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Truncated order-8 octagonal tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U88_01}}

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:

:240px

Related polyhedra and tiling

{{Order 8-8 tiling table}}

= 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)].

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|+ Small index subgroups of [(8,8,4)] (*884)

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!Fundamental
domains

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|80px
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|80px
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|80px
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|valign=top|80px

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!Subgroup index

!1

!colspan=3|2

!colspan=2|4

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!Coxeter

|[(8,8,4)]
{{CDD|node|split1-88|branch|label4}}

|[(1+,8,8,4)]
{{CDD|node_c1|split1-88|branch_h0c2|label4}}

|[(8,8,1+,4)]
{{CDD|node_c1|split1-88|branch_c3h0|label4}}

|[(8,1+,8,4)]
{{CDD|labelh|node|split1-88|branch_c3-2|label4}}

|[(1+,8,8,1+,4)]
{{CDD|labelh|node|split1-88|branch_c3h0|label4}}

|[(8+,8+,4)]
{{CDD|node_c1|split1-88|branch_h0h0|label4}}

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!orbifold

|*884

|colspan=2|*8482

|*4444

|2*4444

|442×

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!Coxeter

|

|[(8,8+,4)]
{{CDD|node_h2|split1-88|branch_c3h2|label4}}

|[(8+,8,4)]
{{CDD|node_h2|split1-88|branch_h2c2|label4}}

|[(8,8,4+)]
{{CDD|node_c1|split1-88|branch_h2h2|label4}}

|[(8,1+,8,1+,4)]
{{CDD|labelh|node|split1-88|branch_h0c2|label4}}

|[(1+,8,1+,8,4)]
{{CDD|node_h4|split1-88|branch_h2h2|label4}}

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!Orbifold

|

|colspan=2|8*42

|4*44

|colspan=2|4*4242

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!colspan=9|Direct subgroups

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!Subgroup index

!2

!colspan=3|4

!colspan=2|8

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!Coxeter

|[(8,8,4)]+
{{CDD|node_h2|split1-88|branch_h2h2|label4}}

|[(1+,8,8+,4)]
{{CDD|node_h2|split1-88|branch_h0h2|label4}}

|[(8+,8,1+,4)]
{{CDD|node_h2|split1-88|branch_h2h0|label4}}

|[(8,1+,8,4+)]
{{CDD|labelh|node|split1-88|branch_h2h2|label4}}

|colspan=2|[(1+,8,1+,8,1+,4)] = [(8+,8+,4+)]
{{CDD|node_h4|split1-88|branch_h4h4|label4}}

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!Orbifold

|844

|colspan=2|8482

|4444

|colspan=2|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)
  • {{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

External links

  • {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
  • {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
  • [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
  • [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
  • [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Order-8 tilings

Category:Truncated tilings

Category:Uniform tilings

Category:Octagonal tilings