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 Order 5-6 kisrhombille
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[edit]

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.


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
[6,5] = CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 5.pngCDel node c2.png
[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
[6,5+] = CDel node c1.pngCDel 6.pngCDel node h2.pngCDel 5.pngCDel node h2.png
[6,5*] = CDel node c1.pngCDel 6.pngCDel node g.pngCDel 5.pngCDel 3sg.pngCDel node g.png
Direct subgroups
Index 2 4 12
Diagram 652 symmetry aaa.png 652 symmetry abb.png 652 symmetry azz.png
[6,5]+ = CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 5.pngCDel node h2.png
[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
[6,5*]+ = CDel node h2.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.png

Related polyhedra and tilings[edit]

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.

See also[edit]


  • 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[edit]