Order-4 hexagonal tiling

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.

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
24 34 44 54 64 74 84 ...∞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
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
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
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
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

See also
Wikimedia Commons has media related to Order-4 hexagonal tiling.

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.
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