Truncated tetraoctagonal tiling
| Truncated tetraoctagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex figure | 4.8.16 |
| Schläfli symbol | tr{8,4} |
| Wythoff symbol | 2 8 4 | |
| Coxeter diagram |    |
| Symmetry group | [8,4], (*842) |
| Dual | Order-4-8 kisrhombille tiling |
| Properties | Vertex-transitive |
In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.
Dual tiling
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| The dual tiling is called an order-4-8 kisrhombille tiling, made as a complete bisection of the order-4 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,4] (*842) symmetry. | |
Symmetry
, mirror linesThere are 15 subgroups constructed from [8,4] by mirror removal and alternation. 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 subgroup index-8 group, [1+,8,1+,4,1+] (4242) is the commutator subgroup of [8,4].
A larger subgroup is constructed as [8,4*], index 8, as [8,4+], (4*4) with gyration points removed, becomes (*4444) or (*44), and another [8*,4], index 16 as [8+,4], (8*2) with gyration points removed as (*22222222) or (*28). And their direct subgroups [8,4*]+, [8*,4]+, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).
| Index | 1 | 2 | 4 | |||
|---|---|---|---|---|---|---|
| Diagram |  |
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| Coxeter | [8,4] =    |
[1+,8,4] =    |
[8,4,1+] =   = |
[8,1+,4] =    |
[1+,8,4,1+] = |
[8+,4+]  |
| Orbifold | *842 | *444 | *882 | *4222 | *4242 | 42× |
| Semidirect subgroups | ||||||
| Diagram |  |
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| Coxeter | [8,4+] |
[8+,4] |
[(8,4,2+)]  |
[8,1+,4,1+] = = = = |
[1+,8,1+,4] = = = = | |
| Orbifold | 4*4 | 8*2 | 2*42 | 2*44 | 4*22 | |
| Direct subgroups | ||||||
| Index | 2 | 4 | 8 | |||
| Diagram |  |
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| Coxeter | [8,4]+ = |
[8,4+]+ = |
[8+,4]+ = |
[8,1+,4]+  =  |
[8+,4+]+ = [1+,8,1+,4,1+] = = = | |
| Orbifold | 842 | 444 | 882 | 4222 | 4242 | |
| Radical subgroups | ||||||
| Index | 8 | 16 | 32 | |||
| Diagram |  |
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| Coxeter | [8,4*]  =  |
[8*,4] |
[8,4*]+ = |
[8*,4]+ | ||
| Orbifold | *4444 | *22222222 | 4444 | 22222222 | ||
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 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 7 forms with full [8,4] symmetry, and 7 with subsymmetry.
| [8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) | ||||||
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=   =  = |
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= = = |
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| {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 | ||||||
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| 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) |
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| h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} |
| Alternation duals | ||||||
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| 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 |
| Symmetry *n42 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||
|---|---|---|---|---|---|---|---|---|
| *242 [2,4] D4h |
*342 [3,4] Oh |
*442 [4,4] P4m |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] | |
| Omnitruncated figure |
 4.8.4 |
 4.8.6 |
 4.8.8 |
 4.8.10 |
 4.8.12 |
 4.8.14 |
4.8.16 |
 4.8.∞ |
| Coxeter Schläfli |
tr{2,4} |
 tr{3,4} |
tr{4,4} |
 tr{5,4} |
 tr{6,4} |
 tr{7,4} |
tr{8,4} |
 tr{∞,4} |
| Omnitruncated duals |
 V4.8.4 |
 V4.8.6 |
 V4.8.8 |
 V4.8.10 |
 V4.8.12 |
 V4.8.14 |
V4.8.16 |
 V4.8.∞ |
| Coxeter | |
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| Symmetry *nn2 [n,n] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||
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| *222 [2,2] D2h |
*332 [3,3] Td |
*442 [4,4] P4m |
*552 [5,5] |
*662 [6,6] |
*772 [7,7] |
*882 [8,8]... |
*∞∞2 [∞,∞] | |
| Figure |  4.4.4 |
 4.6.6 |
4.8.8 |
 4.10.10 |
 4.12.12 |
 4.14.14 |
 4.16.16 |
 4.∞.∞ |
| Coxeter Schläfli |
tr{2,2} |
tr{3,3} |
tr{4,4} |
tr{5,5} |
tr{6,6} |
tr{7,7} |
tr{8,8} |
tr{∞,∞} |
| Dual |  V4.4.4 |
 V4.6.6 |
V4.8.8 |
V4.10.10 |
V4.12.12 |
V4.14.14 |
V4.16.16 |
V4.∞.∞ |
| Coxeter | |
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See also
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Wikimedia Commons has media related to Uniform tiling 4-8-16. |
- Tilings of regular polygons
- List of uniform planar tilings
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.
External links
- Weisstein, Eric W., "Hyperbolic tiling", MathWorld.
- Weisstein, Eric W., "Poincaré hyperbolic disk", MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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