Order-4 hexagonal tiling

Order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	6⁴
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 *4422 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		=	=		= =

Regular maps

The regular map {6,4}₃ or {6,4}_(4,0) can be seen as a 4-coloring on the {6,4} tiling. It also has a representation as a petrial octahedron, {3,4}^$π$, an abstract polyhedron with vertices and edges of an octahedron, but instead connected by 4 Petrie polygon faces.

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} v t e
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} v t e
Spherical		Euclidean	Hyperbolic tilings

2⁴	3⁴	4⁴	5⁴	6⁴	7⁴	8⁴	...∞⁴

Symmetry mutation of quasiregular tilings: (6.n)² v t e
Symmetry *6n2 [n,6]	Euclidean	Compact hyperbolic					Paracompact	Noncompact
Symmetry *6n2 [n,6]	*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 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}

Similar H2 tilings in *3232 symmetry v t e
Coxeter diagrams


Vertex figure	6⁶	(3.4.3.4)²	3.4.6.6.4	6.4.6.4
Image
Dual

Uniform tilings in symmetry *3222 v t e
6⁴	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	4⁶

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.

External links