Truncated order-6 square tiling

Truncated order-6 square tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	8.8.6
Schläfli symbol	t{4,6}
Wythoff symbol	2 6 | 4
Coxeter diagram
Symmetry group	[6,4], (*642) [(3,3,4)], (*334)
Dual	Order-4 hexakis hexagonal tiling
Properties	Vertex-transitive

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

Uniform colorings

The half symmetry [1+,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons.

Symmetry

Truncated order-6 square tiling with *443 symmetry mirror lines

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)

Related polyhedra and tiling

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  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

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)} {(4,4,3)}	h2{6,4} t0,1{(4,4,3)} r{(3,4,4)}	{4,6} t1{(4,4,3)} {(4,3,4)}	h2{6,4} t1,2{(4,4,3)} r{(4,4,3)}	h{6,4} t2{(4,4,3)} {(3,4,4)}	r{6,4} t0,2{(4,4,3)} r{(4,3,4)}	t{4,6} t0,1,2{(4,4,3)} t{(4,3,4)}	s{4,6}   s{(4,4,3)}	hr{6,4}   hr{(4,3,4)}	h{4,6}   h{(4,3,4)}	q{4,6}   h2{(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

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

See also

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

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)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. 

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

Tessellation   Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff   Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagonal Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet   Other Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal Packing Problems Domino Wang Heesch's Squaring Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg   By vertex type Spherical 2n 33.n V33.n 42.n V42.n Regular 2∞ 36 44 63 Semi- regular 32.4.3.4 V32.4.3.4 33.42 33.∞ 34.6 V34.6 3.4.6.4 (3.6)2 3.122 42.∞ 4.6.12 4.82 Hyper- bolic 32.4.3.5 32.4.3.6 32.4.3.7 32.4.3.8 32.4.3.∞ 32.5.3.5 32.5.3.6 32.6.3.6 32.6.3.8 32.7.3.7 32.8.3.8 33.4.3.4 32.∞.3.∞ 34.7 34.8 34.∞ 35.4 37 38 3∞ (3.4)3 (3.4)4 3.4.62.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7)2 (3.8)2 3.142 3.162 (3.∞)2 3.∞2 42.5.4 42.6.4 42.7.4 42.8.4 42.∞.4 45 46 47 48 4∞ (4.5)2 (4.6)2 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7)2 (4.8)2 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.102 4.10.12 4.122 4.12.16 4.142 4.162 4.∞2 (4.∞)2 54 55 56 5∞ 5.4.6.4 (5.6)2 5.82 5.102 5.122 (5.∞)2 64 65 66 68 6.4.8.4 (6.8)2 6.82 6.102 6.122 6.162 73 74 77 7.62 7.82 7.142 83 84 86 88 812 8.62 8.162 ∞3 ∞4 ∞5 ∞∞ ∞.62 ∞.82