagonal tiling Template:CDD t_0,1{6,3,4}	(1) Template:CDD Error creating thumbnail: (3.3.3.3)	-	-	(4) Template:CDD (3.12.12)
21	Bitruncated order-6 cubic Template:CDD t_1,2{6,3,4}	(2) Template:CDD Error creating thumbnail: (4.6.6)	-	-	(2) Template:CDD File:Uniform tiling 63-t12.png (6.6.6)
22	Truncated order-6 cubic Template:CDD t_0,1{4,3,6}	(6) Template:CDD File:Truncated hexahedron.png (3.8.8)	-	-	(1) Template:CDD (3.3.3.3.3.3)
23	Cantellated order-4 hexagonal tiling Template:CDD t_0,2{6,3,4}	(1) Template:CDD Error creating thumbnail: (3.4.3.4)	(2) Template:CDD File:Tetragonal prism.png (4.4.4)	-	(2) Template:CDD (3.4.6.4)
24	Cantellated order-6 cubic Template:CDD t_0,2{4,3,6}	(2) Template:CDD File:Small rhombicuboctahedron.png (3.4.4.4)	-	(2) Template:CDD (4.4.6)	(1) Template:CDD (3.6.3.6)
25	Runcinated order-6 cubic Template:CDD t_0,3{6,3,4}	(1) Template:CDD Error creating thumbnail: (4.4.4)	(3) Template:CDD File:Tetragonal prism.png (4.4.4)	(3) Template:CDD (4.4.6)	(1) Template:CDD (6.6.6)
26	Cantitruncated order-4 hexagonal tiling Template:CDD t_0,1,2{6,3,4}	(1) Template:CDD Error creating thumbnail: (4.6.6)	(1) Template:CDD File:Tetragonal prism.png (4.4.4)	-	(2) Template:CDD File:Uniform tiling 63-t012.png (4.6.12)
27	Cantitruncated order-6 cubic Template:CDD t_0,1,2{4,3,6}	(2) Template:CDD File:Great rhombicuboctahedron.png (4.6.8)	-	(1) Template:CDD (4.4.6)	(1) Template:CDD File:Uniform tiling 63-t12.png (6.6.6)
28	Runcitruncated order-4 hexagonal tiling Template:CDD t_0,1,3{6,3,4}	(1) Template:CDD File:Small rhombicuboctahedron.png (3.4.4.4)	(1) Template:CDD File:Tetragonal prism.png (4.4.4)	(2) Template:CDD File:Dodecagonal prism.png (4.4.12)	(1) Template:CDD (3.12.12)
29	Runcitruncated order-6 cubic Template:CDD t_0,1,3{4,3,6}	(1) Template:CDD File:Truncated hexahedron.png (3.8.8)	(2) Template:CDD Error creating thumbnail: (4.4.8)	(1) Template:CDD (4.4.6)	(1) Template:CDD (3.4.6.4)
30	Omnitruncated order-6 cubic Template:CDD t_0,1,2,3{6,3,4}	(1) Template:CDD File:Great rhombicuboctahedron.png (4.6.8)	(1) Template:CDD Error creating thumbnail: (4.4.8)	(1) Template:CDD File:Dodecagonal prism.png (4.4.12)	(1) Template:CDD File:Uniform tiling 63-t012.png (4.6.12)
[?]	alternated order-6 cubic Template:CDD = Template:CDD h{4,3,6}	Template:CDD Error creating thumbnail: (3.3.3)				Template:CDD (3.3.3.3.3.3)	Template:CDD (3.6.3.6)
[?]	Template:CDD = Template:CDD h{6,3,4}
Nonuniform	Alternated cantitruncated order-6 cubic honeycomb Template:CDD sr{4,3,6}
Nonuniform	Alternated cantitruncated order-4 hexagonal tiling honeycomb Template:CDD sr{6,3,4}	Template:CDD File:Uniform polyhedron-43-h01.svg (3.3.3.3.3.3)	Template:CDD Error creating thumbnail: (3.3.3)	-	Template:CDD File:Uniform tiling 63-snub.png (3.3.3.3.6)	Error creating thumbnail: +(3.3.3)
Nonuniform	Full snub order-6 cubic Template:CDD ht_0,1,2,3{6,3,4}	Template:CDD File:Snub hexahedron.png (3.3.3.3.4)	Template:CDD Error creating thumbnail: (3.3.3.4)	Template:CDD File:Hexagonal antiprism.png (3.3.3.6)	Template:CDD File:Uniform tiling 63-snub.png (3.3.3.3.6)	Error creating thumbnail: +(3.3.3)

