Truncated order-4 octagonal tiling

Truncated order-4 octagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.16.16
Schläfli symbol	t{8,4} tr{8,8}
Wythoff symbol	2 8 | 8 2 8 8 |
Coxeter diagram
Symmetry group	[8,4], (*842) [8,8], (*882)
Dual	Order-8 tetrakis square tiling
Properties	Vertex-transitive

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_0,1{8,4}. A secondary construction t_0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

Constructions

There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1⁺], gives [8,8], (*882).

Two uniform constructions of 4.8.4.8

Name	Tetraoctagonal	Truncated octaoctagonal
Image
Symmetry	[8,4] (*842)	[8,8] = [8,4,1+] (*882) =
Symbol	t{8,4}	tr{8,8}
Coxeter diagram

Dual tiling

The dual tiling, Order-8 tetrakis square tiling has face configuration V4.16.16, and represents the fundamental domains of the [8,8] symmetry group.

Symmetry

File:Truncated order-4 octagonal tiling with 882 mirrors.png

Truncated order-4 octagonal tiling with *882 mirror lines

The dual of the tiling represents the fundamental domains of (*882) orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup 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 unique mirrors are colored red, green, and blue, and alternatedly colored triangles show the location of gyration points. The [8⁺,8⁺], (44×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1⁺,8,1⁺,8,1⁺] (4444) is the commutator subgroup of [8,8].

One larger subgroup is constructed as [8,8*], removing the gyration points of (8*4), index 16 becomes (*44444444), and its direct subgroup [8,8*]⁺, index 32, (44444444).

The [8,8] symmetry can be doubled by a mirror bisecting the fundamental domain, and creating *884 symmetry.

Small index subgroups of [8,8] (*882)

Index	1	2			4
Diagram	120px	120px	120px	120px	120px	120px
Coxeter	[8,8]	[1+,8,8] =	[8,8,1+] =	[8,1+,8] =	[1+,8,8,1+] =	[8+,8+]
Orbifold	*882	*884		*4242	*4444	44×
Semidirect subgroups
Diagram		120px	120px	120px	120px	120px
Coxeter		[8,8+]	[8+,8]	[(8,8,2+)]	[8,1+,8,1+] = = = =	[1+,8,1+,8] = = = =
Orbifold		8*4		2*44	4*44
Direct subgroups
Index	2	4			8
Diagram	120px	120px	120px	120px	120px
Coxeter	[8,8]+	[8,8+]+ =	[8+,8]+ =	[8,1+,8]+ =	[8+,8+]+ = [1+,8,1+,8,1+] = = =
Orbifold	882	884		4242	4444
Radical subgroups
Index		16		32
Diagram		120px	120px	120px	120px
Coxeter		[8,8*]	[8*,8]	[8,8*]+	[8*,8]+
Orbifold		*44444444		44444444

Related polyhedra and tiling

*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	*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 octagonal/square tilings
[8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
= = =	=	= = =	=	= =	=
					65px
{8,4}	t{8,4}	r{8,4}	2t{8,4}=t{4,8}	2r{8,4}={4,8}	rr{8,4}	tr{8,4}
Uniform duals
	65px		65px	65px
V84	V4.16.16	V(4.8)2	V8.8.8	V48	V4.4.4.8	V4.8.16
Alternations
[1+,8,4] (*444)	[8+,4] (8*2)	[8,1+,4] (*4222)	[8,4+] (4*4)	[8,4,1+] (*882)	[(8,4,2+)] (2*42)	[8,4]+ (842)
=	=	=	=	=	=
	60px
h{8,4}	s{8,4}	hr{8,4}	s{4,8}	h{4,8}	hrr{8,4}	sr{8,4}
Alternation duals
V(4.4)4	V3.(3.8)2	V(4.4.4)2	V(3.4)3	V88	V4.44	V3.3.4.3.8

Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
= =	= =	= =	= =	= =	= =	= =
65px	65px		65px	65px	65px
{8,8}	t{8,8}	r{8,8}	2t{8,8}=t{8,8}	2r{8,8}={8,8}	rr{8,8}	tr{8,8}
Uniform duals
65px	65px	65px	65px	65px		65px
V88	V8.16.16	V8.8.8.8	V8.16.16	V88	V4.8.4.8	V4.16.16
Alternations
[1+,8,8] (*884)	[8+,8] (8*4)	[8,1+,8] (*4242)	[8,8+] (8*4)	[8,8,1+] (*884)	[(8,8,2+)] (2*44)	[8,8]+ (882)
=		=		=	= =	= =
60px				60px
h{8,8}	s{8,8}	hr{8,8}	s{8,8}	h{8,8}	hrr{8,8}	sr{8,8}
Alternation duals
V(4.8)8	V3.4.3.8.3.8	V(4.4)4	V3.4.3.8.3.8	V(4.8)8	V46	V3.3.8.3.8

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)

See also

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

External links

• 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