Truncated infinite-order triangular tiling

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Infinite-order truncated triangular tiling
Truncated infinite-order triangular tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration ∞.6.6
Schläfli symbol t{3,∞}
Wythoff symbol 2 ∞ | 3
Coxeter diagram CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node 1.png
Symmetry group [∞,3], (*∞32)
Dual apeirokis apeirogonal tiling
Properties Vertex-transitive

In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.


Truncated infinite-order triangular tiling with mirror lines, CDel node c1.pngCDel split1.pngCDel branch c1.pngCDel labelinfin.png.

The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.

Small index subgroups of [(∞,3,3)], (*∞33)
Type Reflectional Rotational
Index 1 2
Diagram I33 symmetry 000.png I33 symmetry aaa.png
CDel node c1.pngCDel split1.pngCDel branch c1.pngCDel labelinfin.png
CDel node h2.pngCDel split1.pngCDel branch h2h2.pngCDel labelinfin.png

Related polyhedra and tiling[edit]

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.

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]