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) File:Uniform tiling 63-t1.png (3.6.3.6)		File:Rectified order-3 hexagonal tiling honeycomb verf.png Template:CDD Triangular prism	File:Hyperbolic 3d rectified hexagonal tiling.png
3	Rectified order-6 tetrahedral Template:CDD t₁{3,3,6}	(6) File:Uniform polyhedron-33-t1.png (3.3.3.3)	-	-	(2) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)		File:Rectified order-6 tetrahedral honeycomb verf.png Template:CDD Hexagonal prism	File:Hyperbolic rectified order-6 tetrahedral honeycomb.png
4	Order-6 tetrahedral Regular Template:CDD {3,3,6}	(∞) File:Uniform polyhedron-33-t2.png (3.3.3)	-	-	-		File:Uniform tiling 63-t2.png Template:CDD Triangular tiling	File:H3 336 CC center.png
5	Truncated hexagonal tiling Template:CDD t_0,1{6,3,3}	(1) Error creating thumbnail: (3.3.3)	-	-	(3) File:Uniform tiling 63-t01.png (3.12.12)		File:Truncated order-3 hexagonal tiling honeycomb verf.png Triangular pyramid
6	Cantellated hexagonal tiling Template:CDD t_0,2{6,3,3}	(1) File:Uniform polyhedron-33-t1.png 3.3.3.3	(2) File:Triangular prism.png (4.4.3)	-	(2) File:Uniform tiling 63-t02.png (3.4.6.4)		File:Cantellated order-3 hexagonal tiling honeycomb verf.png
7	Runcinated hexagonal tiling Template:CDD t_0,3{6,3,3}	(1) File:Uniform polyhedron-33-t2.png (3.3.3)	(3) File:Triangular prism.png (4.4.3)	(3) File:Hexagonal prism.png (4.4.6)	(1) Error creating thumbnail: (6.6.6)		File:Runcinated order-3 hexagonal tiling honeycomb verf.png
8	Cantellated order-6 tetrahedral Template:CDD t_0,2{3,3,6}	(1) File:Uniform polyhedron-33-t02.png (3.4.3.4)	-	(2) File:Hexagonal prism.png (4.4.6)	(2) File:Uniform tiling 63-t1.png (3.6.3.6)		File:Cantellated order-6 tetrahedral honeycomb verf.png
9	Bitruncated hexagonal tiling Template:CDD t_1,2{6,3,3}	(2) File:Uniform polyhedron-33-t01.png (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) File:Uniform tiling 63-t2.png (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) File:Uniform polyhedron-33-t01.png (3.6.6)	(1) File:Triangular prism.png (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) File:Uniform polyhedron-33-t02.png (3.4.3.4)	(2) File:Triangular prism.png (4.4.3)	(1) File:Dodecagonal prism.png (4.4.12)	(1) File:Uniform tiling 63-t01.png (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) File:Hexagonal prism.png (4.4.6)	(2) File:Hexagonal prism.png (4.4.6)	(1) File:Uniform tiling 63-t02.png (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) File:Hexagonal prism.png (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) File:Hexagonal prism.png (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	File:Tetrahedron.png (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	File:Tetrahedron.png 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	File:Tetrahedron.png 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 Error creating thumbnail: (6.6.6)		File:Octahedron.png Template:CDD (3.3.3.3)	Error creating thumbnail:
17	Rectified order-4 hexagonal tiling Template:CDD t₁{6,3,4}	(2) Template:CDD File:Octahedron.png (3.3.3.3)	-	-	(4) Template:CDD File:Uniform tiling 63-t1.png (3.6.3.6)		File:Hexahedron.png Template:CDD (4.4.4)
18	Rectified order-6 cubic Template:CDD t₁{4,3,6}	(6) Template:CDD File:Cuboctahedron.png (3.4.3.4)	-	-	(2) Template:CDD File:Uniform tiling 63-t2.png (3.3.3.3.3.3)		File:Hexagonal prism.png Template:CDD (6.4.4)
