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}
Wythoff symbol 2 8 3 |
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.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

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).

Small index subgroups of [8,3], (*832)
Index 1 2 6
Diagrams 832 symmetry 000.png 832 symmetry a00.png 832 symmetry 0bb.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 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 12
Diagrams 832 symmetry aaa.png 832 symmetry abb.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 g.pngCDel 3sg.pngCDel node g.png = CDel label4.pngCDel branch h2h2.pngCDel split2-44.pngCDel node h2.png
(444)

Order 3-8 kisrhombille

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

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

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

References

  • 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)
  • Lua error in package.lua at line 80: module 'strict' not found.

External links


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