geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an apeirogonal tiling is a

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

of the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...

, or some other two-dimensional space by

apeirogon In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an in ...

s. 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 In geometry, the order-4 apeirogonal tiling is a List of regular polytopes#Hyperbolic tilings, regular Tessellation, tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of . Symmetry This tiling represents the mirror li ...

, 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-

k

apeirogonal tiling form a

Bethe lattice In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite symmetric regular tree where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature by ...

, a regular infinite tree.

References

{{set index article, mathematics Apeirogonal tilings

See also

References