[6,3,5] family

#	Honeycomb name Coxeter diagram Schläfli symbol	Cells by location (and count around each vertex)					Vertex figure	Picture
#	Honeycomb name Coxeter diagram Schläfli symbol	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
31	Order-5 hexagonal tiling Template:CDD {6,3,5}	-	-	-	Error creating thumbnail: (6)³		Error creating thumbnail: Template:CDD Icosahedron	Error creating thumbnail:
32	Recified order-5 hexagonal tiling Template:CDD t₁{6,3,5}	Error creating thumbnail: (3.3.3.3.3)	-	-	Error creating thumbnail: (3.6)²		Error creating thumbnail: Template:CDD (5.4.4)
33	Rectified order-6 dodecahedral Template:CDD t₁{5,3,6}	(3.5.3.5)	-	-	Error creating thumbnail: (3)⁶		Template:CDD (6.4.4)
34	Order-6 dodecahedral Template:CDD {5,3,6}	Error creating thumbnail: (5.5.5)	-	-	-		Error creating thumbnail: Template:CDD (3)⁶	Error creating thumbnail:
35	Truncated order-6 dodecahedral Template:CDD t_0,1{6,3,5}	Error creating thumbnail: (3.3.3.3.3)	-	-	Error creating thumbnail: 3.12.12
36	Cantellated order-6 dodecahedral Template:CDD t_0,2{6,3,5}	(3.5.3.5)	Error creating thumbnail: (5.4.4)	-	Error creating thumbnail: 3.4.6.4
37	Runcinated order-6 dodecahedral Template:CDD t_0,3{6,3,5}	Error creating thumbnail: (5.5.5)	-	(6.4.4)	Error creating thumbnail: (6)³
38	Template:CDD t_0,2{5,3,6}	File:Uniform polyhedron-53-t02.png (4.3.4.5)	-	(6.4.4)	Error creating thumbnail: (3.6)²
39	Template:CDD t_1,2{6,3,5}	File:Uniform polyhedron-53-t12.png (5.6.6)	-	(6.4.4)	File:Uniform tiling 63-t12.png (6)³
40	Template:CDD t_0,1{5,3,6}	File:Uniform polyhedron-53-t01.png (3.10.10)	-	-	Error creating thumbnail: (3)⁶
41	Template:CDD t_0,1,2{6,3,5}	File:Uniform polyhedron-53-t12.png (5.6.6)	Error creating thumbnail: (5.4.4)	-	File:Uniform tiling 63-t012.png 4.6.10
42	Template:CDD t_0,1,3{6,3,5}	File:Uniform polyhedron-53-t02.png (4.3.4.5)	Error creating thumbnail: (5.4.4)	File:Dodecagonal prism.png (12.4.4)	Error creating thumbnail: 3.12.12
43	Template:CDD t_0,1,3{5,3,6}	File:Uniform polyhedron-53-t01.png (3.10.10)	File:Decagonal prism.png (10.4.4)	(6.4.4)	Error creating thumbnail: 3.4.6.4
44	Template:CDD t_0,1,2{5,3,6}	File:Uniform polyhedron-53-t012.png (4.6.10)	-	(6.4.4)	File:Uniform tiling 63-t12.png (6)³
45	Template:CDD t_0,1,2,3{6,3,5}	File:Uniform polyhedron-53-t012.png (4.6.10)	File:Decagonal prism.png (10.4.4)	File:Dodecagonal prism.png (12.4.4)	File:Uniform tiling 63-t012.png 4.6.12
[?]	Alternated order-5 hexagonal tiling honeycomb Template:CDD = Template:CDD h{6,3,5}	-	-	-	File:Uniform tiling 333-t1.png (3)⁶	Error creating thumbnail: (3)⁵
Nonuniform	Snub rectified order-6 dodecahedral honeycomb Template:CDD sr{5,3,6}
Nonuniform	Full snub order-5 hexagonal tiling honeycomb Template:CDD ht_0,1,2,3{6,3,5}

[6,3,6] family

There are 9 forms, generated by ring permutations of the Coxeter group: [6,3,6] or Template:CDD

#	Name of honeycomb Coxeter diagram Schläfli symbol	Cells by location and count per vertex					Vertex figure	Picture
#	Name of honeycomb Coxeter diagram Schläfli symbol	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
46	(Regular) Order-6 hexagonal tiling Template:CDD t₀{6,3,6}	-	-	-	(20) Template:CDD (6.6.6)		Template:CDD (3.3.3.3.3.3)	Error creating thumbnail:
47	rectified order-6 hexagonal tiling Template:CDD t₁{6,3,6}	(2) Template:CDD (3.3.3.3.3.3)	-	-	(6) (3.6.3.6)		Template:CDD (6.4.4)
48	truncated order-6 hexagonal tiling Template:CDD t_0,1{6,3,6}	(1) (3.3.3.3.3.3)	-	-	(6) (3.12.12)
49	cantellated order-6 hexagonal tiling Template:CDD t_0,2{6,3,6}	(1) (3.6.3.6)	(2) (4.4.6)	-	(2) File:Uniform tiling 63-t012.png (3.6.4.6)
50	Runcinated order-6 hexagonal tiling Template:CDD t_0,3{6,3,6}	(1) (6.6.6)	(3) (4.4.6)	(3) (4.4.6)	(1) (6.6.6)
[1]	bitruncated order-6 hexagonal tiling Template:CDD = Template:CDD t_1,2{6,3,6}	(2) File:Uniform tiling 63-t12.png (6.6.6)	-	-	(2) File:Uniform tiling 63-t12.png (6.6.6)			Error creating thumbnail:
51	cantitruncated order-6 hexagonal tiling Template:CDD t_0,1,2{6,3,6}	(1) File:Uniform tiling 63-t12.png (6.6.6)	(1) (4.4.6)	-	(2) File:Uniform tiling 63-t012.png (4.6.12)
52	runcitruncated order-6 hexagonal tiling Template:CDD t_0,1,3{6,3,6}	(1) File:Uniform tiling 63-t012.png (3.6.4.6)	(1) (4.4.6)	(2) File:Decagonal prism.png (4.4.12)	(1) (3.12.12)
53	omnitruncated order-6 hexagonal tiling Template:CDD t_0,1,2,3{6,3,6}	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Decagonal prism.png (4.4.12)	(1) File:Decagonal prism.png (4.4.12)	(1) File:Uniform tiling 63-t012.png (4.6.12)
[?]	Half order-6 hexagonal tiling Template:CDD = Template:CDD h{6,3,6}
Nonuniform	Template:CDD h2t{6,3,6}
Nonuniform	Template:CDD sr{6,3,6}
	Template:CDD ht_0,3{6,3,6}
	Quarter order-6 hexagonal tiling Template:CDD ht₀ht₃{6,3,6}
Nonuniform	Full snub order-6 hexagonal tiling Template:CDD ht_0,1,2,3{6,3,6}	Template:CDD File:Uniform tiling 63-snub.png (3.3.3.3.6)	Template:CDD File:Hexagonal antiprism.png (3.3.3.6)	Template:CDD File:Hexagonal antiprism.png (3.3.3.6)	Template:CDD File:Uniform tiling 63-snub.png (3.3.3.3.6)	Error creating thumbnail: +(3.3.3)

[3,6,3] family

There are 9 forms, generated by ring permutations of the Coxeter group: [3,6,3] or Template:CDD

