Login
Login

truncated tetrapentagonal tiling

{{Short description|A uniform tiling of the hyperbolic plane}}

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U54_012}}

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.

Symmetry

File:Truncated_tetrapentagonal_tiling_with_mirrors.png

There are four small index subgroup constructed from [5,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

{{-}}

A radical subgroup is constructed [5*,4], index 10, as [5+,4], (5*2) with gyration points removed, becoming orbifold (*22222), and its direct subgroup [5*,4]+, index 20, becomes orbifold (22222).

class="wikitable collapsible collapsed"

!colspan=12| Small index subgroups of [5,4]

align=center

!Index

!1

!colspan=2|2

!10

align=center

!Diagram

|160px

|160px

|160px

|160px

Coxeter
(orbifold)

![5,4] = {{CDD|node_c1|5|node_c1|4|node_c2}}
(*542)

![5,4,1+] = {{CDD|node_c1|5|node_c1|4|node_h0}} = {{CDD|node_c1|split1-55|nodeab_c1}}
(*552)

![5+,4] = {{CDD|node_h2|5|node_h2|4|node_c2}}
(5*2)

![5*,4] = {{CDD|node_g|5g|3sg|node_g|4|node_c2}}
(*22222)

align=center

!Colspan=5|Direct subgroups

align=center

!Index

!2

!colspan=2|4

!20

align=center

!Diagram

|160px

|colspan=2|160px

|160px

Coxeter
(orbifold)

![5,4]+ = {{CDD|node_h2|5|node_h2|4|node_h2}}
(542)

!colspan=2|[5+,4]+ = {{CDD|node_h2|5|node_h2|4|node_h0}} = {{CDD|node_h2|split1-55|branch_h2h2|label2}}
(552)

![5*,4]+ = {{CDD|node_g|5g|3sg|node_g|4|node_h0}}
(22222)

Related polyhedra and tiling

{{Omnitruncated4 table}}

{{Omnitruncated_symmetric_table}}

{{Order 5-4 tiling table}}

See also

{{Commons category|Uniform tiling 4-8-10}}

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|first=H. S. M.|last=Coxeter|authorlink=H. S. M. Coxeter|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.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Truncated tilings

Category:Uniform tilings

{{hyperbolic-geometry-stub}}