Sébastien Labbé’s Research Code


This is an optional package for SageMath containing code I wrote for research. It contains modules on the following topics.

Discrete dynamical systems

diophantine approximation, Markov transformations, Wang tilings, Lyapunov exponents, matrix cocycles, multidimensional continued fraction algorithms, polyhedron exchange transformations, GIFS.


2d substitutions, bispecial factors, bond percolation, Dyck word in 3D, words, Joyal bijection, languages, Oldenburger sequence, ostrowski numeration, partial injections,

Digital geometry

Christoffel graph, discrete subset, discrete plane, double square tiles, polyhedron partitions,




analyze Sage build time, fruit Python classes example, ranking scale


Prerequisites - System packages (easy to install)

Some functionalities of slabbe package depend on the installation of packages on the system such as libpari-dev, pdflatex, lualatex (lualatex compiles large tikzpicture exceeding the memory limits of pdflatex), pdf2svg as well as softwares including ImageMagick (to convert pdf to png so that tikzpicture appear directly in the Jupyter notebook), Graphviz (to draw graphs).

On Debian or Ubuntu, one may do:

sudo apt update
sudo apt install libpari-dev -y
sudo apt install texlive texlive-latex-extra texlive-luatex texlive-fonts-extra -qy
sudo apt install graphviz imagemagick pdf2svg -y

The installation of imagemagick done above provides the command convert but it does not allow to convert pdf to png unless you edit the Imagemagick’s policy.xml file which can be done as follows (thanks to Naveed’s comment on stackoverflow):

sudo sed -i '/PDF/s/none/read|write/' /etc/ImageMagick-6/policy.xml

On OSX, one should first install Homebrew. Then one can install the above packages similarly as above using brew instead of apt:

sudo brew install graphviz imagemagick pdf2svg

Prerequisites - SageMath optional packages

Installing slabbe requires a working SageMath installation (with Cython and gcc). Depending on the usage, it might be necessary to install the optional sagemath packages dot2tex (translate dot file to tikz to draw nice graphs), glucose (SAT solver) and latte_int:

sage -i dot2tex glucose latte_int

Note that graphviz must be installed before dot2tex.


The module is distributed on PyPI and is easily installed through the Python package manager pip:

sage -pip install slabbe

To install the module in your user space (which does not require administrator rights):

sage -pip install slabbe --user

To install the most recent development version:

sage -pip install --upgrade git+https://gitlab.com/seblabbe/slabbe


sage: from slabbe import *

Other Python packages you may want to install

Some functionalities of slabbe package depend on the installation of few Python libraries such as pytimeparse (time expression parser) and roman (integer to Roman numerals converter):

sage -pip install pytimeparse roman

Other System packages you may want to install

Some functionalities of slabbe package depend on the installation of a linear program solver such as Gurobi. See the thematic tutorial to setup the installation of Gurobi in SageMath`.

It builds on SageMath

It depends heavily on the SageMath library as it uses the following modules: combinat, functions, geometry, graphs, matrix, misc, modules, numerical, parallel, plot, probability, rings, sat, sets, structure, symbolic.

SageMath is free open source math software that supports research and teaching in algebra, geometry, number theory, cryptography, and related areas.

Follows the Best practices for scientific computing

It follows as much as possible the SageMath general conventions and the Best Practices for Scientific Computing. Each module is fully documented and doctested. Before each new release, we make sure that all examples are working. As the ReScience Journal says: “Reproducible science is good. Replicated Science is better”.

Future inclusion into Sage

Some modules may have a wider interest to the SageMath community (tikz_picture.py for example) and could be included in SageMath at some point. Please contact the author if you want to act as a reviewer for some module(s) and I will create a ticket on trac for its inclusion into SageMath.

