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truncated tetraoctagonal tiling

{{short description|Semiregular tiling in geometry}}

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U84_012}}

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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colspan=2|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=

File:Truncated_tetraoctagonal_tiling_with_mirrors.png

There 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).

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!colspan=12| Small index subgroups of [8,4] (*842)

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!Index

!1

!colspan=3|2

!colspan=2|4

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!Diagram

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!Coxeter

|[8,4]
{{CDD|node_c2|8|node_c3|4|node_c1}} = {{CDD|node_c3|split1-84|branch_c2-1|label2}}

|[1+,8,4]
{{CDD|node_h0|8|node_c3|4|node_c1}} = {{CDD|label4|branch_c3|split2-44|node_c1}}

|[8,4,1+]
{{CDD|node_c2|8|node_c3|4|node_h0}} = {{CDD|node_c2|split1-88|nodeab_c3}} = {{CDD|node_c2|split1-88|branch_c3|label2}}

|[8,1+,4]
{{CDD|node_c2|8|node_h0|4|node_c1}} = {{CDD|label4|branch_c2|2a2b-cross|nodeab_c1}}

|[1+,8,4,1+]
{{CDD|node_h0|8|node_c3|4|node_h0}} = {{CDD|label4|branch_c3|2a2b-cross|branch_c3|label4}}

|[8+,4+]
{{CDD|node_h2|8|node_h4|4|node_h2}}

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!Orbifold

|*842

|*444

|*882

|*4222

|*4242

|42×

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!colspan=7|Semidirect subgroups

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!Diagram

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!Coxeter

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|[8,4+]
{{CDD|node_c2|8|node_h2|4|node_h2}}

|[8+,4]
{{CDD|node_h2|8|node_h2|4|node_c1}}

|[(8,4,2+)]
{{CDD|node_c3|split1-48|branch_h2h2}}

|[8,1+,4,1+]
{{CDD|node_c2|8|node_h0|4|node_h0}} = {{CDD|node_c2|8|node_h2|4|node_h0}} = {{CDD|node_c2|split1-88|branch_h2h2|label2}}
= {{CDD|node_c2|8|node_h0|4|node_h2}} = {{CDD|label4|branch_c2|2a2b-cross|branch_h2h2|label2}}

|[1+,8,1+,4]
{{CDD|node_h0|8|node_h0|4|node_c1}} = {{CDD|node_h0|8|node_h2|4|node_c1}} = {{CDD|label4|branch_h2h2|split2-44|node_c1}}
= {{CDD|node_h2|8|node_h0|4|node_c1}} = {{CDD|label4|branch_h2h2|2a2b-cross|nodeab_c1}}

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!Orbifold

|

|4*4

|8*2

|2*42

|2*44

|4*22

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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

|[8,4]+
{{CDD|node_h2|8|node_h2|4|node_h2}} = {{CDD|node_h2|split1-84|branch_h2h2|label2}}

|[8,4+]+
{{CDD|node_h0|8|node_h2|4|node_h2}} = {{CDD|label4|branch_h2h2|split2-44|node_h2}}

|[8+,4]+
{{CDD|node_h2|8|node_h2|4|node_h0}} = {{CDD|node_h2|split1-88|branch_h2h2|label2}}

|[8,1+,4]+
{{CDD|labelh|node|split1-48|branch_h2h2}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label2}}

|colspan=2|[8+,4+]+ = [1+,8,1+,4,1+]
{{CDD|node_h4|split1-48|branch_h4h4|label2}} = {{CDD|node_h0|8|node_h0|4|node_h0}} = {{CDD|node_h0|8|node_h2|4|node_h0}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}

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!Orbifold

|842

|444

|882

|4222

|colspan=2|4242

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!colspan=6|Radical subgroups

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!Index

!

!8

!colspan=2|16

!32

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!Diagram

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!Coxeter

|

|[8,4*]
{{CDD|node_c2|8|node_g|4sg|node_g}} = {{CDD|label4|branch_c2|4a4b-cross|branch_c2|label4}}

|[8*,4]
{{CDD|node_g|8g|3sg|node_g|4|node_c1}}

|[8,4*]+
{{CDD|node_h0|8|node_g|4sg|node_g}} = {{CDD|label4|branch_h2h2|4a4b-cross|branch_h2h2|label4}}

|[8*,4]+
{{CDD|node_g|8g|3sg|node_g|4|node_h0}}

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!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.

{{Order 8-4 tiling table}}

{{Omnitruncated4 table}}

{{Omnitruncated_symmetric_table}}

See also

{{Commons category|Uniform tiling 4-8-16}}

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