View source for Truncated order-8 triangular tiling

In [[Geometry|geometry]], the '''truncated order-8 triangular tiling''' is a semiregular tiling of the hyperbolic plane. There are two [[Hexagon|hexagon]]s and one [[Octagon|octagon]] on each [[Vertex (geometry)|vertex]]. It has [[Schläfli symbol]] of t{3,8}.

== Uniform colors==

{| class=wikitable width=480
|- valign=top align=center
|[[File:H2 tiling 334-7.png|240px]]<BR>The half symmetry [1<sup>+</sup>,8,3] = [(4,3,3)] can be shown with alternating two colors of hexagons
|[[File:H2-8-3-kis-dual.svg|240px]]<BR>Dual tiling
|}

== Symmetry ==
The dual of this tiling represents the fundamental domains of *443 symmetry. It only has one subgroup 443, replacing mirrors with gyration points.

This symmetry can be doubled to 832 symmetry by adding a bisecting mirror to the fundamental domain.
{| class=wikitable
|+ Small index subgroups of [(4,3,3)], (*433)
|- align=center
!Type
!Reflectional 
!Rotational
|- align=center
!Index
!1
!2
|- align=center
!Diagram
|[[File:433_symmetry_000.png|160px]]
|[[File:433_symmetry_aaa.png|160px]]
|- align=center
!Coxeter<BR>([[Orbifold notation|orbifold]])
|[(4,3,3)] = {{CDD|node_c1|split1|branch_c1|label4}}<BR>(*433)
|[(4,3,3)]<sup>+</sup> = {{CDD|node_h2|split1|branch_h2h2|label4}}<BR>(433)
|}

== Related tilings ==
From a [[Wythoff construction]] there are ten hyperbolic [[Uniform tilings in hyperbolic plane|uniform tilings]] that can be based from the regular octagonal tiling. 

{{Octagonal tiling table}}

It can also be generated from the (4 3 3) hyperbolic tilings:

This hyperbolic tiling is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[Vertex configuration|vertex configuration]]s (n.6.6), and [n,3] [[Coxeter group]] symmetry.

== See also ==
* [[Triangular tiling]]
* [[Order-3 octagonal tiling]]
* [[Order-8 triangular tiling]]
* Tilings of regular polygons
* List of uniform tilings

==References==

<references />
* [[Biography:John Horton Conway|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:Triangular tilings]]
[[Category:Truncated tilings]]



{{Sourceattribution|Truncated order-8 triangular tiling}}