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 ...
, the order-5 pentagonal 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
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 , constructed from five pentagons around every vertex. As such, it is
self-dual.
Related tilings
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5
n).
See also
*
Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille.
Structure and properties
The square tili ...
*
Uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as Face (geometry), faces and is vertex-transitive (Tran ...
*
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 H. Conway, 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
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-5 tilings
Pentagonal tilings
Regular tilings
Self-dual tilings
{{hyperbolic-geometry-stub