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Truncated order-6 square tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U64_12}}

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

Uniform colorings

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The half symmetry [1+,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons, with as Coxeter diagram {{CDD|branch_11|split2-44|node_1}}.

Symmetry

File:Truncated_order-6_square_tiling_with_mirrors.png

The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.

A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).

The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain.

{{-}}

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!colspan=12| Small index subgroups of [(4,4,3)] (*443)

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!Index

!1

!colspan=2|2

!6

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!Diagram

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!Coxeter
(orbifold)

|[(4,4,3)] = {{CDD|node_c1|split1-44|branch_c2}}
(*443)

|[(4,1+,4,3)] = {{CDD|labelh|node|split1-44|branch_c2}} = {{CDD|branch_c2|2a2b-cross|branch_c2}}
(*3232)

|[(4,4,3+)] = {{CDD|node_c1|split1-44|branch_h2h2}}
(3*22)

|[(4,4,3*)] = {{CDD|node_c1|split1-44|branch|labels}}
(*222222)

colspan=5|Direct subgroups
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!Index

!2

!colspan=2|4

!12

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!Diagram

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|colspan=2|150px

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!Coxeter
(orbifold)

|[(4,4,3)]+ = {{CDD|node_h2|split1-44|branch_h2h2}}
(443)

|colspan=2|[(4,4,3+)]+ = {{CDD|labelh|node|split1-44|branch_h2h2}} = {{CDD|branch_h2h2|2xa2xb-cross|branch_h2h2}}
(3232)

|[(4,4,3*)]+ = {{CDD|node_h2|split1-44|branch|labels}}
(222222)

Related polyhedra and tilings

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

{{Order 6-4 tiling table}}

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

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

{{Truncated figure4 table}}

{{Omnitruncated34 table}}

See also

{{Commonscat|Uniform tiling 6-8-8}}

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

{{Tessellation}}

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Order-6 tilings

Category:Square tilings

Category:Truncated tilings

Category:Uniform tilings