| 释义 |
- Constructions
- Dual tiling
- Symmetry
- Related polyhedra and tiling
- References
- See also
- External links
{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U84_01}}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 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 |
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| Image |
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| Symmetry | (*842)
{{CDD>node_c1|8|node_c2|4|node_c3}} | +]
(*882)
{{CDD>node_c1|split1-88|nodeab_c2}} = {{CDD|node_c1|8|node_c2|4|node_h0}} |
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| Symbol | t{8,4} | tr{8,8} |
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| Coxeter diagram | node_1|8|node_1|4|node}} | node_1|8|node_1|8|node_1}} |
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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. |
SymmetryThe 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 alternatedly 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 |
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| Diagram |
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| Coxeter | {{CDD>node_c1|8|node_c3|8|node_c2}} | +,8,8]
{{CDD>node_h0|8|node_c3|8|node_c2}} = {{CDD|label4|branch_c3|split2-88|node_c2}} | +]
{{CDD>node_c1|8|node_c3|8|node_h0}} = {{CDD|node_c1|split1-88|branch_c3|label4}} | +,8]
{{CDD>node_c1|8|node_h0|8|node_c2}} = {{CDD|label4|branch_c1|2a2b-cross|branch_c2|label4}} | +,8,8,1+]
{{CDD>node_h0|8|node_c3|8|node_h0}} = {{CDD|label4|branch_c3|4a4b-cross|branch_c3|label4}} | +,8+]
{{CDD>node_h2|8|node_h4|8|node_h2}} |
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| Orbifold | *882 | *884 | *4242 | *4444 | 44× |
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| Semidirect subgroups |
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| Diagram |
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| Coxeter | +]
{{CDD>node_c1|8|node_h2|8|node_h2}} | +,8]
{{CDD>node_h2|8|node_h2|8|node_c2}} | +)]
{{CDD>node_c3|split1-88|branch_h2h2|label2}} | +,8,1+]
{{CDD>node_c1|8|node_h0|8|node_h0}} = {{CDD|node_c1|8|node_h2|8|node_h0}} = {{CDD|node_c1|split1-88|branch_h2h2|label4}}
= {{CDD|node_c1|8|node_h0|8|node_h2}} = {{CDD|label4|branch_c1|2a2b-cross|branch_h2h2|label4}} | +,8,1+,8]
{{CDD>node_h0|8|node_h0|8|node_c2}} = {{CDD|node_h0|8|node_h2|8|node_c2}} = {{CDD|label4|branch_h2h2|split2-88|node_c2}}
= {{CDD|node_h2|8|node_h0|8|node_c2}} = {{CDD|label4|branch_h2h2|2a2b-cross|branch_c2|label4}} |
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| Orbifold | 8*4 | 2*44 | 4*44 |
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| Direct subgroups |
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| Index | 2 | 4 | 8 |
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| Diagram | |
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| Coxeter | +
{{CDD>node_h2|8|node_h2|8|node_h2}} | +]+
{{CDD>node_h0|8|node_h2|8|node_h2}} = {{CDD|label4|branch_h2h2|split2-88|node_h2}} | +,8]+
{{CDD>node_h2|8|node_h2|8|node_h0}} = {{CDD|node_h2|split1-88|branch_h2h2|label4}} | +,8]+
{{CDD>labelh|node|split1-88|branch_h2h2|label2}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}} | [8+,8+]+ = [1+,8,1+,8,1+]
{{CDD|node_h4|split1-88|branch_h4h4|label2}} = {{CDD|node_h0|8|node_h0|8|node_h0}} = {{CDD|node_h0|8|node_h2|8|node_h0}} = {{CDD|label4|branch_h2h2|4a4b-cross|branch_h2h2|label4}} |
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| Orbifold | 882 | 884 | 4242 | 4444 |
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| Radical subgroups |
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| Index | 16 | 32 |
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| Diagram |
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| Coxeter | {{CDD>node_c1|8|node_g|8|3sg|node_g}} | {{CDD>node_g|8|3sg|node_g|8|node_c2}} | +
{{CDD>node_h0|8|node_g|8|3sg|node_g}} | +
{{CDD>node_g|8|3sg|node_g|8|node_h0}} |
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| Orbifold | *44444444 | 44444444 |
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Related polyhedra and tiling {{Truncated figure3 table}}{{Order 8-4 tiling table}}{{Order 8-8 tiling table}}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)
- {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}
See also{{Commonscat|Uniform tiling 4-16-16}}- Square tiling
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes
External links - {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
- {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
{{Tessellation}} 6 : Hyperbolic tilings|Isogonal tilings|Order-4 tilings|Truncated tilings|Uniform tilings|Octagonal tilings |