Apeirogonal tiling

In geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this type include:

• Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces
• Order-3 apeirogonal tiling, hyperbolic tiling with 3 apeirogons around a vertex
• Order-4 apeirogonal tiling, hyperbolic tiling with 4 apeirogons around a vertex
• Order-5 apeirogonal tiling, hyperbolic tiling with 5 apeirogons around a vertex
• Infinite-order apeirogonal tiling, hyperbolic tiling with an infinite number of apeirogons around a vertex

The vertices of an order-${\displaystyle k}$ apeirogonal tiling form a Bethe lattice, a regular infinite tree.^[1]

See also

• Apeirogonal antiprism
• Apeirogonal prism
• Apeirohedron

References

1. Mosseri, R.; Sadoc, J.F. (1982), "The Bethe lattice: a regular tiling of the hyperbolic plane" (PDF), Journal de Physique Lettres, 43 (8): 249–252, doi:10.1051/jphyslet:01982004308024900