19	(Regular) order-6 cubic Template:CDD {4,3,6}	(20) Template:CDD File:Hexahedron.png (4.4.4)	-	-	-		File:Uniform tiling 63-t2.png 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 File:Octahedron.png (3.3.3.3)	-	-	(4) Template:CDD File:Uniform tiling 63-t01.png (3.12.12)
21	Bitruncated order-6 cubic Template:CDD t_1,2{6,3,4}	(2) Template:CDD File:Truncated octahedron.png (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 File:Uniform tiling 63-t2.png (3.3.3.3.3.3)
23	Cantellated order-4 hexagonal tiling Template:CDD t_0,2{6,3,4}	(1) Template:CDD File:Cuboctahedron.png (3.4.3.4)	(2) Template:CDD File:Tetragonal prism.png (4.4.4)	-	(2) Template:CDD File:Uniform tiling 63-t02.png (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 File:Hexagonal prism.png (4.4.6)	(1) Template:CDD File:Uniform tiling 63-t1.png (3.6.3.6)
25	Runcinated order-6 cubic Template:CDD t_0,3{6,3,4}	(1) Template:CDD File:Hexahedron.png (4.4.4)	(3) Template:CDD File:Tetragonal prism.png (4.4.4)	(3) Template:CDD File:Hexagonal prism.png (4.4.6)	(1) Template:CDD Error creating thumbnail: (6.6.6)
26	Cantitruncated order-4 hexagonal tiling Template:CDD t_0,1,2{6,3,4}	(1) Template:CDD File:Truncated octahedron.png (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 File:Hexagonal prism.png (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 File:Uniform tiling 63-t01.png (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 File:Octagonal prism.png (4.4.8)	(1) Template:CDD File:Hexagonal prism.png (4.4.6)	(1) Template:CDD File:Uniform tiling 63-t02.png (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 File:Octagonal prism.png (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 File:Tetrahedron.png (3.3.3)				Template:CDD File:Uniform tiling 63-t2.png (3.3.3.3.3.3)	Template:CDD File:Uniform tiling 63-t1.png (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 File:Tetrahedron.png (3.3.3)	-	Template:CDD File:Uniform tiling 63-snub.png (3.3.3.3.6)	File:Tetrahedron.png +(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 File:Square antiprism.png (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)	File:Tetrahedron.png +(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		File:Icosahedron.png Template:CDD Icosahedron	File:H3 635 FC boundary.png
32	Recified order-5 hexagonal tiling Template:CDD t₁{6,3,5}	File:Uniform polyhedron-53-t2.png (3.3.3.3.3)	-	-	File:Uniform tiling 63-t1.png (3.6)²		File:Pentagonal prism.png Template:CDD (5.4.4)
33	Rectified order-6 dodecahedral Template:CDD t₁{5,3,6}	File:Uniform polyhedron-53-t1.png (3.5.3.5)	-	-	File:Uniform tiling 63-t2.png (3)⁶		File:Hexagonal prism.png Template:CDD (6.4.4)
34	Order-6 dodecahedral Template:CDD {5,3,6}	File:Uniform polyhedron-53-t0.png (5.5.5)	-	-	-		File:Uniform tiling 63-t2.png Template:CDD (3)⁶	File:H3 536 CC center.png
35	Truncated order-6 dodecahedral Template:CDD t_0,1{6,3,5}	File:Uniform polyhedron-53-t2.png (3.3.3.3.3)	-	-	File:Uniform tiling 63-t01.png 3.12.12
36	Cantellated order-6 dodecahedral Template:CDD t_0,2{6,3,5}	File:Uniform polyhedron-53-t1.png (3.5.3.5)	File:Pentagonal prism.png (5.4.4)	-	File:Uniform tiling 63-t02.png 3.4.6.4
37	Runcinated order-6 dodecahedral Template:CDD t_0,3{6,3,5}	File:Uniform polyhedron-53-t0.png (5.5.5)	-	File:Hexagonal prism.png (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)	-	File:Hexagonal prism.png (6.4.4)	File:Uniform tiling 63-t1.png (3.6)²
