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order-8 triangular tiling

{{Short description|Concept in geometry}}

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U83_2}}

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

Uniform colorings

The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles:

:240px

Symmetry

File:Octagonal_tiling_with_444_mirror_lines.png

From [(4,4,4)] symmetry, there are 15 small index subgroups (7 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. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, [(1+,4,1+,4,1+,4)] (222222) is the commutator subgroup of [(4,4,4)].

A larger subgroup is constructed [(4,4,4*)], index 8, as (2*2222) with gyration points removed, becomes (*22222222).

The symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as 832 symmetry, by 3 bisecting mirrors per domain.

{{-}}

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|+ Small index subgroups of [(4,4,4)] (*444)

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!Index

!1

!colspan=3|2

!colspan=2|4

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!Diagram

|120px

|120px

|120px

|120px

|120px

|120px

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!Coxeter

|[(4,4,4)]
{{CDD|node_c1|split1-44|branch_c3-2|label4}}

|[(1+,4,4,4)]
{{CDD|labelh|node|split1-44|branch_c3-2|label4}} = {{CDD|label4|branch_c3-2|2a2b-cross|branch_c3-2|label4}}

|[(4,1+,4,4)]
{{CDD|node_c1|split1-44|branch_h0c2|label4}} = {{CDD|label4|branch_c1-2|2a2b-cross|branch_c1-2|label4}}

|[(4,4,1+,4)]
{{CDD|node_c1|split1-44|branch_c3h0|label4}} = {{CDD|label4|branch_c1-3|2a2b-cross|branch_c1-3|label4}}

|[(1+,4,1+,4,4)]
{{CDD|labelh|node|split1-44|branch_h0c2|label4}}

|[(4+,4+,4)]
{{CDD|node_h4|split1-44|branch_h2h2|label4}}

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!Orbifold

|*444

|colspan=3|*4242

|2*222

|222×

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!Diagram

|

|120px

|120px

|120px

|120px

|120px

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!Coxeter

|

|[(4,4+,4)]
{{CDD|node_c1|split1-44|branch_h2h2|label4}}

|[(4,4,4+)]
{{CDD|node_h2|split1-44|branch_c3h2|label4}}

|[(4+,4,4)]
{{CDD|node_h2|split1-44|branch_h2c2|label4}}

|[(4,1+,4,1+,4)]
{{CDD|node_c1|split1-44|branch_h0h0|label4}}

|[(1+,4,4,1+,4)]
{{CDD|labelh|node|split1-44|branch_c3h2|label4}} = {{CDD|label4|branch_c3h2|2a2b-cross|branch_c3h2|label4}}

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!Orbifold

|

|colspan=3|4*22

|colspan=2|2*222

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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

|120px

|120px

|120px

|120px

|colspan=2|120px

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!Coxeter

|[(4,4,4)]+
{{CDD|node_h2|split1-44|branch_h2h2|label4}}

|[(4,4+,4)]+
{{CDD|labelh|node|split1-44|branch_h2h2|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}

|[(4,4,4+)]+
{{CDD|node_h2|split1-44|branch_h0h2|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}

|[(4+,4,4)]+
{{CDD|node_h2|split1-44|branch_h2h0|label4}} = {{CDD|label4|branch_h2h2|2xa2xb-cross|branch_h2h2|label4}}

|colspan=2|[(4,1+,4,1+,4)]+
{{CDD|labelh|node|split1-44|branch_h0h0|label4}} = {{CDD|node_h4|split1-44|branch_h4h4|label4}}

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!Orbifold

|444

|colspan=3|4242

|colspan=2|222222

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!colspan=7|Radical subgroups

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!Index

!colspan=3|8

!colspan=3|16

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!Diagram

|120px

|120px

|120px

|120px

|120px

|120px

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!Coxeter

|[(4,4*,4)]

|[(4,4,4*)]

|[(4*,4,4)]

|[(4,4*,4)]+

|[(4,4,4*)]+

|[(4*,4,4)]+

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!Orbifold

|colspan=3|*22222222

|colspan=3|22222222

Related polyhedra and tilings

File:H3_338_UHS_plane_at_infinity.png honeycomb has {3,8} vertex figures.]]

{{Triangular regular tiling}}

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

{{Octagonal tiling table}}

{{Order-8 regular tilings}}

It can also be generated from the (4 3 3) hyperbolic tilings:

{{Order 4-3-3 tiling table}}

{{Order 4-4-4 tiling table}}

See also

{{Commonscat|Order-8 triangular tiling}}

References

{{reflist}}

{{refbegin}}

  • 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}}

{{refend}}

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:Isohedral tilings

Category:Order-8 tilings

Category:Regular tilings

Category:Triangular tilings