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 order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedron
[Conway (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
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...
Two apeirogons, joined along all their
edges, can completely fill the entire plane as an apeirogon is infinite in size and has an
interior angle of 180°, which is half of a full 360°.
Related tilings and polyhedra
The apeirogonal tiling is the arithmetic limit of the family of
dihedra
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat f ...
, as ''p'' tends to
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
, thereby turning the dihedron into a Euclidean tiling.
Similarly to the
uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also f ...
and the
uniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
Uniform tilings can exist in both the Euclidean plane and Hyperbolic space, hyperbolic plane. Uniform tilings ar ...
s, eight uniform tilings may be based from the regular apeirogonal tiling. The
rectified and
cantellated forms are duplicated, and as two times infinity is also infinity, the
truncated and
omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the
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 ...
, and the
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 ...
.
See also
*
Order-3 apeirogonal tiling - hyperbolic tiling
*
Order-4 apeirogonal tiling - hyperbolic tiling
Notes
References
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
External links
Jim McNeill: Tessellations of the Plane
Apeirogonal tilings
Euclidean tilings
Isogonal tilings
Isohedral tilings
Order-2 tilings
Regular tilings
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