Order-4 hexagonal tiling

Order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex figure	6^4
Schläfli symbol	{6,4}
Wythoff symbol	4 | 6 2
Coxeter diagram
Symmetry group	[6,4], (*642)
Dual	Order-6 square tiling
Properties	Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

Symmetry

This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *222222 with 6 order-2 mirror intersections. In Coxeter notation can be represented as [6^*,4], removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *3322 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry. Adding all 6 bisectors leads to full *642 symmetry.

*222222

*443

*3222

*642

Uniform colorings

There are 7 distinct uniform colorings for the order-4 hexagonal tiling. They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions and Coxeter diagrams while three of them are undercolorings.

Uniform constructions of 6.6.6.6

	1 color	2 colors	3 and 2 colors		4, 3 and 2 colors
Uniform Coloring	(1111)	(1212)	(1213)	(1113)	(1234)	(1123)	(1122)
Symmetry	[6,4] (*642)	[6,6] (*662) =	[(6,6,3)] = [6,6,1^+] (*663) =		[1^+,6,6,1^+] (*3333) = =
Symbol	{6,4}	r{6,6} = {6,4}^1/2	r(6,3,6) = r{6,6}^1/2		r{6,6}^1/4
Coxeter diagram		=	=		= =

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.

*n62 symmetry mutation of regular tilings: {6,n}
Spherical	Euclidean	Hyperbolic tilings
{6,2}	{6,3}	{6,4}	{6,5}	{6,6}	{6,7}	{6,8}	...	{6,∞}

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

*n42 symmetry mutation of regular tilings: {n,4}
Spherical		Euclidean	Hyperbolic tilings
2^4	3^4	4^4	5^4	6^4	7^4	8^4	...∞^4

Symmetry mutation of quasiregular tilings: 6.n.6.n
Symmetry *6n2 [n,6]	Euclidean	Compact hyperbolic					Paracompact	Noncompact
	*632 [3,6]	*642 [4,6]	*652 [5,6]	*662 [6,6]	*762 [7,6]	*862 [8,6]...	*∞62 [∞,6]	[iπ/λ,6]
Quasiregular figures configuration	6.3.6.3	6.4.6.4	6.5.6.5	6.6.6.6	6.7.6.7	6.8.6.8	6.∞.6.∞	6.∞.6.∞
Dual figures
Rhombic figures configuration	V6.3.6.3	V6.4.6.4	V6.5.6.5	V6.6.6.6	V6.7.6.7	V6.8.6.8	V6.∞.6.∞

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
V6^4	V4.12.12	V(4.6)^2	V6.8.8	V4^6	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
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
V6^6	V6.12.12	V6.6.6.6	V6.12.12	V6^6	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}

Similar H2 tilings in *3232 symmetry
Coxeter diagrams
Vertex figure	6^6		(3.4.3.4)^2		3.4.6.6.4		6.4.6.4
Image
Dual

Uniform tilings in symmetry *3222
6^4	6.6.4.4	(3.4.4)^2	4.3.4.3.3.3
6.6.4.4	6.4.4.4	3.4.4.4.4
(3.4.4)^2	3.4.4.4.4	4^6

See also

• 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