#	Honeycomb name Coxeter diagram and Schläfli symbol	Cell counts/vertex and positions in honeycomb					Vertex figure	Picture
#	Honeycomb name Coxeter diagram and Schläfli symbol	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
54	(Regular) triangular tiling Template:CDD {3,6,3}	-	-	-	(∞) (3)⁶		Template:CDD (6)³	Error creating thumbnail:
55	rectified triangular tiling Template:CDD t₁{3,6,3}	(2) (6)³	-	-	(3) (3.6)²		Template:CDD (3.4.4)
[1]	truncated triangular tiling Template:CDD t_0,1{3,6,3}	(1) (6)³	-	-	(3) File:Uniform tiling 63-t12.png (6)³
56	cantellated triangular tiling Template:CDD t_0,2{3,6,3}	(1) (3.6)²	(2) (4.4.3)	-	(2) (3.6.4.6)
57	Runcinated triangular tiling Template:CDD t_0,3{3,6,3}	(1) (3)⁶	(6) (4.4.3)	(6) (4.4.3)	(1) (3)⁶
58	bitruncated triangular tiling Template:CDD t_1,2{3,6,3}	(2) (3.12.12)	-	-	(2) (3.12.12)
59	cantitruncated triangular tiling Template:CDD t_0,1,2{3,6,3}	(1) (3.12.12)	(1) (4.4.3)	-	(2) File:Uniform tiling 63-t012.png (4.6.12)
60	runcitruncated triangular tiling Template:CDD t_0,1,3{3,6,3}	(1) (3.6.4.6)	(1) (4.4.3)	(2) (4.4.6)	(1) (6)³
61	omnitruncated triangular tiling Template:CDD t_0,1,2,3{3,6,3}	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) (4.4.6)	(1) (4.4.6)	(1) File:Uniform tiling 63-t012.png (4.6.12)
Nonuniform	Alternated truncated triangular tiling Template:CDD s{3,6,3}	File:Uniform tiling 333-t1.png (3)⁶	-	-	File:Uniform tiling 63-h12.png (3)⁶	Error creating thumbnail: +(3)³
Nonuniform	Template:CDD s₃{3,6,3}
Nonuniform	Full snub triangular tiling Template:CDD ht_0,1,2,3{3,6,3}	File:Uniform tiling 63-snub.png (3.3.3.3.6)	Error creating thumbnail: (3)⁴	Error creating thumbnail: (3)⁴	File:Uniform tiling 63-snub.png (3.3.3.3.6)	Error creating thumbnail: +(3)³

[4,4,3] family

There are 15 forms, generated by ring permutations of the Coxeter group: [4,4,3] or Template:CDD

#	Honeycomb name Coxeter diagram and Schläfli symbol	Cell counts/vertex and positions in honeycomb					Vertex figure	Picture
#	Honeycomb name Coxeter diagram and Schläfli symbol	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
62	Square tiling honeycomb Template:CDD {4,4,3}	-	-	-	Template:CDD Error creating thumbnail:		Template:CDD Cube	Error creating thumbnail:
63	Template:CDD t₁{4,4,3}	Template:CDD	-	-	Template:CDD Error creating thumbnail:		Template:CDD Triangular prism
64	Template:CDD t₁{3,4,4}	Template:CDD	-	-	Template:CDD Error creating thumbnail:
65	Order-4 octahedral honeycomb Template:CDD {3,4,4}	Template:CDD Error creating thumbnail:	-	-	-			Error creating thumbnail:
66	Template:CDD t_0,1{4,4,3}	Template:CDD	-	-	Template:CDD File:Uniform tiling 44-t01.png
67	Template:CDD t_0,1{3,4,4}	Template:CDD File:Uniform polyhedron-43-t12.png	-	-	Template:CDD Error creating thumbnail:
68	Template:CDD t_1,2{4,4,3}	Template:CDD File:Uniform polyhedron-43-t01.png	-	-	Template:CDD File:Uniform tiling 44-t12.png
69	Template:CDD t_0,2{4,4,3}	Template:CDD	Template:CDD	-	Template:CDD File:Uniform tiling 44-t02.png
70	Template:CDD t_0,2{3,4,4}	Template:CDD	-	Template:CDD File:Tetragonal prism.png	Template:CDD Error creating thumbnail:
71	Template:CDD t_0,3{4,4,3}	Template:CDD Error creating thumbnail:	Template:CDD	Template:CDD File:Tetragonal prism.png	Template:CDD Error creating thumbnail:
72	Template:CDD t_0,1,2{4,4,3}	Template:CDD File:Uniform polyhedron-43-t01.png	Template:CDD	-	Template:CDD File:Uniform tiling 44-t012.png
73	Template:CDD t_0,1,2{3,4,4}	Template:CDD File:Uniform polyhedron-43-t012.png	-	Template:CDD File:Tetragonal prism.png	Template:CDD File:Uniform tiling 44-t12.png
74	Template:CDD t_0,1,3{4,4,3}	Template:CDD	Template:CDD	Template:CDD Error creating thumbnail:	Template:CDD File:Uniform tiling 44-t01.png
75	Template:CDD t_0,1,3{3,4,4}	Template:CDD File:Uniform polyhedron-43-t12.png	Template:CDD	Template:CDD File:Tetragonal prism.png	Template:CDD File:Uniform tiling 44-t02.png
76	Template:CDD t_0,1,2,3{4,4,3}	Template:CDD File:Uniform polyhedron-43-t012.png	Template:CDD	Template:CDD Error creating thumbnail:	Template:CDD File:Uniform tiling 44-t012.png
[?]	Template:CDD = Template:CDD h{4,4,3}	-	-	-	Template:CDD
[?]	Template:CDD = Template:CDD hr{4,4,3}	Template:CDD	-	-	Template:CDD
Nonuniform	Template:CDD s{4,4,3}	Template:CDD	-	-	Template:CDD
Nonuniform	Template:CDD s{3,4,4}	Template:CDD	-	-	Template:CDD
Nonuniform	Template:CDD sr{3,4,4}	Template:CDD	-	Template:CDD	Template:CDD
Nonuniform	Template:CDD ht_0,1,3{3,4,4}	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Nonuniform	Template:CDD ht_0,1,2,3{4,4,3}	Template:CDD	Template:CDD	Template:CDD	Template:CDD

