Infinite-order apeirogonal tiling on:
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: a or = ∪
Hyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Apeirogonal tilings Hyperbolic tilings Infinite-order tilings Isogonal tilings Isohedral tilings Regular tilings Self-dual tilings
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 ...
, the infinite-order apeirogonal tiling is a regular tiling of the 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' ...
. It has Schläfli symbol of , which means it has countably infinitely many 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 th ...
s around all its ideal vertices.
Symmetry
This tiling represents the fundamental domains of *∞∞ symmetry.Uniform colorings
This tiling can also be alternately colored in the ∞,∞,∞)symmetry from 3 generator positions.Related polyhedra and tiling
The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain. :
: a or = ∪
See also
* Tilings of regular polygons * List of uniform planar tilings *List of regular polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli ...
References
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) *External links
* *Hyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Apeirogonal tilings Hyperbolic tilings Infinite-order tilings Isogonal tilings Isohedral tilings Regular tilings Self-dual tilings