A regular tesselation or tiling is a covering of a plane by regular polygons (the tiles), so that polygons overlap only on their boundary and such the same number of tiles meet at each vertex.
A tesselation by regular $n$-gons such that $k$ of them meet at each vertex is denoted by the Schläfli symbol $\{n,k\}$.
For example $\{3,6\}$, $\{4,4\}$ and $\{6,3\}$ denote the familiar tilings of the Euclidean plane by triangles, squares and hexagons, respectively. In fact these are all tesselations of the Euclidean plane which exist.
Similarly a hyperbologic tesselation is such a tiling of the hyperbolic plane by regular polygons. In contrast to the Euclidean case, there are infinitely many different such tesselations.
For example the hyperbolic tesselation $\{5,4\}$ by pentagons:
In discussion of holographic entanglement entropy and quantum error correcting codes this controls the HaPPY code, see also JGPE 19.
Or the hyperbolic tesselation $\{4,6\}$ by squares:
Introduction and survey:
See also:
Wikipedia, Tesselation
Wikipedia, Uniform tilings in hyperbolic plane
Claim of a hyperbolic tessalation approximated by a photonic crystal:
On iterative constructions of hyperbolic tesselations from “inflation rules”:
following:
See also:
Last revised on February 13, 2023 at 14:02:37. See the history of this page for a list of all contributions to it.