Truncated trioctagonal tiling

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}.

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

Symmetry

Truncated trioctagonal tiling with mirrors
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 000 832 symmetry a00 832 symmetry 0bb 842 symmetry mirrors 832 symmetry 0zz
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 832 symmetry abb 842 symmetry aaa 832 symmetry azz
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

Truncated trioctagonal tiling
Order-3 octakis octagonal tiling
TypeDual semiregular hyperbolic tiling
FacesRight triangle
EdgesInfinite
VerticesInfinite
Coxeter diagramCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 8.pngCDel node f1.png
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.

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)
  • "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

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}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node.png or CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel label4.pngCDel branch hh.pngCDel split2.pngCDel node h.png
Uniform tiling 83-t0.png Uniform tiling 83-t01.png Uniform tiling 83-t1.png
Uniform tiling 433-t02.png
Uniform tiling 83-t12.png
Uniform tiling 433-t012.png
Uniform tiling 83-t2.png Uniform tiling 83-t02.png Uniform tiling 83-t012.png Uniform tiling 83-snub.png Uniform tiling 433-t0.pngUniform tiling 433-t1.png Uniform tiling 433-t02.pngUniform tiling 433-t12.png Uniform tiling 433-snub1.png
Uniform tiling 433-snub2.png
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
CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 83-t2.png Ord8 triakis triang til.png Uniform dual tiling 433-t01-yellow.png Uniform dual tiling 433-t012.png Uniform tiling 83-t0.png Deltoidal trioctagonal til.png Order-3 octakis octagonal tiling.png Uniform dual tiling 433-t0.png Uniform dual tiling 433-t01.png Uniform dual tiling 433-snub.png
*n32 symmetry mutations 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 Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png H2 tiling 237-7.png H2 tiling 238-7.png H2 tiling 23i-7.png H2 tiling 23j12-7.png H2 tiling 23j9-7.png H2 tiling 23j6-7.png H2 tiling 23j3-7.png
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 Spherical hexagonal bipyramid.png Spherical tetrakis hexahedron.png Spherical disdyakis dodecahedron.png Spherical disdyakis triacontahedron.png Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg H2checkers 237.png H2checkers 238.png H2checkers 23i.png H2 checkers 23j12.png H2 checkers 23j9.png H2 checkers 23j6.png H2 checkers 23j3.png
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

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