Order-3 bisected heptagonal tiling

Order-3 bisected heptagonal tiling
Type	Dual semiregular hyperbolic tiling
Faces	Right triangle
Face configuration	V4.6.14
Symmetry group	*732
Dual	Great rhombitriheptagonal tiling
Properties	face-transitive

In geometry, the order-3 bisected heptagonal 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 great rhombitriheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex.

Naming

An alternative name is 3-7 kisrhombille 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:

3-7 rhombic tiling (rhombille)

Related polyhedra and tilings

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.

V4.6.6

V4.6.8

V4.6.10

V4.6.12

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

See also

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

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