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 (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

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 of ...

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 the ...

s. Tilings of this type include: *

Order-2 apeirogonal tiling In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedronConway (2008), p. 263 is a tiling of the plane consisting of two apeirogons. It may be considered an improper regular tiling of the Euclidean plane, with Schl ...

, Euclidean tiling of two half-spaces *

Order-3 apeirogonal tiling In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol , having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle. The order-2 ap ...

, hyperbolic tiling with 3 apeirogons around a vertex *

Order-4 apeirogonal tiling In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Symmetry This tiling represents the mirror lines of *2∞ symmetry. It dual to this tiling represents the fundamental domain ...

, hyperbolic tiling with 4 apeirogons around a vertex *

Order-5 apeirogonal tiling In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Symmetry The dual to this tiling represents the fundamental domains of ��,5*symmetry, orbifold notation *∞∞∞∞∞ sym ...

, hyperbolic tiling with 5 apeirogons around a vertex *

Infinite-order apeirogonal tiling In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of , which means it has countably infinitely many apeirogons around all its ideal vertices. Symmetry This tiling represents ...

, hyperbolic tiling with an infinite number of apeirogons around a vertex

See also