Truncated order-4 octagonal tiling

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

Truncated order-4 octagonal tiling
Truncated order-4 octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.16.16
Schläfli symbolt{8,4}
tr{8,8} or
Wythoff symbol2 8 | 8
2 8 8 |
Coxeter diagram
or
Symmetry group[8,4], (*842)
[8,8], (*882)
DualOrder-8 tetrakis square tiling
PropertiesVertex-transitive

Constructions

There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1+], gives [8,8], (*882).

Two uniform constructions of 4.8.4.8
Name Tetraoctagonal Truncated octaoctagonal
Image
Symmetry [8,4]
(*842)
 [8,8] = [8,4,1+]
(*882)
=
Symbol t{8,4} tr{8,8}
Coxeter diagram

Dual tiling
The dual tiling, Order-8 tetrakis square tiling has face configuration V4.16.16, and represents the fundamental domains of the [8,8] symmetry group.

Symmetry
Truncated order-4 octagonal tiling with *882 mirror lines

The dual of the tiling represents the fundamental domains of (*882) orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup 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 unique mirrors are colored red, green, and blue, and alternatively colored triangles show the location of gyration points. The [8+,8+], (44×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1+,8,1+,8,1+] (4444) is the commutator subgroup of [8,8].

One larger subgroup is constructed as [8,8*], removing the gyration points of (8*4), index 16 becomes (*44444444), and its direct subgroup [8,8*]+, index 32, (44444444).

The [8,8] symmetry can be doubled by a mirror bisecting the fundamental domain, and creating *884 symmetry.

Small index subgroups of [8,8] (*882)
Index 1 2 4
Diagram
Coxeter [8,8]
 [1+,8,8]
= 		[8,8,1+]
= 		[8,1+,8]
= 		[1+,8,8,1+]
= 		[8+,8+]
Orbifold *882 *884 *4242 *4444 44×
Semidirect subgroups
Diagram
Coxeter [8,8+]
 [8+,8]
 [(8,8,2+)]
 [8,1+,8,1+]
= =
= = 		[1+,8,1+,8]
= =
= =
Orbifold 8*4 2*44 4*44
Direct subgroups
Index 2 4 8
Diagram
Coxeter [8,8]+
 [8,8+]+
= 		[8+,8]+
= 		[8,1+,8]+
= 		[8+,8+]+ = [1+,8,1+,8,1+]
= = =
Orbifold 882 884 4242 4444
Radical subgroups
Index 16 32
Diagram
Coxeter [8,8*]
 [8*,8]
 [8,8*]+
 [8*,8]+
Orbifold *44444444 44444444
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]		 Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]		 *342
[3,4]		 *442
[4,4]		 *542
[5,4]		 *642
[6,4]		 *742
[7,4]		 *842
[8,4]...		 *∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
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

References
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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
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