Truncated order-8 octagonal tiling

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Truncated order-8 octagonal tiling
Truncated order-8 octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 8.16.16
Schläfli symbol t{8,8}
t(8,8,4)
Wythoff symbol 2 8 | 4
Coxeter diagram
Symmetry group [8,8], (*882)
[(8,8,4)], (*884)
Dual Order-8 octakis octagonal tiling
Properties
Vertex-transitive

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.

Uniform colorings

This tiling can also be constructed in *884 symmetry with 3 colors of faces:

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

Symmetry

The dual of the tiling represents the fundamental domains of (*884)

subgroup index-8 group, [(1+,8,1+,8,1+,4)] (442442) is the commutator subgroup
of [(8,8,4)].

Small index subgroups of [(8,8,4)] (*884)
Fundamental
domains



Subgroup index 1 2 4
Coxeter [(8,8,4)]
 [(1+,8,8,4)]
 [(8,8,1+,4)]
 [(8,1+,8,4)]
 [(1+,8,8,1+,4)]
 [(8+,8+,4)]
orbifold *884 *8482
*4444
 2*4444 442×
Coxeter [(8,8+,4)]
 [(8+,8,4)]
 [(8,8,4+)]
 [(8,1+,8,1+,4)]
 [(1+,8,1+,8,4)]
Orbifold 8*42 4*44 4*4242
Direct subgroups
Subgroup index 2 4 8
Coxeter [(8,8,4)]+
 [(1+,8,8+,4)]
 [(8+,8,1+,4)]
 [(8,1+,8,4+)]
 [(1+,8,1+,8,1+,4)] = [(8+,8+,4+)]
Orbifold 844 8482 4444 442442

References

See also
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