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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
Wythoff symbol 2 | 8 3|
3 3 | 4
Coxeter diagram or
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

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The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles, by Coxeter diagram .
Dual tiling
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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
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

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

Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
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
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
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
Figure
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
Dual uniform figures
Dual
conf.
V(3.3)2
V(3.4)2
V(3.5)2
V(3.6)2
V(3.7)2
V(3.8)2
V(3.∞)2
Dimensional family of quasiregular polyhedra and tilings: (8.n)2
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
Quasiregular
figures
configuration
3.8.3.8
4.8.4.8
8.5.8.5
8.6.8.6
8.7.8.7
8.8.8.8
8.∞.8.∞ 		 
8.∞.8.∞

See also

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Wikimedia Commons logo
Wikimedia Commons has media related to Uniform tiling 3-8-3-8.

References

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  • 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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