Truncated order-8 octagonal tiling

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) orbifold symmetry. From [(8,8,4)] (*884) symmetry, there are 15 small index subgroup (11 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 882 symmetry by adding a bisecting mirror across the fundamental domains. The 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

  • 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.

See also