In
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 order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedron
[Conway (2008), p. 263] 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 ...
(gap-free filling with repeated shapes) of the
plane consisting of two
apeirogons. It may be considered an improper
regular tiling
Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his (Latin language, Latin: ''The Har ...
of the
Euclidean plane, with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
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
In geometry, an angle of a polygon is formed by two adjacent sides. For a simple polygon (non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point withi ...
of 180°, which is half of a full 360°.
Related tilings and polyhedra
Similarly to the
uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fac ...
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 plane. Uniform tilings are related to t ...
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
In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each Flag (geometry), flag of the original polytope and a Facet (geometry), facet for each face of any dimension of the original pol ...
forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the
apeirogonal prism, and the
apeirogonal antiprism.
See also
*
Order-3 apeirogonal tiling - hyperbolic tiling
*
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
Notes
References
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,
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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