[4,4,4] family

There are 9 forms, generated by ring permutations of the Coxeter group: [4,4,4] or Template:CDD

#	Honeycomb name Coxeter diagram and Schläfli symbol	Cell counts/vertex and positions in honeycomb					Vertex figure	Picture
#	Honeycomb name Coxeter diagram and Schläfli symbol	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
77	Order-4 square tiling honeycomb Template:CDD {4,4,4}	-	-	-	Template:CDD Error creating thumbnail:		Template:CDD Error creating thumbnail: Cube	Error creating thumbnail:
[62]	Template:CDD = Template:CDD t₁{4,4,4}	Template:CDD Error creating thumbnail:	-	-	Template:CDD Error creating thumbnail:		Template:CDD File:Tetragonal prism.png Square prism	Error creating thumbnail:
78	Template:CDD t_0,1{4,4,4}	Template:CDD Error creating thumbnail:	-	-	Template:CDD File:Uniform tiling 44-t01.png
79	Template:CDD t_1,2{4,4,4}	Template:CDD File:Uniform tiling 44-t01.png	-	-	Template:CDD File:Uniform tiling 44-t12.png
[63]	Template:CDD = Template:CDD t_0,2{4,4,4}	Template:CDD Error creating thumbnail:	Template:CDD File:Tetragonal prism.png	-	Template:CDD File:Uniform tiling 44-t02.png
80	Template:CDD t_0,3{4,4,4}	Template:CDD Error creating thumbnail:	Template:CDD File:Tetragonal prism.png	Template:CDD File:Tetragonal prism.png	Template:CDD Error creating thumbnail:
[66]	Template:CDD = Template:CDD t_0,1,2{4,4,4}	Template:CDD File:Uniform tiling 44-t01.png	Template:CDD File:Tetragonal prism.png	-	Template:CDD File:Uniform tiling 44-t012.png
81	Template:CDD t_0,1,3{4,4,4}	Template:CDD File:Uniform tiling 44-t02.png	Template:CDD File:Tetragonal prism.png	Template:CDD Error creating thumbnail:	Template:CDD File:Uniform tiling 44-t01.png
82	Template:CDD t_0,1,2,3{4,4,4}	Template:CDD File:Uniform tiling 44-t012.png	Template:CDD Error creating thumbnail:	Template:CDD Error creating thumbnail:	Template:CDD File:Uniform tiling 44-t012.png
[?]	Template:CDD = Template:CDD h{4,4,4}	-	-	-	Template:CDD Error creating thumbnail:
[?]	Template:CDD = Template:CDD hr{4,4,4}	Template:CDD Error creating thumbnail:	-	-	Template:CDD Error creating thumbnail:
Nonuniform	Template:CDD s{4,4,4}	Template:CDD Error creating thumbnail:	-	-	Template:CDD File:Uniform tiling 44-h01.png
Nonuniform	Template:CDD 2s{4,4,4}	Template:CDD File:Uniform tiling 44-h01.png	-	-	Template:CDD File:Uniform tiling 44-t12.png
Nonuniform	Template:CDD = Template:CDD hrr{4,4,4}	Template:CDD Error creating thumbnail:	Template:CDD Error creating thumbnail:	-	Template:CDD File:Uniform tiling 44-t02.png
Nonuniform	Template:CDD ht_0,3{4,4,4}	Template:CDD Error creating thumbnail:	Template:CDD Error creating thumbnail:	Template:CDD Error creating thumbnail:	Template:CDD Error creating thumbnail:
Nonuniform	Template:CDD = Template:CDD sr{4,4,4}	Template:CDD File:Uniform tiling 44-h01.png	Template:CDD Error creating thumbnail:	-	Template:CDD File:Uniform tiling 44-snub.png
Nonuniform	Template:CDD ht_0,1,3{4,4,4}	Template:CDD File:Uniform tiling 44-t02.png	Template:CDD Error creating thumbnail:	Template:CDD Error creating thumbnail:	Template:CDD File:Uniform tiling 44-h01.png
Nonuniform	Template:CDD ht_0,1,2,3{4,4,4}	Template:CDD File:Uniform tiling 44-snub.png	Template:CDD Error creating thumbnail:	Template:CDD Error creating thumbnail:	Template:CDD File:Uniform tiling 44-snub.png

