Order-3 heptagonal tiling

Order-3 heptagonal tiling
Order-3 heptagonal tiling
Order-3 heptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Regular hyperbolic tiling
Vertex figure 7.7.7
Schläfli symbol(s) {7,3}
Wythoff symbol(s) 3 | 7 2
Coxeter-Dynkin(s) CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
Coxeter group [7,3]
Dual Order-7 triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-3 heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons around each vertex.

Contents

Images

Poincare halfplane eptagonal hb.svg
Heptagonal tiling (black lines) in a Poincaré half-plane model

Related polyhedra and tilings

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli_symbol {n,3}.

Uniform polyhedron-33-t0.png
{3,3}		 Uniform polyhedron-43-t0.png
{4,3}		 Uniform polyhedron-53-t0.png
{5,3}		 Uniform polyhedron-63-t0.png
{6,3}
Uniform tiling 73-t0.png
{7,3}		 Uniform tiling 83-t0.png
{8,3}		 Uniform tiling 93-t0.png
{9,3}		 H2 tiling 23i-1.png
(∞,3}

The dual tiling is the order-7 triangular tiling.

Uniform tiling 73-t0.png
order-3 heptagonal tiling		 Uniform tiling 73-t2.png
order-7 triangular tiling

Wythoff constructions from heptagonal and triangular tilings

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.

Tiling Schläfli
symbol		 Wythoff
symbol		 Vertex
figure		 Image
Order-3 heptagonal tiling t0{7,3} 3 | 7 2 73 Uniform tiling 73-t0.png
Order-3 truncated heptagonal tiling t0,1{7,3} 2 3 | 7 3.14.14 Uniform tiling 73-t01.png
Rectified order-3 heptagonal tiling
(Triheptagonal tiling)		 t1{7,3} 2 | 7 3 (3.7)2 Uniform tiling 73-t1.png
Bitruncated order-3 heptagonal tiling
(Order-7 truncated triangular tiling)		 t1,2{7,3} 2 7 | 3 7.6.6 Uniform tiling 73-t12.png
Order-7 triangular tiling t2{7,3} 7 | 3 2 37 Uniform tiling 73-t2.png
Cantellated order-3 heptagonal tiling
(Rhombitriheptagonal tiling)		 t0,2{7,3} 7 3 | 2 3.4.7.4 Uniform tiling 73-t02.png
Order-3 omnitruncated heptagonal tiling
(Truncated triheptagonal tiling)		 t0,1,2{7,3} 7 3 2 | 4.7.14 Uniform tiling 73-t012.png
Order-3 snub heptagonal tiling s{7,3} | 7 3 2 3.3.3.3.7 Uniform tiling 73-snub.png

Hurwitz surfaces

The symmetry group of the order-3 heptagonal tiling has fundamental domain the (2,3,7) Schwarz triangle, which yields this tiling.
Further information: Hurwitz surface

The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a tiling by hepatgons whose symmetry group equals their automorphism group as Riemann surfaces. The smallest Hurwitz surface is the Klein quartic (genus 3, automorphism group of order 168), and the induced tiling has 24 heptagons, meeting at 56 vertices.

The dual order-7 triangular tiling has the same symmetry group, and thus yields triangulations of Hurwitz surfaces.

References

See also

External links

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