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.

*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*42
[,4]
Omnitruncated
figure

4.8.4

4.8.6

4.8.8

4.8.10

4.8.12

4.8.14

4.8.16

4.8.
Omnitruncated
duals

V4.8.4

V4.8.6

V4.8.8

V4.8.10

V4.8.12

V4.8.14

V4.8.16

V4.8.
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55

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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