[4,3^1,1] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: Template:CDD

[3,4^1,1] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: Template:CDD

[4,4^1,1] family

There are 7 forms, 0 unique, generated by ring permutations of the Coxeter group: Template:CDD

[6,3^1,1] family

There are 11 forms (and only 4 not shared with [6,3,4] family), generated by ring permutations of the Coxeter group: [6,3^1,1] or Template:CDD

#	Honeycomb name Coxeter–Dynkin diagram	Cells by location (and count around each vertex)					vertex figure
#	Honeycomb name Coxeter–Dynkin diagram	0 Template:CDD	1 Template:CDD	0' Template:CDD	3 Template:CDD	Alt	vertex figure
	alternated order-6 cubic Template:CDD = Template:CDD	-	-	(12) (3.3.3.3.3)	(20) Error creating thumbnail: (3.3.3)
	Cantic order-6 cubic Template:CDD = Template:CDD	(1) (3.6.3.6)	-	(2) File:Uniform tiling 63-t12.png (6.6.6)	(2) Error creating thumbnail: (3.6.6)
	runcic order-6 cubic Template:CDD = Template:CDD	(1) (6.6.6)	-	(3) (3.4.6.4)	(1) Error creating thumbnail: (3.3.3)
	runcicantic order-6 cubic Template:CDD = Template:CDD	(1) (3.12.12)	-	(2) File:Uniform tiling 63-t012.png (4.6.12)	(1) Error creating thumbnail: (3.6.6)
[?]	rectified order-4 hexagonal tiling Template:CDD = Template:CDD	(2) (3.6.3.6)	-	(2) (3.6.3.6)	(2) (3.3.3.3)
[?]	rectified order-6 cubic Template:CDD = Template:CDD	(1) (3.3.3.3.3)	-	(1) (3.3.3.3.3)	(6) (3.4.3.4)
[?]	bitruncated order-6 cubic Template:CDD = Template:CDD	(1) File:Uniform tiling 63-t12.png (6.6.6)	-	(1) File:Uniform tiling 63-t12.png (6.6.6)	(2) File:Uniform polyhedron-33-t012.png (4.6.6)
[?]	truncated order-4 hexagonal tiling Template:CDD = Template:CDD	(2) (3.12.12)	-	(2) (3.12.12)	(1) (3.3.3.3)
[?]	Order-4 hexagonal tiling Template:CDD = Template:CDD	(4) (6.6.6)	-	(4) (6.6.6)	-
[?]	cantellated order-4 hexagonal tiling Template:CDD = Template:CDD	(1) (3.4.6.4)	(2) File:Uniform polyhedron 222-t012.png (4.4.4)	(1) (3.4.6.4)	(1) (3.4.3.4)
[?]	cantitruncated order-4 hexagonal tiling Template:CDD = Template:CDD	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Uniform polyhedron 222-t012.png (4.4.4)	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Uniform polyhedron-33-t012.png (4.6.6)
Nonuniform	Alternated cantitruncated order-4 hexagonal tiling Template:CDD = Template:CDD	File:Uniform tiling 63-snub.png (3.3.3.3.6)	(3.3.3)	File:Uniform tiling 63-snub.png (3.3.3.3.6)	File:Uniform polyhedron-33-s012.png (3.3.3.3.3)	+(3.3.3)
[?]	Template:CDD = Template:CDD = Template:CDD
Nonuniform	Template:CDD = Template:CDD

