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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

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 CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png.
Uniform dual tiling 433-t01.png
Dual tiling

Dual tiling
Related polyhedra and tilings

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}
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 h0.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch.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
H2-8-3-dual.svg H2-8-3-trunc-dual.svg H2-8-3-rectified.svg
Uniform tiling 433-t01.png
H2-8-3-trunc-primal.svg
Uniform tiling 433-t012.png
H2-8-3-primal.svg
Uniform tiling 433-t2.png
H2-8-3-cantellated.svg H2-8-3-omnitruncated.svg H2-8-3-snub.svg 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
H2-8-3-primal.svg H2-8-3-kis-primal.svg H2-8-3-rhombic.svg H2-8-3-kis-dual.svg H2-8-3-dual.svg H2-8-3-deltoidal.svg H2-8-3-kisrhombille.svg H2-8-3-floret.svg Uniform dual tiling 433-t0.png Uniform dual tiling 433-t01.png Uniform dual tiling 433-snub.png

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)
CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch hh.pngCDel split2.pngCDel node h.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
H2 tiling 334-1.png H2 tiling 334-3.png H2 tiling 334-2.png H2 tiling 334-6.png H2 tiling 334-4.png H2 tiling 334-5.png H2 tiling 334-7.png 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
Uniform dual tiling 433-t0.png Uniform dual tiling 433-t01.png Uniform dual tiling 433-t0.png Uniform dual tiling 433-t12.png H2-8-3-dual.svg Uniform dual tiling 433-t12.png H2-8-3-kis-dual.svg 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
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
Quasiregular fundamental domain.png
Uniform tiling 332-t1-1-.png Uniform tiling 432-t1.png Uniform tiling 532-t1.png Uniform tiling 63-t1.svg Triheptagonal tiling.svg H2-8-3-rectified.svg H2 tiling 23i-2.png H2 tiling 23j12-2.png H2 tiling 23j9-2.png H2 tiling 23j6-2.png
Figure
Half quasiregular fundamental domain.png
Uniform tiling 332-t02.png Uniform tiling 333-t12.png H2 tiling 334-3.png 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
CDel node.pngCDel n.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel labelp.pngCDel branch 11.pngCDel split2.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node.png CDel branch 11.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.png
Dual uniform figures
Dual
conf.
Uniform tiling 432-t0.png
V(3.3)2
Spherical rhombic dodecahedron.png
V(3.4)2
Spherical rhombic triacontahedron.png
V(3.5)2
Rhombic star tiling.png
V(3.6)2
7-3 rhombille tiling.svg
V(3.7)2
H2-8-3-rhombic.svg
V(3.8)2
Ord3infin qreg rhombic til.png
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 CDel node.pngCDel 3.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 8.pngCDel node.png
Quasiregular
figures
configuration
H2-8-3-rectified.svg
3.8.3.8
H2 tiling 248-2.png
4.8.4.8
H2 tiling 258-2.png
8.5.8.5
H2 tiling 268-2.png
8.6.8.6
H2 tiling 278-2.png
8.7.8.7
H2 tiling 288-2.png
8.8.8.8
H2 tiling 25i-2.png
8.∞.8.∞
 
8.∞.8.∞

See also

Trihexagonal tiling - 3.6.3.6 tiling
Rhombille tiling - dual V3.6.3.6 tiling
Tilings of regular polygons
List of uniform tilings

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.

Undergraduate Texts in Mathematics

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Graduate Studies in Mathematics

Mathematics Encyclopedia

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