Contents

# Contents

## 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:

## References

### General

Introduction and survey:

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

### 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)