Rhombitrioctagonal tiling

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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
s2{3,8}
Wythoff symbol 3 | 8 2
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png or CDel node.pngCDel split1-83.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel 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[edit]

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 CDel node h.pngCDel 3.pngCDel node h.pngCDel 8.pngCDel 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, CDel node h.pngCDel 3.pngCDel node h.pngCDel 8.pngCDel node.png.

Related polyhedra and tilings[edit]

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[edit]

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.

See also[edit]

References[edit]

  • 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.

External links[edit]