Order-4 hexagonal tiling

In this article, the topic of Order-4 hexagonal tiling will be addressed, which has sparked great interest and debate in different areas. Order-4 hexagonal tiling is a topic that has captured the attention of academics, specialists, and the general public due to its relevance in today's society. From its origins to the present, Order-4 hexagonal tiling has been the object of study, analysis and reflection, which has led to the generation of different theories, approaches and opinions on the matter. Throughout this article, the most relevant aspects of Order-4 hexagonal tiling will be explored, as well as its impact on various aspects of daily life. Likewise, the different perspectives that exist around Order-4 hexagonal tiling will be examined, with the aim of offering a broad and comprehensive vision of the topic.

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 , (*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 , 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
(*642)
(*662)
= 		=
(*663)
=
(*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)2
Symmetry
*6n2
 Euclidean Compact hyperbolic Paracompact Noncompact
*632
 *642
 *652
 *662
 *762
 *862
... 		*∞62
  
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: , (*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
V64 V4.12.12 V(4.6)2 V6.8.8 V46 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}
Uniform hexahexagonal tilings
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
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 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}
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-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.