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truncated tetraheptagonal tiling

{{Short description|Hyperbolic tiling}}

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U74_012}}

In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.

Images

Poincaré disk projection, centered on 14-gon:

:160px

Symmetry

File:Truncated_tetraheptagonal_tiling_with_mirrors.png

The dual to this tiling represents the fundamental domains of [7,4] (*742) symmetry. There are three small index subgroups constructed from [7,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

{{-}}

class="wikitable collapsible collapsed"

!colspan=12| Small index subgroups of [7,4] (*742)

align=center

!Index

!1

!colspan=2|2

!14

align=center

!Diagram

!160px

!160px

!160px

!160px

Coxeter
(orbifold)

![7,4] = {{CDD|node_c1|7|node_c1|4|node_c2}}
(*742)

![7,4,1+] = {{CDD|node_c1|7|node_c1|4|node_h0}} = {{CDD|node_c1|split1-77|nodeab_c1}}
(*772)

![7+,4] = {{CDD|node_h2|7|node_h2|4|node_c2}}
(7*2)

![7*,4] = {{CDD|node_g|7|3sg|node_g|4|node_c2}}
(*2222222)

align=center

!Index

!2

!colspan=2|4

!28

align=center

!Diagram

!160px

!colspan=2|160px

!160px

Coxeter
(orbifold)

![7,4]+ = {{CDD|node_h2|7|node_h2|4|node_h2}}
(742)

!colspan=2|[7+,4]+ = {{CDD|node_h2|7|node_h2|4|node_h0}} = {{CDD|node_h2|split1-77|branch_h2h2|label2}}
(772)

![7*,4]+ = {{CDD|node_g|7|3sg|node_g|4|node_h2}}
(2222222)

Related polyhedra and tiling

{{Order 7-4 tiling table}}

{{Omnitruncated4 table}}

{{Omnitruncated_symmetric_table}}

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|Uniform tiling 4-8-14}}

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:Hyperbolic tilings

Category:Isogonal tilings

Category:Truncated tilings

Category:Uniform tilings

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