39	Template:CDD t_1,2{6,3,5}	File:Uniform polyhedron-53-t12.png (5.6.6)	-	File:Hexagonal prism.png (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)	-	-	File:Uniform tiling 63-t2.png (3)⁶
41	Template:CDD t_0,1,2{6,3,5}	File:Uniform polyhedron-53-t12.png (5.6.6)	File:Pentagonal prism.png (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)	File:Pentagonal prism.png (5.4.4)	File:Dodecagonal prism.png (12.4.4)	File:Uniform tiling 63-t01.png 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)	File:Hexagonal prism.png (6.4.4)	File:Uniform tiling 63-t02.png 3.4.6.4
44	Template:CDD t_0,1,2{5,3,6}	File:Uniform polyhedron-53-t012.png (4.6.10)	-	File:Hexagonal prism.png (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	File:Icosahedron.png (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) Error creating thumbnail: Template:CDD (6.6.6)		File:Uniform tiling 63-t2.png Template:CDD (3.3.3.3.3.3)	Error creating thumbnail:
47	rectified order-6 hexagonal tiling Template:CDD t₁{6,3,6}	(2) File:Uniform tiling 63-t2.png Template:CDD (3.3.3.3.3.3)	-	-	(6) File:Uniform tiling 63-t1.png (3.6.3.6)		File:Hexagonal prism.png Template:CDD (6.4.4)
48	truncated order-6 hexagonal tiling Template:CDD t_0,1{6,3,6}	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)	-	-	(6) File:Uniform tiling 63-t01.png (3.12.12)
49	cantellated order-6 hexagonal tiling Template:CDD t_0,2{6,3,6}	(1) File:Uniform tiling 63-t1.png (3.6.3.6)	(2) File:Hexagonal prism.png (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) Error creating thumbnail: (6.6.6)	(3) File:Hexagonal prism.png (4.4.6)	(3) File:Hexagonal prism.png (4.4.6)	(1) Error creating thumbnail: (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) File:Hexagonal prism.png (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) File:Hexagonal prism.png (4.4.6)	(2) File:Decagonal prism.png (4.4.12)	(1) File:Uniform tiling 63-t01.png (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)	File:Tetrahedron.png +(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}	-	-	-	(∞) File:Uniform tiling 63-t2.png (3)⁶		Error creating thumbnail: Template:CDD (6)³	Error creating thumbnail:
55	rectified triangular tiling Template:CDD t₁{3,6,3}	(2) Error creating thumbnail: (6)³	-	-	(3) File:Uniform tiling 63-t1.png (3.6)²		File:Triangular prism.png Template:CDD (3.4.4)
[1]	truncated triangular tiling Template:CDD t_0,1{3,6,3}	(1) Error creating thumbnail: (6)³	-	-	(3) File:Uniform tiling 63-t12.png (6)³			Error creating thumbnail:
56	cantellated triangular tiling Template:CDD t_0,2{3,6,3}	(1) File:Uniform tiling 63-t1.png (3.6)²	(2) File:Triangular prism.png (4.4.3)	-	(2) File:Uniform tiling 63-t02.png (3.6.4.6)
57	Runcinated triangular tiling Template:CDD t_0,3{3,6,3}	(1) File:Uniform tiling 63-t2.png (3)⁶	(6) File:Triangular prism.png (4.4.3)	(6) File:Triangular prism.png (4.4.3)	(1) File:Uniform tiling 63-t2.png (3)⁶
58	bitruncated triangular tiling Template:CDD t_1,2{3,6,3}	(2) File:Uniform tiling 63-t01.png (3.12.12)	-	-	(2) File:Uniform tiling 63-t01.png (3.12.12)
