Truncated order-6 square tiling

The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.

A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).

The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain.

Small index subgroups of [(4,4,3)] (*443)
Index	1	2		6
Diagram
Coxeter (orbifold)	[(4,4,3)] = (*443)	[(4,1+,4,3)] = = (*3232)	[(4,4,3+)] = (3*22)	[(4,4,3*)] = (*222222)
Direct subgroups
Index	2	4		12
Diagram
Coxeter (orbifold)	[(4,4,3)]+ = (443)	[(4,4,3+)]+ = = (3232)		[(4,4,3*)]+ = (222222)

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform tetrahexagonal tilings v t e
Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
= = =	=	= = =	=	= = =	=
{6,4}	t{6,4}	r{6,4}	t{4,6}	{4,6}	rr{6,4}	tr{6,4}
Uniform duals
V64	V4.12.12	V(4.6)2	V6.8.8	V46	V4.4.4.6	V4.8.12
Alternations
[1+,6,4] (*443)	[6+,4] (6*2)	[6,1+,4] (*3222)	[6,4+] (4*3)	[6,4,1+] (*662)	[(6,4,2+)] (2*32)	[6,4]+ (642)
=	=	=	=	=	=
h{6,4}	s{6,4}	hr{6,4}	s{4,6}	h{4,6}	hrr{6,4}	sr{6,4}

It can also be generated from the (4 4 3) hyperbolic tilings:

Uniform (4,4,3) tilings v t e
Symmetry: [(4,4,3)] (*443)							[(4,4,3)]+ (443)	[(4,4,3+)] (3*22)	[(4,1+,4,3)] (*3232)
h{6,4} t0(4,4,3)	h2{6,4} t0,1(4,4,3)	{4,6}1/2 t1(4,4,3)	h2{6,4} t1,2(4,4,3)	h{6,4} t2(4,4,3)	r{6,4}1/2 t0,2(4,4,3)	t{4,6}1/2 t0,1,2(4,4,3)	s{4,6}1/2 s(4,4,3)	hr{4,6}1/2 hr(4,3,4)	h{4,6}1/2 h(4,3,4)	q{4,6} h1(4,3,4)
Uniform duals
V(3.4)4	V3.8.4.8	V(4.4)3	V3.8.4.8	V(3.4)4	V4.6.4.6	V6.8.8	V3.3.3.4.3.4	V(4.4.3)2	V66	V4.3.4.6.6

*n42 symmetry mutation of truncated tilings: n.8.8 v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracompact
	*242 [2,4]	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]
Truncated figures
Config.	2.8.8	3.8.8	4.8.8	5.8.8	6.8.8	7.8.8	8.8.8	∞.8.8
n-kis figures
Config.	V2.8.8	V3.8.8	V4.8.8	V5.8.8	V6.8.8	V7.8.8	V8.8.8	V∞.8.8

*n32 symmetry mutation of omnitruncated tilings: 6.8.2n v t e
Sym. *n43 [(n,4,3)]	Spherical	Compact hyperbolic						Paraco.
	*243 [4,3]	*343 [(3,4,3)]	*443 [(4,4,3)]	*543 [(5,4,3)]	*643 [(6,4,3)]	*743 [(7,4,3)]	*843 [(8,4,3)]	*∞43 [(∞,4,3)]
Figures
Config.	4.8.6	6.8.6	8.8.6	10.8.6	12.8.6	14.8.6	16.8.6	∞.8.6
Duals
Config.	V4.8.6	V6.8.6	V8.8.6	V10.8.6	V12.8.6	V14.8.6	V16.8.6	V6.8.∞

Wikimedia Commons has media related to Uniform tiling 6-8-8.

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch