Truncated order-8 triangular tiling

Truncated order-8 triangular tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.6.6
Schläfli symbolt{3,8}
Wythoff symbol2 8 | 3
4 3 3 |
Coxeter diagram
Symmetry group[8,3], (*832)
[(4,3,3)], (*433)
DualOctakis octagonal tiling
PropertiesVertex-transitive

In geometry, the Truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex. It has Schläfli symbol of t{3,8}.

Uniform colors


The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of hexagons
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.

Small index subgroups of [(4,3,3)], (*433)
Type Reflectional Rotational
Index 1 2
Diagram
Coxeter
(orbifold)		 [(4,3,3)] =
(*433)		 [(4,3,3)]+ =
(433)

Related tilings

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.

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

Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
h{8,3}
t0{(4,3,3)}
{(4,3,3)}		 r{8,3}
t0,1{(4,3,3)}		 h{8,3}
t1{(4,3,3)}
{(3,3,4)}		 h2{8,3}
t1,2{(4,3,3)}		 {3,8}
t2{(4,3,3)}
{(3,4,3)}		 h2{8,3}
t0,2{(4,3,3)}		 t{3,8}
t0,1,2{(4,3,3)}
t{(4,3,3)}		 s{3,8}
 
s{(4,3,3)}
Uniform duals
V(3.4)3 V3.8.3.8 V(3.4)3 V3.6.4.6 V(3.3)4 V3.6.4.6 V6.6.8 V3.3.3.3.3.4

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

See also

Wikimedia Commons has media related to Uniform tiling 6-6-8.

References

    External links

    This article is issued from Wikipedia - version of the Wednesday, November 12, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.