59	cantitruncated triangular tiling Template:CDD t_0,1,2{3,6,3}	(1) File:Uniform tiling 63-t01.png (3.12.12)	(1) File:Triangular prism.png (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) File:Uniform tiling 63-t02.png (3.6.4.6)	(1) File:Triangular prism.png (4.4.3)	(2) File:Hexagonal prism.png (4.4.6)	(1) File:Uniform tiling 63-t01.png (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) File:Hexagonal prism.png (4.4.6)	(1) File:Hexagonal prism.png (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	File:Tetrahedron.png +(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)	File:Octahedron.png (3)⁴	File:Octahedron.png (3)⁴	File:Uniform tiling 63-snub.png (3.3.3.3.6)	File:Tetrahedron.png +(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 File:Uniform tiling 44-t0.png		Template:CDD Cube	Error creating thumbnail:
63	Template:CDD t₁{4,4,3}	Template:CDD File:Uniform polyhedron-43-t0.png	-	-	Template:CDD File:Uniform tiling 44-t1.png		Template:CDD File:Triangular prism.png Triangular prism
64	Template:CDD t₁{3,4,4}	Template:CDD File:Uniform polyhedron-43-t1.png	-	-	Template:CDD File:Uniform tiling 44-t2.png
65	Order-4 octahedral honeycomb Template:CDD {3,4,4}	Template:CDD File:Uniform polyhedron-43-t2.png	-	-	-			Error creating thumbnail:
66	Template:CDD t_0,1{4,4,3}	Template:CDD File:Uniform polyhedron-43-t0.png	-	-	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 File:Uniform tiling 44-t2.png
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 File:Uniform polyhedron-43-t1.png	Template:CDD File:Triangular prism.png	-	Template:CDD File:Uniform tiling 44-t02.png
70	Template:CDD t_0,2{3,4,4}	Template:CDD File:Uniform polyhedron-43-t02.png	-	Template:CDD File:Tetragonal prism.png	Template:CDD File:Uniform tiling 44-t1.png
71	Template:CDD t_0,3{4,4,3}	Template:CDD File:Uniform polyhedron-43-t2.png	Template:CDD File:Triangular prism.png	Template:CDD File:Tetragonal prism.png	Template:CDD File:Uniform tiling 44-t0.png
72	Template:CDD t_0,1,2{4,4,3}	Template:CDD File:Uniform polyhedron-43-t01.png	Template:CDD File:Triangular prism.png	-	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 File:Uniform polyhedron-43-t02.png	Template:CDD File:Triangular prism.png	Template:CDD File:Octagonal prism.png	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 File:Hexagonal prism.png	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 File:Hexagonal prism.png	Template:CDD File:Octagonal prism.png	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 File:Uniform tiling 44-t0.png		Template:CDD File:Hexahedron.png Cube	File:H3 444 FC boundary.png
[62]	Template:CDD = Template:CDD t₁{4,4,4}	Template:CDD File:Uniform tiling 44-t0.png	-	-	Template:CDD File:Uniform tiling 44-t1.png		Template:CDD File:Tetragonal prism.png Square prism	Error creating thumbnail:
78	Template:CDD t_0,1{4,4,4}	Template:CDD File:Uniform tiling 44-t0.png	-	-	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 File:Uniform tiling 44-t1.png	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 File:Uniform tiling 44-t2.png	Template:CDD File:Tetragonal prism.png	Template:CDD File:Tetragonal prism.png	Template:CDD File:Uniform tiling 44-t0.png
[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 File:Octagonal prism.png	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 File:Octagonal prism.png	Template:CDD File:Octagonal prism.png	Template:CDD File:Uniform tiling 44-t012.png
[?]	Template:CDD = Template:CDD h{4,4,4}	-	-	-	Template:CDD File:Uniform tiling 44-t0.png
[?]	Template:CDD = Template:CDD hr{4,4,4}	Template:CDD File:Uniform tiling 44-t0.png	-	-	Template:CDD File:Uniform tiling 44-t1.png
