Triheptagonal tiling

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Triheptagonal tiling
Triheptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (3.7)2
Schläfli symbol r{7,3} or
Wythoff symbol 2 | 7 3
Coxeter diagram CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel split1-73.pngCDel nodes.png
Symmetry group [7,3], (*732)
Dual Order-7-3 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.

Compare to trihexagonal tiling with vertex configuration 3.6.3.6.

Images[edit]

Uniform tiling 73-t1 klein.png
Klein disk model of this tiling preserves straight lines, but distorts angles
Order73 qreg rhombic til.png
The dual tiling is called an Order-7-3 rhombille tiling, made from rhombic faces, alternating 3 and 7 per vertex.

7-3 Rhombille[edit]

Triheptagonal tiling
Order73 qreg rhombic til.png
FacesRhombi
Coxeter diagramCDel node.pngCDel 3.pngCDel node f1.pngCDel 7.pngCDel node.png
Symmetry group[7,3], *732
Rotation group[7,3]+, (732)
Dual polyhedronTriheptagonal tiling
Face configurationV3.7.3.7
Propertiesedge-transitive face-transitive

In geometry, the 7-3 rhombille tiling is a tessellation of identical rhombi on the hyperbolic plane. Sets of three and seven rhombi meet two classes of vertices.

Order 7-3 rhombic tiling in the Band Model.png
7-3 rhombile tiling in band model

Related polyhedra and tilings[edit]

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:

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.

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]