Truncated trioctagonal tiling

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Truncated trioctagonal tiling
Truncated trioctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.6.16
Schläfli symbol tr{8,3} or
Wythoff symbol 2 8 3 |
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png or CDel node 1.pngCDel split1-83.pngCDel nodes 11.png
Symmetry group [8,3], (*832)
Dual Order 3-8 kisrhombille
Properties Vertex-transitive

In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of tr{8,3}.

Symmetry[edit]

Truncated trioctagonal tiling with mirror lines

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.

Small index subgroups of [8,3], (*832)
Index 1 2 3 6
Diagrams 832 symmetry 000.png 832 symmetry a00.png 832 symmetry 0bb.png 842 symmetry mirrors.png 832 symmetry 0zz.png
Coxeter
(orbifold)
[8,3] = CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 3.pngCDel node c2.png
(*832)
[1+,8,3] = CDel node h0.pngCDel 8.pngCDel node c2.pngCDel 3.pngCDel node c2.png = CDel label4.pngCDel branch c2.pngCDel split2.pngCDel node c2.png
(*433)
[8,3+] = CDel node c1.pngCDel 8.pngCDel node h2.pngCDel 3.pngCDel node h2.png
(3*4)
[8,3] = CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 3trionic.pngCDel node c2.png = CDel node c1.pngCDel 4.pngCDel node c1.pngCDel 8.pngCDel node c2.png
(*842)
[8,3*] = CDel node c1.pngCDel 8.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel label4.pngCDel branch c1.pngCDel split2-44.pngCDel node c1.png
(*444)
Direct subgroups
Index 2 4 6 12
Diagrams 832 symmetry aaa.png 832 symmetry abb.png 842 symmetry aaa.png 832 symmetry azz.png
Coxeter
(orbifold)
[8,3]+ = CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 3.pngCDel node h2.png
(832)
[8,3+]+ = CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel label4.pngCDel branch h2h2.pngCDel split2.pngCDel node h2.png
(433)
[8,3]+ = CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 3trionic.pngCDel node h2.png = CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 8.pngCDel node h2.png
(842)
[8,3*]+ = CDel node h2.pngCDel 8.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel label4.pngCDel branch h2h2.pngCDel split2-44.pngCDel node h2.png
(444)

Order 3-8 kisrhombille[edit]

Truncated trioctagonal tiling
Order-3 octakis octagonal tiling.png
Type Dual semiregular hyperbolic tiling
Faces Right triangle
Edges Infinite
Vertices Infinite
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 8.pngCDel node f1.png
Symmetry group [8,3], (*832)
Rotation group [8,3]+, (832)
Dual polyhedron Truncated trioctagonal tiling
Face configuration V4.6.16
Properties 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.

Naming[edit]

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.

Related polyhedra and tilings[edit]

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]+.

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. 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.

See also[edit]

References[edit]

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. 

External links[edit]