Nonuniform	Template:CDD s{4,4,4}	Template:CDD File:Uniform tiling 44-t0.png	-	-	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 File:Uniform tiling 44-t1.png	Template:CDD File:Tetrahedron.png	-	Template:CDD File:Uniform tiling 44-t02.png
Nonuniform	Template:CDD ht_0,3{4,4,4}	Template:CDD File:Uniform tiling 44-t2.png	Template:CDD File:Tetrahedron.png	Template:CDD File:Tetrahedron.png	Template:CDD File:Uniform tiling 44-t0.png
Nonuniform	Template:CDD = Template:CDD sr{4,4,4}	Template:CDD File:Uniform tiling 44-h01.png	Template:CDD File:Tetrahedron.png	-	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 File:Tetrahedron.png	Template:CDD File:Square antiprism.png	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 File:Square antiprism.png	Template:CDD File:Square antiprism.png	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) File:Uniform tiling 63-t2.png (3.3.3.3.3)	(20) File:Tetrahedron.png (3.3.3)
	Cantic order-6 cubic Template:CDD = Template:CDD	(1) File:Uniform tiling 63-t1.png (3.6.3.6)	-	(2) File:Uniform tiling 63-t12.png (6.6.6)	(2) File:Truncated tetrahedron.png (3.6.6)
	runcic order-6 cubic Template:CDD = Template:CDD	(1) Error creating thumbnail: (6.6.6)	-	(3) File:Uniform tiling 63-t02.png (3.4.6.4)	(1) File:Tetrahedron.png (3.3.3)
	runcicantic order-6 cubic Template:CDD = Template:CDD	(1) File:Uniform tiling 63-t01.png (3.12.12)	-	(2) File:Uniform tiling 63-t012.png (4.6.12)	(1) File:Truncated tetrahedron.png (3.6.6)
[?]	rectified order-4 hexagonal tiling Template:CDD = Template:CDD	(2) File:Uniform tiling 63-t1.png (3.6.3.6)	-	(2) File:Uniform tiling 63-t1.png (3.6.3.6)	(2) File:Uniform polyhedron-33-t1.png (3.3.3.3)
[?]	rectified order-6 cubic Template:CDD = Template:CDD	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3)	-	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3)	(6) File:Uniform polyhedron-33-t02.png (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) File:Uniform tiling 63-t01.png (3.12.12)	-	(2) File:Uniform tiling 63-t01.png (3.12.12)	(1) File:Uniform polyhedron-33-t1.png (3.3.3.3)
[?]	Order-4 hexagonal tiling Template:CDD = Template:CDD	(4) Error creating thumbnail: (6.6.6)	-	(4) Error creating thumbnail: (6.6.6)	-
[?]	cantellated order-4 hexagonal tiling Template:CDD = Template:CDD	(1) File:Uniform tiling 63-t02.png (3.4.6.4)	(2) File:Uniform polyhedron 222-t012.png (4.4.4)	(1) File:Uniform tiling 63-t02.png (3.4.6.4)	(1) File:Uniform polyhedron-33-t02.png (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)	Error creating thumbnail: (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)	File:Uniform polyhedron-33-t2.png +(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) File:Octahedron.png (3.3.3.3)	-	(8) Error creating thumbnail: (6.6.6)	(1) File:Uniform tiling 63-t1.png (3.6.3.6)		File:Uniform polyhedron-43-t02.png Template:CDD (3.4.4.4)
92	Template:CDD	(30) File:Cuboctahedron.png (3.4.3.4)	(20) File:Hexahedron.png (4.4.4)	-	(12) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)		File:Uniform tiling 63-t02.png Template:CDD (3.4.6.4)
93	Template:CDD	(3) File:Truncated octahedron.png (4.6.6)	(1) File:Hexahedron.png (4.4.4)	(1) Error creating thumbnail: (6.6.6)	(3) File:Uniform tiling 63-t12.png (6.6.6)
94	Template:CDD	(1) File:Octahedron.png (3.3.3.3)	(1) File:Octahedron.png (3.3.3.3)	(4) File:Uniform tiling 63-t01.png (3.12.12)	(4) File:Uniform tiling 63-t01.png (3.12.12)
