Truncated order-6 pentagonal tiling

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Truncated order-6 pentagonal tiling
Truncated order-6 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 6.10.10
Schläfli symbol t{5,6}
t(5,5,3)
Wythoff symbol 2 6 | 5
3 5 5 |
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node 1.png
CDel branch 11.pngCDel split2-55.pngCDel node 1.png
Symmetry group [6,5], (*652)
[(5,5,3)], (*553)
Dual
Properties Vertex-transitive

In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}.

Uniform colorings[]

H2 tiling 355-7.png
t012(5,5,3) 		H2 tiling 355-7-mirrors.png
With mirrors
An alternate construction exists from the [(5,5,3)] family, as the omnitruncation t012(5,5,3). It is shown with two (colors) of decagons.

Symmetry[]

The dual of this tiling represents the fundamental domains of the *553 symmetry. There are no mirror removal subgroups of [(5,5,3)], but this symmetry group can be doubled to 652 symmetry by adding a bisecting mirror to the fundamental domains.

Small index subgroups of [(5,5,3)]
Type Reflective domains Rotational symmetry
Index 1 2
Diagram 553 symmetry 000.png 553 symmetry aaa.png
Coxeter
(orbifold) 		[(5,5,3)] = CDel node c1.pngCDel split1-55.pngCDel branch c1.png
(*553) 		[(5,5,3)]+ = CDel node h2.pngCDel split1-55.pngCDel branch h2h2.png
(553)

Related polyhedra and tiling[]

Uniform hexagonal/pentagonal tilings
  • v
Symmetry: [6,5], (*652) [6,5]+, (652) [6,5+], (5*3) [1+,6,5], (*553)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 5.pngCDel node h.png CDel node.pngCDel 6.pngCDel node h.pngCDel 5.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 256-1.png H2 tiling 256-3.png H2 tiling 256-2.png H2 tiling 256-6.png H2 tiling 256-4.png H2 tiling 256-5.png H2 tiling 256-7.png Uniform tiling 65-snub.png H2 tiling 355-1.png
{6,5} t{6,5} r{6,5} 2t{6,5}=t{5,6} 2r{6,5}={5,6} rr{6,5} tr{6,5} sr{6,5} s{5,6}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 5.pngCDel node fh.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 5.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
H2chess 256b.png Order-6 pentakis pentagonal tiling.png Order-6-5 quasiregular rhombic tiling.png H2chess 256e.png H2 tiling 256-1.png Deltoidal pentahexagonal tiling.png H2checkers 256.png
V65 V5.12.12 V5.6.5.6 V6.10.10 V56 V4.5.4.6 V4.10.12 V3.3.5.3.6 V3.3.3.5.3.5 V(3.5)5
[(5,5,3)] reflective symmetry uniform tilings
H2 tiling 355-1.png H2 tiling 355-2.png H2 tiling 355-3.png H2 tiling 355-4.png H2 tiling 355-5.png H2 tiling 355-6.png H2 tiling 355-7.png

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[]

  • Square tiling
  • Tilings of regular polygons
  • List of uniform planar tilings
  • List of regular polytopes

External links[]

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