nLab tesselation




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 nn-gons such that kk of them meet at each vertex is denoted by the Schläfli symbol {n,k}\{n,k\}.

Euclidean tesselation

For example {3,6}\{3,6\}, {4,4}\{4,4\} and {6,3}\{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.

Hyperbolic tesselations

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}\{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}\{4,6\} by squares:



Introduction and survey:

See also:

Claim of a hyperbolic tessalation approximated by a photonic crystal:

Via inflation rules

On iterative constructions of hyperbolic tesselations from “inflation rules”:

  • Latham Boyle, Madeline Dickens, Felix Flicker, Section IV of: Conformal Quasicrystals and Holography, Phys. Rev. X 10, 011009 (2020) (arXiv:1805.02665)


  • Latham Boyle, Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices (arXiv:1608.08220)

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.