95	Template:CDD	(6) File:Truncated hexahedron.png (3.8.8)	(6) File:Truncated hexahedron.png (3.8.8)	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)
96	Template:CDD	(1) File:Cuboctahedron.png (3.4.3.4)	(2) File:Small rhombicuboctahedron.png (3.4.4.4)	(1) File:Uniform tiling 63-t1.png (3.6.3.6)	(2) File:Uniform tiling 63-t02.png (3.4.6.4)
97	Template:CDD	(1) File:Truncated octahedron.png (4.6.6)	(1) File:Small rhombicuboctahedron.png (3.4.4.4)	(1) File:Uniform tiling 63-t01.png (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) File:Uniform tiling 63-t02.png (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)	File:Tetrahedron.png +(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) File:Icosahedron.png (3.3.3.3.3)	-	(8) Error creating thumbnail: (6.6.6)	(1) File:Uniform tiling 63-t1.png (3.6.3.6)		File:Uniform polyhedron-53-t02.png 3.4.5.4
101	Template:CDD	(30) File:Icosidodecahedron.png (3.5.3.5)	(20) File:Dodecahedron.png (5.5.5)	-	(12) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)		File:Uniform tiling 63-t02.png (3.4.6.4)
102	Template:CDD	(3) File:Truncated icosahedron.png (5.6.6)	(1) File:Dodecahedron.png (5.5.5)	(1) Error creating thumbnail: (6.6.6)	(3) File:Uniform tiling 63-t12.png (6.6.6)
103	Template:CDD	(1) File:Icosahedron.png (3.3.3.3.3)	(1) File:Icosahedron.png (3.3.3.3.3)	(5) File:Uniform tiling 63-t01.png (3.12.12)	(5) File:Uniform tiling 63-t01.png (3.12.12)
104	Template:CDD	(6) File:Truncated dodecahedron.png (3.10.10)	(6) File:Truncated dodecahedron.png (3.10.10)	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)
105	Template:CDD	(1) File:Icosidodecahedron.png (3.5.3.5)	(2) File:Small rhombicosidodecahedron.png (3.4.5.4)	(1) File:Uniform tiling 63-t1.png (3.6.3.6)	(2) File:Uniform tiling 63-t02.png (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) File:Uniform tiling 63-t01.png (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) File:Uniform tiling 63-t02.png (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)	File:Tetrahedron.png +(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) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)	-	(20) Error creating thumbnail: (6.6.6)	(30) File:Uniform tiling 63-t1.png (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) Error creating thumbnail: (6.6.6)	(1) Error creating thumbnail: (6.6.6)	(3) File:Uniform tiling 63-t12.png (6.6.6)		File:Uniform polyhedron-33-t1.png (3.3.3.3)	Error creating thumbnail:
109	Template:CDD	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)	(1) File:Uniform tiling 63-t2.png (3.3.3.3.3.3)	(3) File:Uniform tiling 63-t01.png (3.12.12)	(3) File:Uniform tiling 63-t01.png (3.12.12)
110	Template:CDD	(1) File:Uniform tiling 63-t1.png (3.6.3.6)	(2) File:Uniform tiling 63-t02.png (3.4.6.4)	(1) File:Uniform tiling 63-t1.png (3.6.3.6)	(2) File:Uniform tiling 63-t02.png (3.4.6.4)
111	Template:CDD	(1) File:Uniform tiling 63-t12.png (6.6.6)	(1) File:Uniform tiling 63-t02.png (3.4.6.4)	(1) File:Uniform tiling 63-t01.png (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)	File:Uniform polyhedron-33-t1.png +(3.3.3.3)	File:Uniform polyhedron-33-t02.png (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)	Error creating thumbnail: +(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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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