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order-4 heptagonal tiling

{{short description|Regular tiling of the hyperbolic plane}}

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U74_0}}

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.

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Related polyhedra and tiling

{{Order 7-4 tiling table}}

{{Order_7-7_tiling_table}}

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 {{CDD|node_1|7|node|n|node}}, progressing to infinity.

class="wikitable"
align=center

|100px
{7,3}
{{CDD|node_1|7|node|3|node}}

|100px
{7,4}
{{CDD|node_1|7|node|4|node}}

|100px
{7,5}
{{CDD|node_1|7|node|5|node}}

|100px
{7,6}
{{CDD|node_1|7|node|6|node}}

|100px
{7,7}
{{CDD|node_1|7|node|7|node}}

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 {{CDD|node_1|n|node|4|node}}, with n progressing to infinity.

{{Order-4_regular_tilings}}

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)
  • {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}

See also

{{Commonscat|Order-4 heptagonal tiling}}

External links

  • {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
  • {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
  • [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
  • [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
  • [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

Category:Heptagonal tilings

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Isohedral tilings

Category:Order-4 tilings

Category:Regular tilings

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