Snub hexahexagonal tiling - Misplaced Pages

Snub hexahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.3.6.3.6
Schläfli symbol	s{6,4} sr{6,6}
Wythoff symbol	\| 6 6 2
Coxeter diagram
Symmetry group	, (662) , (6*2)
Dual	Order-6-6 floret hexagonal tiling
Properties	Vertex-transitive

In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}.

Images

Drawn in chiral pairs, with edges missing between black triangles:

A higher symmetry coloring can be constructed from symmetry as s{6,4}, . In this construction there is only one color of hexagon.

Uniform hexahexagonal tilings v t e
Symmetry: , (*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


V6	V6.12.12	V6.6.6.6	V6.12.12	V6	V4.6.4.6	V4.12.12
Alternations
(*663)	(6*3)	(*3232)	(6*3)	(*663)	(2*33)	(662)
=		=		=


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

Uniform tetrahexagonal tilings v t e
*Symmetry: , (642)** (with (662), (443) , (3222) index 2 subsymmetries) (And (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


V6	V4.12.12	V(4.6)	V6.8.8	V4	V4.4.4.6	V4.8.12
Alternations
(*443)	(6*2)	(*3222)	(4*3)	(*662)	(2*32)	(642)
=	=	=	=	=	=

h{6,4}	s{6,4}	hr{6,4}	s{4,6}	h{4,6}	hrr{6,4}	sr{6,4}

4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry 4n2	Spherical		Euclidean	Compact hyperbolic				Paracompact
Symmetry 4n2	222	322	442	552	662	772	882	∞∞2
Snub figures
Config.	3.3.2.3.2	3.3.3.3.3	3.3.4.3.4	3.3.5.3.5	3.3.6.3.6	3.3.7.3.7	3.3.8.3.8	3.3.∞.3.∞
Gyro figures
Config.	V3.3.2.3.2	V3.3.3.3.3	V3.3.4.3.4	V3.3.5.3.5	V3.3.6.3.6	V3.3.7.3.7	V3.3.8.3.8	V3.3.∞.3.∞

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.

Periodic
Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling and honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff

Aperiodic
Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pinwheel Quaquaversal Rep-tile and Self-tiling Sphinx Socolar Truchet

Other
Anisohedral and Isohedral Architectonic and catoptric Circle Limit III Computer graphics Honeycomb Isotoxal List Packing Pentagonal Problems Domino Wang Heesch's Squaring Dividing a square into similar rectangles Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg