Order-4 heptagonal tiling

Short description: Regular tiling of the hyperbolic plane

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.

Symmetry

This tiling represents a hyperbolic kaleidoscope of 7 mirrors meeting as edges of a regular heptagon. This symmetry by orbifold notation is called *2222222 with 7 order-2 mirror intersections. In Coxeter notation can be represented as [1⁺,7,1⁺,4], removing two of three mirrors (passing through the heptagon center) in the [7,4] symmetry.

The kaleidoscopic domains can be seen as bicolored heptagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t₁{7,7} and as a quasiregular tiling is called a heptaheptagonal tiling.

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular tilings with heptagonal faces, starting with the heptagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.

{7,3}

{7,4}

{7,5}

{7,6}

{7,7}

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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. 

See also

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

External links

• Weisstein, Eric W.. "Hyperbolic tiling". http://mathworld.wolfram.com/HyperbolicTiling.html. 
• Weisstein, Eric W.. "Poincaré hyperbolic disk". http://mathworld.wolfram.com/PoincareHyperbolicDisk.html. 
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch