hypertiling 1.5.1

hypertiling is a high-performance Python library for the generation and visualization of regular hyperbolic lattices embedded in the Poincare disk model. Using highly optimized, efficient algorithms, hyperbolic tilings with millions of vertices can be created in a matter of minutes on a single workstation computer. Facilities including computation of adjacent vertices, dynamic lattice manipulation, refinements, as well as powerful plotting and animation capabilities are provided to support advanced uses of hyperbolic graphs.

Installation

hypertiling is available in the PyPI package index and can be installed using

$ pip install hypertiling

For optimal performance, we highly recommand to use hypertiling together with python-numba, which, if not already present on your system can be installed automatically using the [numba]-suffix, i.e.

$ pip install hypertiling[numba]

The package can also be locally installed from our public git repository via

$ git clone https://git.physik.uni-wuerzburg.de/hypertiling/hypertiling
$ pip install .

Quick Start

In Python, import tiling object from the hypertiling library

from hypertiling import HyperbolicTiling

Set parameters, initialize and generate the tiling

p = 7
q = 3
nlayers = 5

T = HyperbolicTiling(p,q,nlayers) 

Documentation

Further usage examples and a full API reference are available in our documentation.

Authors

• Manuel Schrauth
  manuel.schrauth@iis.fraunhofer.de
• Yanick Thurn
• Florian Goth
• Jefferson S. E. Portela
• Dietmar Herdt
• Felix Dusel

This project is developed at:
Institute for Theoretical Physics and Astrophysics
University of Wuerzburg

Citation

If you use hypertiling, we encourage you to cite or reference this work as you would any other scientific research. The package is a result of a huge amount of time and effort invested by the authors. Citing us allows us to measure the impact of the research and encourages others to use the library.

Codebase:

Manuel Schrauth, Yanick Thurn, Florian Goth, Jefferson Portela, Dietmar Herdt and Felix Dusel. (2023). The hypertiling project. Zenodo. https://doi.org/10.5281/zenodo.7559393

Release publication:

Manuel Schrauth, Yanick Thurn, Florian Goth, Jefferson Portela, Dietmar Herdt and Felix Dusel. (2024). HYPERTILING - a high performance python library for the generation and visualization of hyperbolic lattices project. SciPost Physics Codebases (2024): 034.

Further publications:

Yanick Thurn, Manuel Schrauth, Johanna Erdmenger. Hyperbolic tiling neighborhoods in O(1) time. arXiv preprint

Examples

Tilings

The core functionality of the package is the generation of regular hyperbolic tilings projected onto the Poincare disk. Tilings in the upper row are centered about a cell, in the lower row about a vertex.

Refinements

The hypertiling package allows to perform triangle refinements, such as shown here

Applications

Hyperbolic Magnet

Simulation of a Ising-like Boltzmann spin model with anti-ferromagnetic interactions on a hyperbolic (7,3) tiling, quenched at low temperature. The hyperbolic antiferromagnet (left) exhibits geometrical frustration, whereas on a flat lattice (right) an ordered anti-parallel alignment can be observed.

Helmholtz Equation

Solution of an electrostatic Helmholtz problem on a refined (3,7) tiling, where boundary values have been fixed to either -1 (red) or +1 (blue). One readily recognizes a field value separation according to geodesic arcs in the Poincare disk representation of the hyperbolic plane.

Further information and examples can be found in our Jupyter notebooks in /examples subfolder.

License

Every part of hypertiling is available under the MIT license.