3-7 kisrhombille

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Order 3-7 kisrhombille
Order-3 heptakis heptagonal tiling.png
Type Dual semiregular hyperbolic tiling
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 7.pngCDel node f1.png
Faces Right triangle
Face configuration V4.6.14
Symmetry group [7,3], (*732)
Rotation group [7,3]+, (732)
Dual Truncated triheptagonal tiling
Properties face-transitive

In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 14 triangles meeting at each vertex.

The image shows a Poincaré disk model projection of the hyperbolic plane.

It is labeled V4.6.14 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 14 triangles. It is the dual tessellation of the truncated triheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex.

Naming

The name 3-7 kisrhombille is given by Conway, seeing it as a 3-7 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.

Symmetry

There are no mirror removal subgroups of [7,3]. The only small index subgroup is the alternation, [7,3]+, (732).

Small index subgroups of [7,3], (*732)
Type Reflectional Rotational
index 1 2
Diagram 160px 160px
Coxeter
(orbifold)		 [7,3] = CDel node c1.pngCDel 7.pngCDel node c1.pngCDel 3.pngCDel node c1.png
(*732)		 [7,3]+ = CDel node h2.pngCDel 7.pngCDel node h2.pngCDel 3.pngCDel node h2.png
(732)

Related polyhedra and tilings

Three isohedral (regular or quasiregular) tilings can be constructed from this tiling by combining triangles:

Projections centered on different triangle points
Poincaré
disk
model		 Hyperbolic domains 732.png 150px 150px
Center Heptagon Triangle Rhombic
Klein
disk
model		 150px 150px 150px
Related
tiling		 Uniform tiling 73-t0.png Uniform tiling 73-t2.png Order73 qreg rhombic til.png
Heptagonal tiling Triangular tiling Rhombic tiling

It is topologically related to a polyhedra sequence; see discussion. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and are the reflection domains for the (2,3,n) triangle groups – for the heptagonal tiling, the important (2,3,7) triangle group.

See also the uniform tilings of the hyperbolic plane with (2,3,7) symmetry.

The kisrhombille tilings can be seen as from the sequence of rhombille tilings, starting with the cube, with faces divided or kissed at the corners by a face central point.

Visualization of the map (2,3,∞) → (2,3,7) by morphing the associated tilings.[1]

Just as the (2,3,7) triangle group is a quotient of the modular group (2,3,∞), the associated tiling is the quotient of the modular tiling, as depicted in the video at right.

References

  1. ↑ Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp

See also

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