Release history

Version 0.7 (December 8, 2022)

New module on d-dimensional Sturmian configurations, to draw figures from arXiv:2204.06413. New module on Hypercubic billiard words (written with Mélodie Andrieu). The module TikzPicture got merged in SageMath 9.6 into \(sage.misc.latex_standalone\) through ticket #20343 during which many improvements were made. The line from slabbe import TikzPicture now uses the one in SageMath by default. All tests passed on version 9.7 of SageMath. Up to recently, all tests passed on versions 9.0, 9.1, 9.2, 9.3 and 9.4 of SageMath; can’t test those earlier versions of SageMath anymore because gitlab continuous integration tests is not free anymore.

Version 0.6.3 (November 1, 2021)

New module on Ostrowski numeration system (basic stuff). New module on EkEkStar (written with Milton Minvervino). New modules containing the updated code published in arXiv:1808.07768, arXiv:1903.06137 and arXiv:1906.01104. All tests passed on versions 9.0, 9.1, 9.2, 9.3 and 9.4 of SageMath.

Version 0.6.2 (December 15, 2020)

New module on Graph-directed iterated function systems (GIFS). Fixed TransitiveIdeal import error. Now using gitlab continuous integration automatic tests: installation + import slabbe tested to work on versions 8.7, 8.8, 9.0, 9.1, 9.2 of SageMath. All tests passed on versions 9.0, 9.1, 9.2 of SageMath.

Version 0.6.1 (May 8, 2020)

New modules to deal with the coding of \(Z^d\)-action by PETs, \(d\)-dimensional sturmian configurations. Improved the computation of induced polyhedron partition and induced polyhedron exchange transformation. New modules containing the code for the articles arxiv:1903.06137 and arXiv:1906.01104

Version 0.6 (November 22, 2019)

Make the package work with Python 3. Most of the tests pass with Python 3 now.

Version 0.5.1 (May 30, 2019)

Few fixes for the publication of “Induction of \(Z^2\)-actions on partitions of the 2-torus”. Improved html documentation.

Version 0.5 (April 10, 2019)

Few fixes for the version 2 of “Substitutive structure of Jeandel-Rao aperiodic tilings”. New additions includes solving Wang tilings problem using SAT solvers and a class for Polyhedron exchange transformations.

Version 0.4.4 (September 28, 2018)

Make import slabbe work in Sage with Python 3.

Version 0.4.3 (August 22, 2018)

Few fixes for the publication of “Substitutive structure of Jeandel-Rao aperiodic tilings”.

Version 0.4.2 (July 20, 2018)

Few fixes for the version 2 of “A self-similar aperiodic set of 19 Wang tiles”.

Version 0.4.1 (February 9, 2018)

Few fixes for the publication of “A self-similar aperiodic set of 19 Wang tiles”. New module to solve the Magic hexagon problem.

Version 0.4 (January 20, 2018)

Version 0.4 includes new modules for Wang tilings, 2d substitutions, polyhedron partitions, partial injections, ostrowski numeration and many improvements to other modules.

Version 0.3b2 (December 11, 2016)

Version 0.3b2 includes a new module for diophantine approximations, random point generation inside polytopes, analyzing sage building time, and many improvements to previous modules.

Version 0.3b1 (June 12, 2016)

Version 0.3b1 is now a Python package available in the Python Package Index (PyPI). It was migrated from the previous sage optional spkg old-style format. It also adds code to deal with bispecial factors, some new methods of graphs, substitutions and matrices.

Version 0.2 (November 25, 2015)

slabbe-0.2.spkg (documentation) provides modules on multidimensional continued fraction algorithms, matrix cocycles, languages and tikzpictures.

Version 0.1.1 (June 3, 2015)

slabbe-0.1.1.spkg fixes a bug with gcd import error.

Version 0.1 (August 27, 2014)

slabbe-0.1.spkg (documentation) contains modules on digital geometry, combinatorics on words and more.

Module documentation

This reference manual contains many examples that illustrate the usage of the slabbe package. The examples are all tested at each release and should produce exactly the same output as in this manual.

This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.

Dynamical systems


Combinatorics on words

Digital Geometry


Code in research papers


Indices and Tables