Trioctagonal tiling

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Template:Short description

Trioctagonal tiling
Trioctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (3.8)2
Schläfli symbol r{8,3} or {83}
Wythoff symbol 2 | 8 3|
3 3 | 4
Coxeter diagram Template:CDD or Template:CDD
Template:CDD
Symmetry group [8,3], (*832)
[(4,3,3)], (*433)
Dual Order-8-3 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling. There are two triangles and two octagons alternating on each vertex. It has Schläfli symbol of r{8,3}.

Symmetry

File:H2 tiling 334-3.png
The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles, by Coxeter diagram Template:CDD. 		File:Uniform dual tiling 433-t01.png
Dual tiling

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform octagonal/triangular tilings Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
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}
Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
Template:CDD
Template:CDD 		Template:CDD
Template:CDD 		Template:CDD
Template:CDD 		Template:CDD Template:CDD
Template:CDD or Template:CDD 		Template:CDD
Template:CDD or Template:CDD 		Template:CDD
Template:CDD
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.pngFile:Uniform tiling 433-t1.png File:Uniform tiling 433-t02.pngFile: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
Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
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

It can also be generated from the (4 3 3) hyperbolic tilings:

Uniform (4,3,3) tilings Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
File:H2 tiling 334-1.png File:H2 tiling 334-3.png File:H2 tiling 334-2.png File:H2 tiling 334-6.png File:H2 tiling 334-4.png File:H2 tiling 334-5.png File:H2 tiling 334-7.png File:Uniform tiling 433-snub2.png
h{8,3}
t0(4,3,3) 		r{3,8}1/2
t0,1(4,3,3) 		h{8,3}
t1(4,3,3) 		h2{8,3}
t1,2(4,3,3) 		{3,8}1/2
t2(4,3,3) 		h2{8,3}
t0,2(4,3,3) 		t{3,8}1/2
t0,1,2(4,3,3) 		s{3,8}1/2
s(4,3,3)
Uniform duals
File:Uniform dual tiling 433-t0.png File:Uniform dual tiling 433-t01.png File:Uniform dual tiling 433-t0.png File:Uniform dual tiling 433-t12.png File:H2-8-3-dual.svg File:Uniform dual tiling 433-t12.png File:H2-8-3-kis-dual.svg File:Uniform dual tiling 433-snub.png
V(3.4)3 V3.8.3.8 V(3.4)3 V3.6.4.6 V(3.3)4 V3.6.4.6 V6.6.8 V3.3.3.3.3.4

The trioctagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:

Quasiregular tilings: (3.n)2 Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
Sym.
*n32
[n,3] 		Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*332
[3,3]
Td 		*432
[4,3]
Oh 		*532
[5,3]
Ih 		*632
[6,3]
p6m 		*732
[7,3]
  		*832
[8,3]...
  		*∞32
[∞,3]
  		[12i,3] [9i,3] [6i,3]
Figure
File:Quasiregular fundamental domain.png 		File:Uniform tiling 332-t1-1-.svg File:Uniform tiling 432-t1.svg File:Uniform tiling 532-t1.png File:Uniform tiling 63-t1.svg File:Triheptagonal tiling.svg File:H2-8-3-rectified.svg File:H2 tiling 23i-2.png File:H2 tiling 23j12-2.png File:H2 tiling 23j9-2.png File:H2 tiling 23j6-2.png
Figure
File:Half quasiregular fundamental domain.png 		File:Uniform tiling 332-t02.svg File:Uniform tiling 333-t12.svg File:H2 tiling 334-3.png File:H2 tiling 33i-3.png
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2 (3.12i)2 (3.9i)2 (3.6i)2
Schläfli r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞} r{3,12i} r{3,9i} r{3,6i}
Coxeter
Template:CDD
Template:CDD 		Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
Template:CDD Template:CDD Template:CDD Template:CDD
Dual uniform figures
Dual
conf. 		File:Uniform tiling 432-t0.svg
V(3.3)2 		File:Spherical rhombic dodecahedron.svg
V(3.4)2 		File:Spherical rhombic triacontahedron.svg
V(3.5)2 		File:Rhombic star tiling.svg
V(3.6)2 		File:7-3 rhombille tiling.svg
V(3.7)2 		File:H2-8-3-rhombic.svg
V(3.8)2 		File:Ord3infin qreg rhombic til.png
V(3.∞)2
Dimensional family of quasiregular polyhedra and tilings: (8.n)2 Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
Symmetry
*8n2
[n,8] 		Hyperbolic... Paracompact Noncompact
*832
[3,8] 		*842
[4,8] 		*852
[5,8] 		*862
[6,8] 		*872
[7,8] 		*882
[8,8]... 		*∞82
[∞,8] 		 
[iπ/λ,8]
Coxeter Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
Quasiregular
figures
configuration 		File:H2-8-3-rectified.svg
3.8.3.8 		File:H2 tiling 248-2.png
4.8.4.8 		File:H2 tiling 258-2.png
8.5.8.5 		File:H2 tiling 268-2.png
8.6.8.6 		File:H2 tiling 278-2.png
8.7.8.7 		File:H2 tiling 288-2.png
8.8.8.8 		File:H2 tiling 25i-2.png
8.∞.8.∞ 		 
8.∞.8.∞

See also

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References

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