[(4,4,3,3)] family

There are 11 forms, 4 unique to this family, generated by ring permutations of the Coxeter group: Template:CDD

[(4,4,4,3)] family

There are 9 forms, generated by ring permutations of the Coxeter group: Template:CDD

[(4,4,4,4)] family

There are 9 forms, 1 unique, generated by ring permutations of the Coxeter group: Template:CDD

[(6,3,3,3)] family

There are 9 forms, generated by ring permutations of the Coxeter group: Template:CDD

[(6,3,4,3)] family

There are 9 forms, generated by ring permutations of the Coxeter group: Template:CDD

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure
#	Honeycomb name Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure
91	Template:CDD	(6) Error creating thumbnail: (3.3.3.3)	-	(8) (6.6.6)	(1) (3.6.3.6)		Template:CDD (3.4.4.4)
92	Template:CDD	(30) Error creating thumbnail: (3.4.3.4)	(20) Error creating thumbnail: (4.4.4)	-	(12) (3.3.3.3.3.3)		Template:CDD (3.4.6.4)
93	Template:CDD	(3) Error creating thumbnail: (4.6.6)	(1) Error creating thumbnail: (4.4.4)	(1) (6.6.6)	(3) File:Uniform tiling 63-t12.png (6.6.6)
94	Template:CDD	(1) Error creating thumbnail: (3.3.3.3)	(1) Error creating thumbnail: (3.3.3.3)	(4) (3.12.12)	(4) (3.12.12)
95	Template:CDD	(6) File:Truncated hexahedron.png (3.8.8)	(6) File:Truncated hexahedron.png (3.8.8)	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)
96	Template:CDD	(1) Error creating thumbnail: (3.4.3.4)	(2) File:Small rhombicuboctahedron.png (3.4.4.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
97	Template:CDD	(1) Error creating thumbnail: (4.6.6)	(1) File:Small rhombicuboctahedron.png (3.4.4.4)	(1) (3.12.12)	(2) File:Uniform tiling 63-t012.png (4.6.12)
98	Template:CDD	(2) File:Great rhombicuboctahedron.png (4.6.8)	(1) File:Truncated hexahedron.png (3.8.8)	(1) (3.4.6.4)	(1) File:Uniform tiling 63-t12.png (6.6.6)
99	Template:CDD	(1) File:Great rhombicuboctahedron.png (4.6.8)	(1) File:Great rhombicuboctahedron.png (4.6.8)	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Uniform tiling 63-t012.png (4.6.12)
Nonuniform	Template:CDD
Nonuniform	Template:CDD	File:Snub hexahedron.png (3.3.3.3.4)	File:Snub hexahedron.png (3.3.3.3.4)	File:Uniform tiling 63-snub.png (3.3.3.3.6)	File:Uniform tiling 63-snub.png (3.3.3.3.6)	Error creating thumbnail: +(3.3.3)

[(6,3,5,3)] family

There are 9 forms, generated by ring permutations of the Coxeter group: Template:CDD

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure
#	Honeycomb name Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure
100	Template:CDD	(6) Error creating thumbnail: (3.3.3.3.3)	-	(8) Error creating thumbnail: (6.6.6)	(1) Error creating thumbnail: (3.6.3.6)		File:Uniform polyhedron-53-t02.png 3.4.5.4
101	Template:CDD	(30) Error creating thumbnail: (3.5.3.5)	(20) Error creating thumbnail: (5.5.5)	-	(12) Error creating thumbnail: (3.3.3.3.3.3)		Error creating thumbnail: (3.4.6.4)
102	Template:CDD	(3) File:Truncated icosahedron.png (5.6.6)	(1) Error creating thumbnail: (5.5.5)	(1) Error creating thumbnail: (6.6.6)	(3) File:Uniform tiling 63-t12.png (6.6.6)
103	Template:CDD	(1) Error creating thumbnail: (3.3.3.3.3)	(1) Error creating thumbnail: (3.3.3.3.3)	(5) Error creating thumbnail: (3.12.12)	(5) Error creating thumbnail: (3.12.12)
104	Template:CDD	(6) File:Truncated dodecahedron.png (3.10.10)	(6) File:Truncated dodecahedron.png (3.10.10)	(1) Error creating thumbnail: (3.3.3.3.3.3)	(1) Error creating thumbnail: (3.3.3.3.3.3)
105	Template:CDD	(1) Error creating thumbnail: (3.5.3.5)	(2) File:Small rhombicosidodecahedron.png (3.4.5.4)	(1) Error creating thumbnail: (3.6.3.6)	(2) Error creating thumbnail: (3.4.6.4)
105	Template:CDD	(1) File:Truncated icosahedron.png (5.6.6)	(1) File:Small rhombicosidodecahedron.png (3.5.5.5)	(1) Error creating thumbnail: (3.12.12)	(2) File:Uniform tiling 63-t012.png (4.6.12)
106	Template:CDD	(2) File:Great rhombicosidodecahedron.png (4.6.10)	(1) File:Truncated dodecahedron.png (3.10.10)	(1) Error creating thumbnail: (3.4.6.4)	(1) File:Uniform tiling 63-t12.png (6.6.6)
107	Template:CDD	(1) File:Great rhombicosidodecahedron.png (4.6.10)	(1) File:Great rhombicosidodecahedron.png (4.6.10)	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Uniform tiling 63-t012.png (4.6.12)
Nonuniform	Template:CDD	File:Snub dodecahedron cw.png (3.3.3.3.5)	File:Snub dodecahedron cw.png (3.3.3.3.5)	File:Uniform tiling 63-snub.png (3.3.3.3.6)	File:Uniform tiling 63-snub.png (3.3.3.3.6)	Error creating thumbnail: +(3.3.3)

