4-5 kisrhombille

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

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.

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
  • v
  • t
  • e
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
  • v
  • t
  • e
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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)

See also

  • v
  • t
  • e


Other
Spherical
  • 2n
  • 33.n
  • V33.n
  • 42.n
  • V42.n
Regular
  • 2∞
  • 36
  • 44
  • 63
Semi-
regular
  • 32.4.3.4
  • V32.4.3.4
  • 33.42
  • 33.∞
  • 34.6
  • V34.6
  • 3.4.6.4
  • (3.6)2
  • 3.122
  • 42.∞
  • 4.6.12
  • 4.82
Hyper-
bolic
  • 32.4.3.5
  • 32.4.3.6
  • 32.4.3.7
  • 32.4.3.8
  • 32.4.3.∞
  • 32.5.3.5
  • 32.5.3.6
  • 32.6.3.6
  • 32.6.3.8
  • 32.7.3.7
  • 32.8.3.8
  • 33.4.3.4
  • 32.∞.3.∞
  • 34.7
  • 34.8
  • 34.∞
  • 35.4
  • 37
  • 38
  • 3∞
  • (3.4)3
  • (3.4)4
  • 3.4.62.4
  • 3.4.7.4
  • 3.4.8.4
  • 3.4.∞.4
  • 3.6.4.6
  • (3.7)2
  • (3.8)2
  • 3.142
  • 3.162
  • 3.∞2
  • 42.5.4
  • 42.6.4
  • 42.7.4
  • 42.8.4
  • 42.∞.4
  • 45
  • 46
  • 47
  • 48
  • 4∞
  • (4.5)2
  • (4.6)2
  • 4.6.12
  • 4.6.14
  • V4.6.14
  • 4.6.16
  • V4.6.16
  • 4.6.∞
  • (4.7)2
  • (4.8)2
  • 4.8.10
  • V4.8.10
  • 4.8.12
  • 4.8.14
  • 4.8.16
  • 4.8.∞
  • 4.102
  • 4.10.12
  • 4.122
  • 4.12.16
  • 4.142
  • 4.162
  • 4.∞2
  • 54
  • 55
  • 56
  • 5∞
  • 5.4.6.4
  • (5.6)2
  • 5.82
  • 5.102
  • 5.122
  • 64
  • 65
  • 66
  • 68
  • 6.4.8.4
  • (6.8)2
  • 6.82
  • 6.102
  • 6.122
  • 6.162
  • 73
  • 74
  • 77
  • 7.62
  • 7.82
  • 7.142
  • 83
  • 84
  • 86
  • 88
  • 8.62
  • 8.122
  • 8.162
  • ∞3
  • ∞4
  • ∞5
  • ∞∞
  • ∞.62
  • ∞.82