Truncated order-4 octagonal tiling – Wikipedia
From Wikipedia, the free encyclopedia
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.
Constructions[edit]
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).
Dual tiling[edit]
 |
 |
| 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[edit]
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.
| 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 | ||||
Related polyhedra and tiling[edit]
| 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[edit]
See also[edit]
External links[edit]
Recent Comments