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The infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has
: 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
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 ...
of , which means it has countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
ly many apeirogons 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 in Euclidean, spherical and hyperbolic spaces.
Overview
This table shows a summary of regular polytope counts by rank.
There are no Euclidean regular star tessellations in any number of dimensions.
...
References
*John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...
, Heidi Burgiel, Chaim Goodman-Strauss, ''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