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Trioctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(3.8)
Schläfli symbol	r{8,3} or ${\displaystyle {\begin{Bmatrix}8\\3\end{Bmatrix}}}$
Wythoff symbol	2 | 8 3| 3 3 | 4
Coxeter diagram	or
Symmetry group	, (*832) , (*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

The half symmetry = can be shown with alternating two colors of triangles, by Coxeter diagram .

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 v t e
Symmetry: , (*832)							(832)	(*443)		(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
V8	V3.16.16	V3.8.3.8	V6.6.8	V3	V3.4.8.4	V4.6.16	V3.8	V(3.4)	V8.6.6	V3.4

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

Uniform (4,3,3) tilings

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Symmetry: , (*433)							, (433)
h{8,3} t0(4,3,3)	r{3,8}/2 t0,1(4,3,3)	h{8,3} t1(4,3,3)	h2{8,3} t1,2(4,3,3)	{3,8}/2 t2(4,3,3)	h2{8,3} t0,2(4,3,3)	t{3,8}/2 t0,1,2(4,3,3)	s{3,8}/2 s(4,3,3)
Uniform duals
V(3.4)	V3.8.3.8	V(3.4)	V3.6.4.6	V(3.3)	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) v t e
Sym. *n32	Spherical			Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
	*332 Td	*432 Oh	*532 Ih	*632 p6m	*732	*832 ...	*∞32
Figure
Figure
Vertex	(3.3)	(3.4)	(3.5)	(3.6)	(3.7)	(3.8)	(3.∞)	(3.12i)	(3.9i)	(3.6i)
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)	V(3.4)	V(3.5)	V(3.6)	V(3.7)	V(3.8)	V(3.∞)

Dimensional family of quasiregular polyhedra and tilings: (8.n) v t e
Symmetry *8n2	Hyperbolic...						Paracompact	Noncompact
	*832	*842	*852	*862	*872	*882 ...	*∞82
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

• 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-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.

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch

v t e Tessellation
Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling and honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff
Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pinwheel Quaquaversal Rep-tile and Self-tiling Sphinx Socolar Truchet
Other Anisohedral and Isohedral Architectonic and catoptric Circle Limit III Computer graphics Honeycomb Isotoxal List Packing Pentagonal Problems Domino Wang Heesch's Squaring Dividing a square into similar rectangles Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg
By vertex type Spherical 2 3.n V3.n 4.n V4.n Regular 2 3 4 6 Semi- regular 3.4.3.4 V3.4.3.4 3.4 3.∞ 3.6 V3.6 3.4.6.4 (3.6) 3.12 4.∞ 4.6.12 4.8 Hyper- bolic 3.4.3.5 3.4.3.6 3.4.3.7 3.4.3.8 3.4.3.∞ 3.5.3.5 3.5.3.6 3.6.3.6 3.6.3.8 3.7.3.7 3.8.3.8 3.4.3.4 3.∞.3.∞ 3.7 3.8 3.∞ 3.4 3 3 3 (3.4) (3.4) 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7) (3.8) 3.14 3.16 3.∞ 4.5.4 4.6.4 4.7.4 4.8.4 4.∞.4 4 4 4 4 4 (4.5) (4.6) 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7) (4.8) 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.10 4.10.12 4.12 4.12.16 4.14 4.16 4.∞ 5 5 5 5 5.4.6.4 (5.6) 5.8 5.10 5.12 6 6 6 6 6.4.8.4 (6.8) 6.8 6.10 6.12 6.16 7 7 7 7.6 7.8 7.14 8 8 8 8 8.6 8.12 8.16 ∞ ∞ ∞ ∞ ∞.6 ∞.8

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