Truncated order-8 triangular tiling

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}.

Truncated order-8 triangular tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.6.6
Schläfli symbolt{3,8}
Wythoff symbol3
4 3 3 |
Coxeter diagram
Symmetry group[8,3], (*832)
[(4,3,3)], (*433)
DualOctakis octagonal tiling
PropertiesVertex-transitive

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)

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:

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

References
    • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
    • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
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