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truncated pentahexagonal tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U65_012}}

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon on each vertex. It has Schläfli symbol of t0,1,2{6,5}. Its name is somewhat misleading: literal geometric truncation of pentahexagonal tiling produces rectangles instead of squares.

Dual tiling

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colspan="2" |The dual tiling is called an order-5-6 kisrhombille tiling, made as a complete bisection of the order-5 hexagonal tiling, here with triangles shown in alternating colors. This tiling represents the fundamental triangular domains of [6,5] (*652) symmetry.

Symmetry

There are four small index subgroup from [6,5] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

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|+ Small index subgroups of [6,5], (*652)

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!Index

!1

!colspan=2|2

!6

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!Diagram

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!Coxeter
(orbifold)

|[6,5] = {{CDD|node_c1|6|node_c2|5|node_c2}}
(*652)

|[1+,6,5] = {{CDD|node_h0|6|node_c2|5|node_c2}} = {{CDD|branch_c2|split2-55|node_c2}}
(*553)

|[6,5+] = {{CDD|node_c1|6|node_h2|5|node_h2}}
(5*3)

|[6,5*] = {{CDD|node_c1|6|node_g|5|3sg|node_g}}
(*33333)

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!colspan=5|Direct subgroups

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!Index

!2

!colspan=2|4

!12

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!Diagram

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!Coxeter
(orbifold)

|[6,5]+ = {{CDD|node_h2|6|node_h2|5|node_h2}}
(652)

|colspan=2|[6,5+]+ = {{CDD|node_h0|6|node_h2|5|node_h2}} = {{CDD|branch_h2h2|split2-55|node_h2}}
(553)

|[6,5*]+ = {{CDD|node_h2|6|node_g|3sg|node_g}}
(33333)

Related polyhedra and tilings

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-5 hexagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are seven forms with full [6,5] symmetry, and three with subsymmetry.

{{Order 6-5 tiling table}}

See also

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}}

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.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Semiregular tilings

Category:Truncated tilings