Rhombitriheptagonal tiling

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Rhombitriheptagonal tiling
Rhombitriheptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration
Schläfli symbol rr{7,3} or
Wythoff symbol 3 | 7 2
Coxeter diagram CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png or CDel node.pngCDel split1-73.pngCDel nodes 11.png
Symmetry group [7,3], (*732)
Dual Deltoidal triheptagonal tiling
Properties Vertex-transitive

In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has Schläfli symbol rr{7, 3}. It can be seen as constructed as a rectified triheptagonal tiling, r{7,3}, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling.

Dual tiling[edit]

The dual tiling is called a deltoidal triheptagonal tiling, and consists of congruent kites. It is formed by overlaying an order-3 heptagonal tiling and an order-7 triangular tiling.

Deltoidal triheptagonal til.png

Related polyhedra and tilings[edit]

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal 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.

*n42 symmetry mutation of dual expanded tilings: V3.4.n.4
Spherical Euclid. Compact hyperb. Paraco.
Spherical trigonal bipyramid.png
Spherical rhombic dodecahedron.png
Spherical deltoidal icositetrahedron.png
Spherical deltoidal hexecontahedron.png
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
Deltoidal triheptagonal til.png
Deltoidal trioctagonal til.png
Deltoidal triapeirogonal til.png

See also[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]