Truncated order-8 triangular tiling

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Truncated order-8 triangular tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	8.6.6
Schläfli symbol	t{3,8}
Wythoff symbol	2 8 \| 34 3 3 \|
Coxeter diagram
Symmetry group	[8,3], (832)[(4,3,3)], (433)
Dual	Octakis octagonal tiling
Properties	Vertex-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}.

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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 v t e
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}1/2t0,1(4,3,3)	h{8,3}t1(4,3,3)	h2{8,3}t1,2(4,3,3)	{3,8}1/2t2(4,3,3)	h2{8,3}t0,2(4,3,3)	t{3,8}1/2t0,1,2(4,3,3)	s{3,8}1/2s(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 v t e
Sym.*n42[n,3]	Spherical				Euclid.	Compact		Parac.	Noncompact hyperbolic
Sym.*n42[n,3]	*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]
Truncatedfigures
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-kisfigures
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* v t e
Sym.*n43 [(n,4,3)]	Spherical	Compact hyperbolic						Paraco.
Sym.*n43 [(n,4,3)]	*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.∞

Uniform colors

Symmetry

See also

External links

Uniform colors

Symmetry

Related tilings

See also

External links