Order-4 octagonal tiling

Order-4 octagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex figure8.8.8.8
Schläfli symbol{8,4}
r{8,8}
Wythoff symbol4 | 8 2
Coxeter diagram
Symmetry group[8,4], (*842)
[8,8], (*882)
DualOrder-8 square tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}.

Uniform constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.

Three uniform constructions of 8.8.8.8
Uniform
Coloring
Symmetry [8,4]
(*842)
 [8,8]
(*882)
= 		[(8,4,8)] = [8,8,1+]
(*884)
=

=

[1+,8,8,1+]
(*4444)
=
Symbol {8,4} r{8,8} r{(8,4,8)}
Coxeter
diagram		 =

=

=

Symmetry

This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of a octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.


*444
*4222
*832

The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.

Spherical Euclidean Hyperbolic tilings

{2,4}
{3,4}
{4,4}
{5,4}
{6,4}
{7,4}
{8,4}
 ...
{∞,4}
Spherical Hyperbolic tilings

{2,8}
{3,8}
{4,8}
{5,8}
{6,8}
{7,8}
{8,8}
 ...
{∞,8}

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.


{3,4}
{4,4}
{5,4}
{6,4}
{7,4}
{8,4}
 ...
{∞,4}
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)

=

=
=
=
=
=

=

=

=
=


=
{8,4} t{8,4}
 r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)		 [8+,4]
(8*2)		 [8,1+,4]
(*4222)		 [8,4+]
(4*4)		 [8,4,1+]
(*882)		 [(8,4,2+)]
(2*42)		 [8,4]+
(842)

=
=
=
=
=
=
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
=
= 		=
= 		=
= 		=
= 		=
= 		=
= 		=
=
{8,8} t{8,8}
 r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16
Alternations
[1+,8,8]
(*884)		 [8+,8]
(8*4)		 [8,1+,8]
(*4242)		 [8,8+]
(8*4)		 [8,8,1+]
(*884)		 [(8,8,2+)]
(2*44)		 [8,8]+
(882)
= = = =
= 		=
=
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8

See also

Wikimedia Commons has media related to Order-4 octagonal tiling.

References

External links