User:Tomruen/nonsimplex domain honeycombs

Fundamental domain

Full domain

Half domain

Example compact hyperbolic honeycombs in nonsimplectic domain, a trigonal trapezohedron, with a hexagonal coxeter diagram. the domain is constructed from an index 6 subgroup of [(4,3,4,3)] as [(4,3,4,3*)]:

↔ ↔

If two pairs of mirrors have the same ring state, they can be mapped into an extended symmetry with a half domain:

↔

[(4,3,4,3)]

Name	Honeycomb	Cells			Subgroup tiling	Vertex figure	Perspective
Symmetry template	↔	↔ ↔	↔	↔
Cubic-octahedral	↔ ↔	↔ =	↔ =	↔ =
Cyclotruncated octahedral-cubic	↔ ↔	↔ =	↔ =

Trigonal trapezohedron

Honeycomb	Extended symmetry	Cells						Subgroup tilings	Vertex figure
		4.4.4	4.6.6	3.4.3.4	3.3.3.3	3.6.6	3.3.3
↔	[ ]
↔
↔		(2)	(6)	(1)		(2)
↔	[2]+
↔ ↔	[3]	(8)		(12)	(6)
↔ ↔	[6,2+]	(2)	(6)

*3232 tilings

H2 subgroup *3232 symmetry tiling mirrors with rhombic fundamental domains
↔

Similar H2 tilings in *3232 symmetry

Coxeter diagram	=	=	=	=
Vertex figure	66	(3.4.3.4)2	3.4.6.6.4	6.4.6.4
Image
Dual

Half trigonal trapezohedron

Half domain

#	Honeycomb	Cells						Subgroup tiling	Vertex figure	Perspective
1
2
3	=
4							-
5
6						-
7							-
8					-		-
9	=						-
10	=					-
11	=		-			-	-
12	=					-