3-7 kisrhombille

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.

3-7 kisrhombille
Type	Dual semiregular hyperbolic tiling
Faces	Right triangle
Edges	Infinite
Vertices	Infinite
Coxeter diagram
Symmetry group	[7,3], (*732)
Rotation group	[7,3]+, (732)
Dual polyhedron	Truncated triheptagonal tiling
Face configuration	V4.6.14
Properties	face-transitive

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
Coxeter (orbifold)	[7,3] = (*732)	[7,3]+ = (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
Center	Heptagon	Triangle	Rhombic
Klein disk model
Related tiling
	Heptagonal tiling	Triangular tiling	Rhombic tiling

Uniform heptagonal/triangular tilings v t e
Symmetry: [7,3], (*732)							[7,3]+, (732)
{7,3}	t{7,3}	r{7,3}	t{3,7}	{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

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.

*n32 symmetry mutation of omnitruncated tilings: 4.6.2n v t e
Sym. *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	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]	[3i,3]
Figures
Config.	4.6.4	4.6.6	4.6.8	4.6.10	4.6.12	4.6.14	4.6.16	4.6.∞	4.6.24i	4.6.18i	4.6.12i	4.6.6i
Duals
Config.	V4.6.4	V4.6.6	V4.6.8	V4.6.10	V4.6.12	V4.6.14	V4.6.16	V4.6.∞	V4.6.24i	V4.6.18i	V4.6.12i	V4.6.6i

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

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)

See also

• Hexakis triangular tiling
• Tilings of regular polygons
• List of uniform tilings
• Uniform tilings in hyperbolic plane