In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 ge ...
of the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
,
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 generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to ...
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, 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
See also
*
Apeirogonal antiprism
In geometry, an apeirogonal antiprism or infinite antiprismConway (2008), p. 263 is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane.
If the sides are equilateral triangles, i ...
*
Apeirogonal prism
In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane.Conway (2008), p.263
Thorold Gosset called it a ''2-dimensional semi-check ...
*
Apeirohedron
In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.
Skew apeirohedra have also been ...
{{set index article, mathematics
Apeirogonal tilings