Order-5 hexagonal tiling

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Order-5 hexagonal tiling
Order-5 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 65
Schläfli symbol {6,5}
Wythoff symbol 5 | 6 2
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
Symmetry group [6,5], (*652)
Dual Order-6 pentagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-5 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,5}.

Related polyhedra and tiling[]

This tiling is topologically related as a part of sequence of regular tilings with order-5 vertices with Schläfli symbol {n,5}, and Coxeter diagram CDel node 1.pngCDel n.pngCDel node.pngCDel 5.pngCDel node.png, progressing to infinity.

Spherical Hyperbolic tilings
  • v
Spherical pentagonal hosohedron.png
{2,5}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png 		Uniform tiling 532-t2.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png 		H2-5-4-primal.svg
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 255-1.png
{5,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 256-1.png
{6,5}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 257-1.png

CDel node 1.pngCDel 7.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 258-1.png

CDel node 1.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png 		... H2 tiling 25i-1.png
{∞,5}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.png

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel n.pngCDel node.png, progressing to infinity.

*n62 symmetry mutation of regular tilings: {6,n}
  • v
Spherical Euclidean Hyperbolic tilings
Hexagonal dihedron.svg
{6,2} 		Uniform tiling 63-t0.svg
{6,3} 		H2 tiling 246-1.png
{6,4} 		H2 tiling 256-1.png
{6,5} 		H2 tiling 266-4.png
{6,6} 		H2 tiling 267-4.png
 H2 tiling 268-4.png
{6,8} 		... H2 tiling 26i-4.png
{6,∞}
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

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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