Order-3 apeirogonal tiling

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Order-3 apeirogonal tiling
Order-3 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 3
Schläfli symbol {∞,3}
Wythoff symbol 3 | ∞ 2
2 ∞ | ∞
∞ ∞ ∞ |
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.png
Symmetry group [∞,3], (*∞32)
[∞,∞], (*∞∞2)
[(∞,∞,∞)], (*∞∞∞)
Dual Infinite-order triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as {∞,2}.


Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary.

Order-3 apeirogonal tiling one cell horocycle.png

Uniform colorings[edit]

Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the order-3 apeirogonal tiling, each from different reflective triangle group domains:

Regular Truncations
H2 tiling 23i-1.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 2ii-3.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
H2 tiling 2ii-6.png
CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
H2 tiling iii-7.png
CDel node 1.pngCDel split1-ii.pngCDel branch 11.pngCDel labelinfin.png
Hyperbolic triangle groups
H2checkers 23i.png
H2checkers 2ii.png
H2checkers iii.png


The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. There are 15 small index subgroups (7 unique) constructed from [(∞,∞,∞)] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as ∞∞2 symmetry by adding a mirror bisecting the fundamental domain. Dividing a fundamental domain by 3 mirrors creates a ∞32 symmetry.

A larger subgroup is constructed [(∞,∞,∞*)], index 8, as (∞*∞) with gyration points removed, becomes (*∞).

Related polyhedra and tilings[edit]

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.

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]