[(6,3,6,3)] family

There are 6 forms, generated by ring permutations of the Coxeter group: Template:CDD.

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure	Picture
#	Honeycomb name Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
108	Template:CDD	(12) (3.3.3.3.3.3)	-	(20) (6.6.6)	(30) (3.6.3.6)
[16]	Template:CDD =Template:CDD Order-4 hexagonal tiling honeycomb	(3) File:Uniform tiling 63-t12.png (6.6.6)	(1) (6.6.6)	(1) (6.6.6)	(3) File:Uniform tiling 63-t12.png (6.6.6)		(3.3.3.3)	Error creating thumbnail:
109	Template:CDD	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)	(3) (3.12.12)	(3) (3.12.12)
110	Template:CDD	(1) (3.6.3.6)	(2) (3.4.6.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
111	Template:CDD	(1) File:Uniform tiling 63-t12.png (6.6.6)	(1) (3.4.6.4)	(1) (3.12.12)	(2) File:Uniform tiling 63-t012.png (4.6.12)
112	Template:CDD	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Uniform tiling 63-t012.png (4.6.12)
113	Template:CDD	File:Uniform tiling 63-h12.png (3.3.3.3.3.3)	File:Uniform tiling 333-t1.png (3.3.3.3.3.3)	File:Uniform tiling 333-t1.png (3.3.3.3.3.3)	File:Uniform tiling 63-h12.png (3.3.3.3.3.3)	+(3.3.3.3)	(3.4.3.4)
Nonuniform	Template:CDD	File:Uniform tiling 63-snub.png (3.3.3.3.6)	File:Uniform tiling 63-snub.png (3.3.3.3.6)	File:Uniform tiling 63-snub.png (3.3.3.3.6)	File:Uniform tiling 63-snub.png (3.3.3.3.6)	+(3.3.3)

[3^{[ ]×[ ]}] family

There are 8 forms, 1 unique, generated by ring permutations of the Coxeter group: Template:CDD.

[3^[3,3]] family

There are 4 forms, 0 unique, generated by ring permutations of the Coxeter group: Template:CDD.

[3,3^[3]] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: Template:CDD.

[4,3^[3]] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: Template:CDD.

[5,3^[3]] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: Template:CDD.

[6,3^[3]] family

There are 11 forms, 2 unique, generated by ring permutations of the Coxeter group: Template:CDD.

Notes

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References

James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Template:LCCN, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
Coxeter Decompositions of Hyperbolic Tetrahedra, arXiv/PDF, A. Felikson, December 2002
C. W. L. Garner, Regular Skew Polyhedra in Hyperbolic Three-Space Canad. J. Math. 19, 1179-1186, 1967. PDF [1]
Norman Johnson, Geometries and Transformations, Chapters 11,12,13, preprint 2011
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [2] [3]
N.W. Johnson, R. Kellerhals, J.G. Ratcliffe,S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [4]

↑ P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets (2003)

[1] P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets (2003)

[1]

Étale homotopy type

Contents