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truncated infinite-order triangular tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|Ui3_12}}

In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.

Symmetry

File:Truncated_infinite-order_triangular_tiling_with_mirrors.png

The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.

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|+ Small index subgroups of [(∞,3,3)], (*∞33)

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!Type

!Reflectional

!Rotational

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!Index

!1

!2

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!Diagram

|160px

|160px

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!Coxeter
(orbifold)

|[(∞,3,3)]
{{CDD|node_c1|split1|branch_c1|labelinfin}}
(*∞33)

|[(∞,3,3)]+
{{CDD|node_h2|split1|branch_h2h2|labelinfin}}
(∞33)

Related polyhedra and tiling

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.

{{Truncated figure2 table}}

{{Order i-3 tiling table}}

{{Order_i-3-3_tiling_table}}

See also

{{Commons category|Uniform tiling 6-6-i}}

References

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  • 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 }}

{{Tessellation}}

Category:Hyperbolic tilings

Category:Infinite-order tilings

Category:Isogonal tilings

Category:Triangular tilings

Category:Truncated tilings

Category:Uniform tilings