Truncated trioctagonal tiling

File:Truncated trioctagonal tiling with mirrors.png

The dual of this tiling, the order 3-8 kisrhombille, represents the fundamental domains of [8,3] (*832) symmetry. There are 3 small index subgroups constructed from [8,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

A larger index 6 subgroup constructed as [8,3^*], becomes [(4,4,4)], (*444). An intermediate index 3 subgroup is constructed as [8,3^⅄], with 2/3 of blue mirrors removed.

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{{Infobox face-uniform tiling |

Name=Order 3-8 kisrhombille|

Image_File=H2-8-3-kisrhombille.svg|

Type=Dual semiregular hyperbolic tiling|

Cox={{CDD|node_f1|3|node_f1|8|node_f1}} |

Face_List=Right triangle|

Edge_Count=Infinite|

Vertex_Count=Infinite|

Symmetry_Group=[8,3], (*832)|

Rotation_Group=[8,3]⁺, (832)|

Face_Type=V4.6.16|

Dual=Truncated trioctagonal tiling|

Property_List=face-transitive|

}}

The order 3-8 kisrhombille is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 16 triangles meeting at each vertex.

The image shows a Poincaré disk model projection of the hyperbolic plane.

It is labeled V4.6.16 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 16 triangles. It is the dual tessellation of the truncated trioctagonal tiling which has one square and one octagon and one hexakaidecagon at each vertex.

An alternative name is 3-8 kisrhombille by Conway, seeing it as a 3-8 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.

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This tiling is one of 10 uniform tilings constructed from [8,3] hyperbolic symmetry and three subsymmetries [1⁺,8,3], [8,3⁺] and [8,3]⁺.

{{Octagonal_tiling_table}}

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram {{CDD|node_1|p|node_1|3|node_1}}. For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

{{Omnitruncated table}}

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{{Commons category|Uniform tiling 4-6-16}}

{{Commons category|Uniform dual tiling V 4-6-16}}

• Tilings of regular polygons
• Hexakis triangular tiling
• List of uniform tilings
• Uniform tilings in hyperbolic plane

• 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}}

• {{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