Rhombitrioctagonal tiling

Rhombitrioctagonal tiling
Rhombitrioctagonal tiling Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.4.8.4
Schläfli symbol	rr{8,3} or $r\left\{\begin{array}{c}8\\ 3\end{array}\right\}$ s2{3,8}
Wythoff symbol	3 | 8 2
Coxeter diagram	File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png or File:CDel node.pngFile:CDel split1-83.pngFile:CDel nodes 11.png File:CDel node 1.pngFile:CDel 8.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png
Symmetry group	[8,3], (*832) [8,3+], (3*4)
Dual	Deltoidal trioctagonal tiling
Properties	Vertex-transitive

In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.

Symmetry

This tiling has [8,3], (*832) symmetry. There is only one uniform coloring.

Similar to the Euclidean rhombitrihexagonal tiling, by edge-coloring there is a half symmetry form (3*4) orbifold notation. The octagons can be considered as truncated squares, t{4} with two types of edges. It has Coxeter diagram File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 8.pngFile:CDel node 1.png, Schläfli symbol s₂{3,8}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an order-8 triangular tiling results, constructed as a snub tritetratrigonal tiling, File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 8.pngFile:CDel node.png.

Related polyhedra and tilings

From a Wythoff construction there are ten 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.

Symmetry mutations

This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]
Figure	File:Spherical triangular prism.svg	File:Uniform tiling 332-t02.png	File:Uniform tiling 432-t02.png	File:Uniform tiling 532-t02.png	File:Uniform polyhedron-63-t02.png	File:Rhombitriheptagonal tiling.svg	File:H2-8-3-cantellated.svg	File:H2 tiling 23i-5.png	File:H2 tiling 23j12-5.png	File:H2 tiling 23j9-5.png	File:H2 tiling 23j6-5.png
Config.	3.4.2.4	3.4.3.4	3.4.4.4	3.4.5.4	3.4.6.4	3.4.7.4	3.4.8.4	3.4.∞.4	3.4.12i.4	3.4.9i.4	3.4.6i.4

See also

• Rhombitrihexagonal tiling
• Order-3 octagonal tiling
• Tilings of regular polygons
• List of uniform tilings
• Kagome lattice

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)

External links

• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch Archived 2011-09-27 at the Wayback Machine