nLab tesselation

Idea

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 $\{n,k\}$.

Euclidean tesselation

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.

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 $\{5,4\}$ 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:

Related concepts

• wallpaper group

• holographic entanglement entropy

References

General

Introduction and survey:

• David Joyce, Hyperbolic Tesselations

• Damon Motz-Storey, Constructing Hyperbolic Polygon Tessellations, 2016 (pdf, pdf)

See also:

• Wikipedia, Tesselation

• Wikipedia, Uniform tilings in hyperbolic plane

• Wikipedia, Hyperbolic Geometry – Hyperbolic Tesselations

Claim of a hyperbolic tessalation approximated by a photonic crystal:

• Alicia J. Kollár, Mattias Fitzpatrick, Andrew A. Houck, Hyperbolic Lattices in Circuit Quantum Electrodynamics, Nature 571 (2019) 45–50 [arXiv:1802.09549, doi:10.1038/s41586-019-1348-3]

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)

following:

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

See also:

• Alexander Jahn, Zoltán Zimborás, Jens Eisert, around Fig. 6 in: Central charges of aperiodic holographic tensor network models, Phys. Rev. A 102, 042407 (arXiv:1911.03485)