Snub triheptagonal tiling

Snub triheptagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex figure3.3.3.3.7
Schläfli symbolsr{7,3}
Wythoff symbol| 7 3 2
Coxeter diagram
Symmetry group[7,3]+, (732)
DualOrder-7-3 floret pentagonal tiling
PropertiesVertex-transitive Chiral

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one heptagon on each vertex. It has Schläfli symbol of sr{7,3}. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol sr{7,4}.

Images

Drawn in chiral pairs, with edges missing between black triangles:

Dual tiling

The dual tiling is called an order-7-3 floret pentagonal tiling, and is related to the floret pentagonal tiling.

Related polyhedra and tilings

This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+		 Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3		 332
[3,3]+
T		 432
[4,3]+
O		 532
[5,3]+
I		 632
[6,3]+
P6		 732
[7,3]+		 832
[8,3]+...		 ∞32
[∞,3]+
Snub
figure
3.3.3.3.2
3.3.3.3.3
3.3.3.3.4
3.3.3.3.5
3.3.3.3.6
3.3.3.3.7
3.3.3.3.8
3.3.3.3.∞
Coxeter
Schläfli
sr{2,3}
sr{3,3}
sr{4,3}
sr{5,3}
sr{6,3}
sr{7,3}
sr{8,3}
sr{∞,3}
Snub
dual
figure
V3.3.3.3.2
V3.3.3.3.3
V3.3.3.3.4
V3.3.3.3.5
V3.3.3.3.6
V3.3.3.3.7		 V3.3.3.3.8
V3.3.3.3.∞
Coxeter

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

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
{7,3} t{7,3} r{7,3} 2t{7,3}=t{3,7} 2r{7,3}={3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7

References

See also

Wikimedia Commons has media related to Uniform tiling 3-3-3-3-7.

External links