- Order-3 bisected heptagonal tiling
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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.
Contents
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
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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
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