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
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

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.

Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)		 [1+,8,3]
(*443)		 [8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}		 tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}


or
or




Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4

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)		 r{3,8}1/2
t0,1(4,3,3)		 h{8,3}
t1(4,3,3)		 h2{8,3}
t1,2(4,3,3)		 {3,8}1/2
t2(4,3,3)		 h2{8,3}
t0,2(4,3,3)		 t{3,8}1/2
t0,1,2(4,3,3)		 s{3,8}1/2
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.

*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]		 Spherical Euclid. Compact Parac. Noncompact hyperbolic
*232
[2,3]		 *332
[3,3]		 *432
[4,3]		 *532
[5,3]		 *632
[6,3]		 *732
[7,3]		 *832
[8,3]...		 *∞32
[∞,3]		 [12i,3] [9i,3] [6i,3]
Truncated
figures
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n
Sym.
*n43
[(n,4,3)]		 Spherical Compact hyperbolic Paraco.
*243
[4,3]		 *343
[(3,4,3)]		 *443
[(4,4,3)]		 *543
[(5,4,3)]		 *643
[(6,4,3)]		 *743
[(7,4,3)]		 *843
[(8,4,3)]		 *∞43
[(∞,4,3)]
Figures
Config. 4.8.6 6.8.6 8.8.6 10.8.6 12.8.6 14.8.6 16.8.6 ∞.8.6
Duals
Config. V4.8.6 V6.8.6 V8.8.6 V10.8.6 V12.8.6 V14.8.6 V16.8.6 V6.8.∞

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

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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