hypertiling

hypertiling is a high-performance Python 3 library for the generation of regular hyperbolic tilings embedded in the Poincaré disk model. Using efficient algorithms and the CPU/SIMD optimization provided by numpy, hyperbolic tilings with hundreds of millions of vertices can be created in a matter of minutes on a single computer. Facilities including optimized search algorithms for adjacent vertices and powerful plotting and animation capabilities are provided to support scientific and other advanced uses of the graphs.

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Installation and Usage

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

$ pip install hypertiling

The package can also be locally installed. First download or clone the package from our public git repository, using

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

Now execute

$ pip install .

in the package’s root directory to install the package in-place.

In order to use the package, in Python do the following import

from hypertiling import HyperbolicTiling

Set parameters, initialize and generate the tiling

p = 7
q = 3
nlayers = 5

T = HyperbolicTiling(p,q,nlayers)

Further examples are available in our documentation and Jupyter demo notebooks.

Authors

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

Examples

In this animation we simulate a classical scalar field Hamiltonian on two different lattice structures. The setting can be interpreted as a simple model for magnetic behaviour in a solid crystal structure. In particular, couplings between adjacent cells (“spins”) are adjusted in a way which energetically favors anti-parallel local alignment. Both systems are initiated in a random “hot” configuration of positive (red) and negative (green) field values, which - in the magnetic picture - can be interpreted as uniaxial spins pointing upwards or downwards, respectively. The animation demonstrates the effect of a Metropolis algorithm being repeatedly applied - until the system reaches thermal equilibrium.

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License

Every part of hypertiling is available under the MIT license.