Truncated order-4 hexagonal tiling

Truncated order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.12.12
Schläfli symbol	t{6,4} tr{6,6} or $t{\begin{Bmatrix}6\\6\end{Bmatrix}}$
Wythoff symbol	2 4 \| 6 2 6 6 \|
Coxeter diagram	or
Symmetry group	[6,4], (642) [6,6], (662)
Dual	Order-6 tetrakis square tiling
Properties	Vertex-transitive

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.

Constructions

There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1⁺], gives [6,6], (*662).

Two uniform constructions of 4.6.4.6
Name	Tetrahexagonal	Truncated hexahexagonal
Image
Symmetry	[6,4] (*642)	[6,6] = [6,4,1⁺] (*662) =
Symbol	t{6,4}	tr{6,6}
Coxeter diagram

Dual tiling


The dual tiling, order-6 tetrakis square tiling has face configuration V4.12.12, and represents the fundamental domains of the [6,6] symmetry group.

Related polyhedra and tiling

*n42 symmetry mutation of truncated tilings: 4.2n.2n v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
Symmetry *n42 [n,4]	*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.	4.4.4	4.6.6	4.8.8	4.10.10	4.12.12	4.14.14	4.16.16	4.∞.∞
n-kis figures
Config.	V4.4.4	V4.6.6	V4.8.8	V4.10.10	V4.12.12	V4.14.14	V4.16.16	V4.∞.∞

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}

Uniform hexahexagonal tilings v t e
Symmetry: [6,6], (*662)
= =	= =	= =	= =	= =	= =	= =

{6,6} = h{4,6}	t{6,6} = h₂{4,6}	r{6,6} {6,4}	t{6,6} = h₂{4,6}	{6,6} = h{4,6}	rr{6,6} r{6,4}	tr{6,6} t{6,4}
Uniform duals


V6⁶	V6.12.12	V6.6.6.6	V6.12.12	V6⁶	V4.6.4.6	V4.12.12
Alternations
[1⁺,6,6] (*663)	[6⁺,6] (6*3)	[6,1⁺,6] (*3232)	[6,6⁺] (6*3)	[6,6,1⁺] (*663)	[(6,6,2⁺)] (2*33)	[6,6]⁺ (662)
=		=		=


h{6,6}	s{6,6}	hr{6,6}	s{6,6}	h{6,6}	hrr{6,6}	sr{6,6}

Symmetry

The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1⁺,6,1⁺,6,1⁺] (3333) is the commutator subgroup of [6,6].

Larger subgroup constructed as [6,6^*], removing the gyration points of (6*3), index 12 becomes (*333333).

The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.

Small index subgroups of [6,6] (*662)
Index	1	2			4
Diagram
Coxeter	[6,6]	[1⁺,6,6] =	[6,6,1⁺] =	[6,1⁺,6] =	[1⁺,6,6,1⁺] =	[6⁺,6⁺]
Orbifold	*662	*663		*3232	*3333	33×
Direct subgroups
Diagram
Coxeter		[6,6⁺]	[6⁺,6]	[(6,6,2⁺)]	[6,1⁺,6,1⁺] = = = =	[1⁺,6,1⁺,6] = = = =
Orbifold		6*3		2*33	3*33
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[6,6]⁺	[6,6⁺]⁺ =	[6⁺,6]⁺ =	[6,1⁺,6]⁺ =	[6⁺,6⁺]⁺ = [1⁺,6,1⁺,6]⁺ = = =
Orbifold	662	663		3232	3333
Radical subgroups
Index		12		24
Diagram
Coxeter		[6,6*]	[6*,6]	[6,6*]⁺	[6*,6]⁺
Orbifold		*333333		333333

References

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.