# Octagonal tiling

For other uses, see truncated square tiling.
Octagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 8.8.8
Schläfli symbol {8,3}
t{4,8}
Wythoff symbol 3 | 8 2
2 8 | 4
4 4 4 |
Coxeter diagram

Symmetry group [8,3], (*832)
[8,4], (*842)
[(4,4,4)], (*444)
Dual Order-8 triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex.

## Uniform colorings

Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.

Regular Truncation Omnitruncation

{8,3}

t1,2{8,4}

t0,1,2(4,4,4)
=
 {3,8} = = f0,1,2(4,4,4) =

## Symmetry

From [(4,4,4)] symmetry, there are 15 small index subgroups (7 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, [(1+,4,1+,4,1+,4)] (222222) is the commutator subgroup of [(4,4,4)].

A larger subgroup is constructed [(4,4,4*)], index 8, as (2*2222) with gyration points removed, becomes (*22222222).

Small index subgroup symmetries of [(4,4,4)] (*444)

Subgroup index 1 2 4
Coxeter
(orbifold)
[(4,4,4)]
(*444)
[(1+,4,4,4)]
(*4242)
[(4,4,1+,4)]
(*4242)
[(4,1+,4,4)]
(*4242)
[(4,1+,4,1+,4)]
2*2222
[(1+,4,4,1+,4)]
(2*2222)
[(4,4+,4)]
(4*22)
[(4+,4,4)]
(4*22)
[(4,4,4+)]
(4*22)
[(1+,4,1+,4,4)]
2*2222
[(4+,4+,4)]
(222×)
Rotational subgroups
Subgroup index 2 4 8
Coxeter
(orbifold)
[(4,4,4)]+
(444)
[(1+,4,4+,4)]
(4242)
[(4+,4,1+,4)]
(4242)
[(4,1+,4,4+)]
(4242)
[(1+,4,1+,4,1+,4)]
(222222)

## Related polyhedra and tilings

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}.

Spherical
Polyhedra
Polyhedra Euclidean Hyperbolic tilings

{2,3}

{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}
...
(∞,3}

And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.

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

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)
[1+,8,3]
(*443)
[8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}
tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}

or

or

Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4
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 (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
t0{(4,4,4)} t0,1{(4,4,4)} t1{(4,4,4)} t1,2{(4,4,4)} t2{(4,4,4)} t0,2{(4,4,4)} t0,1,2{(4,4,4)} s{(4,4,4)} h{(4,4,4)} hr{(4,4,4)}
Uniform duals
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3