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 configuration 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
Infinite-order triangular tiling.svg


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]