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Truncated hexaoctagonal tiling

{{Short description|Semi regularly Tiling}}

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U86_012}}

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

Dual tiling

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colspan=2|The dual tiling is called an order-6-8 kisrhombille tiling, made as a complete bisection of the order-6 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,6] (*862) symmetry.

Symmetry

File:H2_tiling_268-7-mirrors.png

There are six reflective subgroup kaleidoscopic constructed from [8,6] by removing one or two of three mirrors. 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+,6,1+] (4343) is the commutator subgroup of [8,6].

A radical subgroup is constructed as [8,6*], index 12, as [8,6+], (6*4) with gyration points removed, becomes (*444444), and another [8*,6], index 16 as [8+,6], (8*3) with gyration points removed as (*33333333).

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|+ Small index subgroups of [8,6] (*862)

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!Index

!1

!colspan=3|2

!colspan=2|4

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!Diagram

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!Coxeter

|[8,6]
{{CDD|node_c2|8|node_c3|6|node_c1}} = {{CDD|node_c3|split1-86|branch_c2-1}}

|[1+,8,6]
{{CDD|node_h0|8|node_c3|6|node_c1}} = {{CDD|label4|branch_c3|split2-66|node_c1}}

|[8,6,1+]
{{CDD|node_c2|8|node_c3|6|node_h0}} = {{CDD|node_c2|split1-88|branch_c3}} = {{CDD|node_c2|split1-88|branch_c3}}

|[8,1+,6]
{{CDD|node_c2|8|node_h0|6|node_c1}} = {{CDD|label4|branch_c2|2a2b-cross|branch_c1}}

|[1+,8,6,1+]
{{CDD|node_h0|8|node_c3|6|node_h0}} = {{CDD|label4|branch_c3|3a3b-cross|branch_c3|label4}}

|[8+,6+]
{{CDD|node_h2|8|node_h4|6|node_h2}}

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!Orbifold

|*862

|*664

|*883

|*4232

|*4343

|43×

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!colspan=7|Semidirect subgroups

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!Diagram

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!Coxeter

|

|[8,6+]
{{CDD|node_c2|8|node_h2|6|node_h2}}

|[8+,6]
{{CDD|node_h2|8|node_h2|6|node_c1}}

|[(8,6,2+)]
{{CDD|node_c3|split1-86|branch_h2h2}}

|[8,1+,6,1+]
{{CDD|node_c2|8|node_h0|6|node_h0}} = {{CDD|node_c2|8|node_h2|6|node_h0}} = {{CDD|node_c2|split1-88|branch_h2h2}}
= {{CDD|node_c2|8|node_h0|6|node_h2}} = {{CDD|label4|branch_c2|2a2b-cross|branch_h2h2}}

|[1+,8,1+,6]
{{CDD|node_h0|8|node_h0|6|node_c1}} = {{CDD|node_h0|8|node_h2|6|node_c1}} = {{CDD|label4|branch_h2h2|split2-66|node_c1}}
= {{CDD|node_h2|8|node_h0|6|node_c1}} = {{CDD|label4|branch_h2h2|2a2b-cross|branch_c1}}

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!Orbifold

|

|6*4

|8*3

|2*43

|3*44

|4*33

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!colspan=7|Direct subgroups

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!Index

!2

!colspan=3|4

!colspan=2|8

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!Diagram

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!Coxeter

|[8,6]+
{{CDD|node_h2|8|node_h2|6|node_h2}} = {{CDD|node_h2|split1-86|branch_h2h2|label2}}

|[8,6+]+
{{CDD|node_h0|8|node_h2|6|node_h2}} = {{CDD|label4|branch_h2h2|split2-66|node_h2}}

|[8+,6]+
{{CDD|node_h2|8|node_h2|6|node_h0}} = {{CDD|node_h2|split1-88|branch_h2h2}}

|[8,1+,6]+
{{CDD|labelh|node|split1-86|branch_h2h2}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2}}

|colspan=2|[8+,6+]+ = [1+,8,1+,6,1+]
{{CDD|node_h4|split1-86|branch_h4h4|label2}} = {{CDD|node_h0|8|node_h0|6|node_h0}} = {{CDD|node_h0|8|node_h2|6|node_h0}} = {{CDD|label4|branch_h2h2|3a3b-cross|branch_h2h2|label4}}

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!Orbifold

|862

|664

|883

|4232

|colspan=2|4343

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!colspan=6|Radical subgroups

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!Index

!

!12

!24

!16

!32

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!Diagram

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!Coxeter

|

|[8,6*]
{{CDD|node_c2|8|node_g|6g|3sg|node_g}}

|[8*,6]
{{CDD|node_g|8g|3sg|node_g|6|node_c1}}

|[8,6*]+
{{CDD|node_h0|8|node_g|6g|3sg|node_g}}

|[8*,6]+
{{CDD|node_g|8g|3sg|node_g|6|node_h0}}

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!Orbifold

|

|*444444

|*33333333

|444444

|33333333

Related polyhedra and tilings

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 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 [8,6] symmetry, and 7 with subsymmetry.

{{Order 8-6 tiling table}}

See also

{{Commonscat|Uniform tiling 4-12-16}}

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}}

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.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Semiregular tilings

Category:Truncated tilings