Truncated pentahexagonal tiling

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Truncated pentahexagonal tiling
Truncated pentahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.10.12
Schläfli symbol tr{6,5} or
Wythoff symbol 2 6 5 |
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node 1.png
Symmetry group [6,5], (*652)
Dual
Properties Vertex-transitive

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon on each vertex. It has Schläfli symbol of t0,1,2{6,5}. Its name is somewhat misleading: literal geometric truncation of pentahexagonal tiling produces rectangles instead of squares.

Dual tiling[]

H2checkers 256.png Hyperbolic domains 652.png
The dual tiling is called an order-5-6 kisrhombille tiling, made as a complete bisection of the order-5 hexagonal tiling, here with triangles shown in alternating colors. This tiling represents the fundamental triangular domains of [6,5] (*652) symmetry.

Symmetry[]

There are four small index subgroup from [6,5] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

Small index subgroups of [6,5], (*652)
Index 1 2 6
Diagram 652 symmetry 000.png 652 symmetry a00.png 652 symmetry 0bb.png 652 symmetry 0zz.png
Coxeter
(orbifold) 		[6,5] = CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 5.pngCDel node c2.png
(*652) 		[1+,6,5] = CDel node h0.pngCDel 6.pngCDel node c2.pngCDel 5.pngCDel node c2.png = CDel branch c2.pngCDel split2-55.pngCDel node c2.png
(*553) 		[6,5+] = CDel node c1.pngCDel 6.pngCDel node h2.pngCDel 5.pngCDel node h2.png
(5*3) 		[6,5*] = CDel node c1.pngCDel 6.pngCDel node g.pngCDel 5.pngCDel 3sg.pngCDel node g.png
(*33333)
Direct subgroups
Index 2 4 12
Diagram 652 symmetry aaa.png 652 symmetry abb.png 652 symmetry azz.png
Coxeter
(orbifold) 		[6,5]+ = CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 5.pngCDel node h2.png
(652) 		[6,5+]+ = CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 5.pngCDel node h2.png = CDel branch h2h2.pngCDel split2-55.pngCDel node h2.png
(553) 		[6,5*]+ = CDel node h2.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.png
(33333)

Related polyhedra and tilings[]

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-5 hexagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [6,5] symmetry, and 3 with subsymmetry.

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

See also[]

  • Tilings of regular polygons
  • List of uniform planar tilings

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.

External links[]

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