Truncated order-4 pentagonal tiling

From Wikipedia, the free encyclopedia
Truncated pentagonal tiling
Truncated order-4 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.10.10
Schläfli symbol t{5,4}
Wythoff symbol 2 4 | 5
2 5 5 |
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png or CDel node 1.pngCDel split1-55.pngCDel nodes 11.png
Symmetry group [5,4], (*542)
[5,5], (*552)
Dual
Properties Vertex-transitive

In geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,4}.

Uniform colorings[]

A half symmetry [1+,4,5] = [5,5] coloring can be constructed with two colors of decagons. This coloring is called a truncated pentapentagonal tiling.

Uniform tiling 552-t012.png

Symmetry[]

There is only one subgroup of [5,5], [5,5]+, removing all the mirrors. This symmetry can be doubled to 542 symmetry by adding a bisecting mirror.

Small index subgroups of [5,5]
Type Reflective domains Rotational symmetry
Index 1 2
Diagram 552 symmetry 000.png 552 symmetry aaa.png
Coxeter
(orbifold) 		[5,5] = CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 5.pngCDel node c1.png = CDel node c1.pngCDel split1-55.pngCDel branch c1.pngCDel label2.png
(*552) 		[5,5]+ = CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 5.pngCDel node h2.png = CDel node h2.pngCDel split1-55.pngCDel branch h2h2.pngCDel label2.png
(552)

Related polyhedra and tiling[]

*n42 symmetry mutation of truncated tilings: 4.2n.2n
  • v
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]
Truncated
figures 		Spherical square prism.png Uniform tiling 432-t12.png Uniform tiling 44-t01.png H2-5-4-trunc-dual.svg H2 tiling 246-3.png H2 tiling 247-3.png H2 tiling 248-3.png H2 tiling 24i-3.png
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures 		Spherical square bipyramid.png Spherical tetrakis hexahedron.png 1-uniform 2 dual.svg H2-5-4-kis-primal.svg Order-6 tetrakis square tiling.png Hyperbolic domains 772.png Order-8 tetrakis square tiling.png H2checkers 2ii.png
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform pentagonal/square tilings
  • v
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node h.png
H2-5-4-dual.svg H2-5-4-trunc-dual.svg H2-5-4-rectified.svg H2-5-4-trunc-primal.svg H2-5-4-primal.svg H2-5-4-cantellated.svg H2-5-4-omnitruncated.svg H2-5-4-snub.svg Uniform tiling 542-h01.png Uniform tiling 552-t0.png
{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
CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node fh.png
H2-5-4-primal.svg H2-5-4-kis-primal.svg H2-5-4-rhombic.svg H2-5-4-kis-dual.svg H2-5-4-dual.svg H2-5-4-deltoidal.svg H2-5-4-kisrhombille.svg H2-5-4-floret.svg Uniform tiling 552-t2.png
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
Uniform pentapentagonal tilings
  • v
Symmetry: [5,5], (*552) [5,5]+, (552)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png 		CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png 		CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png 		CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node 1.png 		CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png 		CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node.png 		CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node 1.png 		CDel node h.pngCDel 5.pngCDel node h.pngCDel 5.pngCDel node h.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 5.pngCDel node h.png
Uniform tiling 552-t0.png Uniform tiling 552-t01.png Uniform tiling 552-t1.png Uniform tiling 552-t12.png Uniform tiling 552-t2.png Uniform tiling 552-t02.png Uniform tiling 552-t012.png Uniform tiling 552-snub.png
{5,5} t{5,5}
 r{5,5} 2t{5,5}=t{5,5} 2r{5,5}={5,5} rr{5,5} tr{5,5} sr{5,5}
Uniform duals
CDel node f1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 5.pngCDel node fh.png
Uniform tiling 552-t2.png Order5 pentakis pentagonal til.png H2-5-4-primal.svg Order5 pentakis pentagonal til.png Uniform tiling 552-t0.png H2-5-4-rhombic.svg H2-5-4-kis-primal.svg
V5.5.5.5.5 V5.10.10 V5.5.5.5 V5.10.10 V5.5.5.5.5 V4.5.4.5 V4.10.10 V3.3.5.3.5

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)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.

See also[]

External links[]

Stub icon

This geometry-related article is a stub. You can help Wikipedia by .

  • v
  • t
Retrieved from ""