Triheptagonal tiling

Template:Uniform hyperbolic tiling stat table 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

File:Uniform tiling 73-t1 klein.png Klein disk model of this tiling preserves straight lines, but distorts angles

File:7-3 rhombille tiling.svg The dual tiling is called an Order-7-3 rhombille tiling, made from rhombic faces, alternating 3 and 7 per vertex.

7-3 Rhombille

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.

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

Related polyhedra and tilings

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings: Template:Quasiregular3 table

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. Template:Heptagonal tiling table

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, Template:Isbn (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

Template:Hyperbolic-geometry-stub