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 File:Truncated trioctagonal tiling with mirrors.png 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
File:832 symmetry 000.png File:832 symmetry a00.png File:832 symmetry 0bb.png File:842 symmetry mirrors.png File:832 symmetry 0zz.png
Coxeter orbifold )
[8,3] = File:CDel node c1.png File:CDel 8.png File:CDel node c2.png File:CDel 3.png File:CDel node c2.png
[1+ ,8,3] = File:CDel node h0.png File:CDel 8.png File:CDel node c2.png File:CDel 3.png File:CDel node c2.png File:CDel label4.png File:CDel branch c2.png File:CDel split2.png File:CDel node c2.png *433 )
[8,3+ ] = File:CDel node c1.png File:CDel 8.png File:CDel node h2.png File:CDel 3.png File:CDel node h2.png
[8,3⅄ ] = File:CDel node c1.png File:CDel 8.png File:CDel node c2.png File:CDel 3trionic.png File:CDel node c2.png File:CDel node c1.png File:CDel 4.png File:CDel node c1.png File:CDel 8.png File:CDel node c2.png *842 )
[8,3* ] = File:CDel node c1.png File:CDel 8.png File:CDel node g.png File:CDel 3sg.png File:CDel node g.png File:CDel label4.png File:CDel branch c1.png File:CDel split2-44.png File:CDel node c1.png *444 )
Direct subgroups
Index
2
4
6
12
Diagrams
File:832 symmetry aaa.png File:832 symmetry abb.png File:842 symmetry aaa.png File:832 symmetry azz.png
Coxeter
[8,3]+ = File:CDel node h2.png File:CDel 8.png File:CDel node h2.png File:CDel 3.png File:CDel node h2.png
[8,3+ ]+ = File:CDel node h0.png File:CDel 8.png File:CDel node h2.png File:CDel 3.png File:CDel node h2.png File:CDel label4.png File:CDel branch h2h2.png File:CDel split2.png File:CDel node h2.png
[8,3⅄ ]+ = File:CDel node h2.png File:CDel 8.png File:CDel node h2.png File:CDel 3trionic.png File:CDel node h2.png File:CDel node h2.png File:CDel 4.png File:CDel node h2.png File:CDel 8.png File:CDel node h2.png
[8,3* ]+ = File:CDel node h2.png File:CDel 8.png File:CDel node g.png File:CDel 3sg.png File:CDel node g.png File:CDel label4.png File:CDel branch h2h2.png File:CDel split2-44.png File:CDel node h2.png
Order 3-8 kisrhombille 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]+ .
Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832)
[8,3]+
[1+ ,8,3]
[8,3+ ]
{8,3}
t{8,3}
r{8,3}
t{3,8}
{3,8}
rr{8,3} 2 {3,8}
tr{8,3}
sr{8,3}
h{8,3}
h2 {8,3}
s{3,8}
File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 8.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node 1.png File:CDel 8.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node h.png File:CDel 8.png File:CDel node h.png File:CDel 3.png File:CDel node h.png
File:CDel node.png File:CDel 8.png File:CDel node h.png File:CDel 3.png File:CDel node h.png
File:CDel node h0.png File:CDel 8.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel label4.png File:CDel branch 11.png File:CDel split2.png File:CDel node.png File:CDel node h0.png File:CDel 8.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel label4.png File:CDel branch 11.png File:CDel split2.png File:CDel node 1.png File:CDel node h0.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel label4.png File:CDel branch.png File:CDel split2.png File:CDel node 1.png File:CDel node 1.png File:CDel 8.png File:CDel node h.png File:CDel 3.png File:CDel node h.png
File:CDel node h1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel label4.png File:CDel branch 10ru.png File:CDel split2.png File:CDel node.png File:CDel label4.png File:CDel branch 01rd.png File:CDel split2.png File:CDel node.png File:CDel node h1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel label4.png File:CDel branch 10ru.png File:CDel split2.png File:CDel node 1.png File:CDel label4.png File:CDel branch 01rd.png File:CDel split2.png File:CDel node 1.png File:CDel node h0.png File:CDel 8.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel label4.png File:CDel branch hh.png File:CDel split2.png File:CDel node h.png
File:H2-8-3-dual.svg File:H2-8-3-trunc-dual.svg File:H2-8-3-rectified.svg File:Uniform tiling 433-t01.png File:H2-8-3-trunc-primal.svg File:Uniform tiling 433-t012.png File:H2-8-3-primal.svg File:Uniform tiling 433-t2.png File:H2-8-3-cantellated.svg File:H2-8-3-omnitruncated.svg File:H2-8-3-snub.svg File:Uniform tiling 433-t0.png File:Uniform tiling 433-t1.png File:Uniform tiling 433-t02.png File:Uniform tiling 433-t12.png File:Uniform tiling 433-snub1.png File: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
File:CDel node f1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node f1.png File:CDel 8.png File:CDel node f1.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node f1.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node f1.png File:CDel 3.png File:CDel node f1.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node f1.png File:CDel node f1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node f1.png File:CDel node f1.png File:CDel 8.png File:CDel node f1.png File:CDel 3.png File:CDel node f1.png File:CDel node fh.png File:CDel 8.png File:CDel node fh.png File:CDel 3.png File:CDel node fh.png File:CDel node fh.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node fh.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node f1.png File:CDel node.png File:CDel 8.png File:CDel node fh.png File:CDel 3.png File:CDel node fh.png
File:H2-8-3-primal.svg File:H2-8-3-kis-primal.svg File:H2-8-3-rhombic.svg File:H2-8-3-kis-dual.svg File:H2-8-3-dual.svg File:H2-8-3-deltoidal.svg File:H2-8-3-kisrhombille.svg File:H2-8-3-floret.svg File:Uniform dual tiling 433-t0.png File:Uniform dual tiling 433-t01.png File:Uniform dual tiling 433-snub.png
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram File:CDel node 1.png File:CDel p.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 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 .
*n 32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.*n 32 [n ,3]
Spherical
Euclid.
Compact hyperb.
Paraco.
Noncompact hyperbolic
*232
*332
*432
*532
*632
*732
*832
*∞32
Figures
File:Spherical truncated trigonal prism.png File:Uniform tiling 332-t012.png File:Uniform tiling 432-t012.png File:Uniform tiling 532-t012.png File:Uniform polyhedron-63-t012.png File:Truncated triheptagonal tiling.svg File:H2-8-3-omnitruncated.svg File:H2 tiling 23i-7.png File:H2 tiling 23j12-7.png File:H2 tiling 23j9-7.png File:H2 tiling 23j6-7.png File: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
File:Spherical hexagonal bipyramid.svg File:Spherical tetrakis hexahedron.svg File:Spherical disdyakis dodecahedron.svg File:Spherical disdyakis triacontahedron.svg File:Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg File:H2checkers 237.png File:H2checkers 238.png File:H2checkers 23i.png File:H2 checkers 23j12.png File:H2 checkers 23j9.png File:H2 checkers 23j6.png File: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
See also References External links
