Truncated trioctagonal tiling
Truncated trioctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.6.16
Schläfli symboltr{8,3} or
Wythoff symbol2 8 3 |
Coxeter diagram or
Symmetry group[8,3], (*832)
DualOrder 3-8 kisrhombille
PropertiesVertex-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). 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
Coxeter
(orbifold)		 [8,3] =
(*832)		 [1+,8,3] = =
(*433)		 [8,3+] =
(3*4)		 [8,3⅄] = =
(*842)		 [8,3*] = =
(*444)
Direct subgroups
Index 2 4 6 12
Diagrams
Coxeter
(orbifold)		 [8,3]+ =
(832)		 [8,3+]+ = =
(433)		 [8,3⅄]+ = =
(842)		 [8,3*]+ = =
(444)

Order 3-8 kisrhombille

Truncated trioctagonal tiling
TypeDual semiregular hyperbolic tiling
FacesRight triangle
EdgesInfinite
VerticesInfinite
Coxeter diagram
Symmetry group[8,3], (*832)
Rotation group[8,3]+, (832)
Dual polyhedronTruncated trioctagonal tiling
Face configurationV4.6.16
Propertiesface-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.

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]+.

Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)		 [1+,8,3]
(*443)		 [8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}		 tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}


or
or




Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4

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

*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]		 Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]		 *332
[3,3]		 *432
[4,3]		 *532
[5,3]		 *632
[6,3]		 *732
[7,3]		 *832
[8,3]		 *∞32
[∞,3]		  
[12i,3]		  
[9i,3]		  
[6i,3]		  
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i

See also

Wikimedia Commons has media related to Uniform tiling 4-6-16.
Wikimedia Commons has media related to Uniform dual tiling V 4-6-16.

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)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
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