4-5 kisrhombille

In geometry, the 4-5 kisrhombille or order-4 bisected pentagonal tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 8, and 10 triangles meeting at each vertex.

4-5 kisrhombille
TypeDual semiregular hyperbolic tiling
FacesRight triangle
EdgesInfinite
VerticesInfinite
Coxeter diagram
Symmetry group[5,4], (*542)
Rotation group[5,4]+, (542)
Dual polyhedrontruncated tetrapentagonal tiling
Face configurationV4.8.10
Propertiesface-transitive

The name 4-5 kisrhombille is by Conway, seeing it as a 4-5 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.

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

It is labeled V4.8.10 because each right triangle face has three types of vertices: one with 4 triangles, one with 8 triangles, and one with 10 triangles.

Dual tiling

It is the dual tessellation of the truncated tetrapentagonal tiling which has one square and one octagon and one decagon at each vertex.

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
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