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This is a software package for working with flat surfaces in SageMath. The flatsurf documentation is available at

We aim for this package to support the investigation of geometric, algebraic and dynamical questions related to flat surfaces. By flat surface we mean a surface modeled on the plane with monodromy given by similarities of the plane, though current efforts are focused on translation surfaces and half-translation surfaces.

Currently, the package can generate images of flat surfaces, plot straight-line trajectories, deform surfaces through the SL(2,R) action, and compute Delaunay decompositions. SageMath is used to perform exact arithmetic.

This package is open source software, released under GPL v2 (see the COPYING file). We welcome any help to improve the package and especially to broaden the package's mathematical abilities.

The package is currently in active development. If you would like assistance in using it, please contact the authors.


Installing the dependency

Our software depends on the surface-dynamics package, which is distributed on PyPI. To install it, run the following command:

$ sage -pip install surface_dynamics [--user]

The --user option is optional and allows to install the corresponding Python module in your user space rather than system-wide (and thus does not require administrator rights).

Installing the package

Since sage-flatsurf is available on PyPI at, the released version of sage-flatsurf can be installed by running the following command:

$ sage -pip install sage-flatsurf [--user] [--upgrade]

To install the development version of sage-flatsurf, run instead:

$ sage -pip install git+ [--user] [--upgrade]

The options --user and --upgrade are optional. The option --user make the installation in your home directory instead of in the Sage sources. The option --upgrade allows you to upgrade if the package is already installed.

This might fail if Git is not installed on your computer (which could happen for example with certain versions of Sage in Windows). In this situation you have two options. Either you install Git. Or you download this project from the "Clone or download" drop-down menu above (you should get a zip file). Then you need to run the command:

$ sage -pip install TARBALL_NAME [--user] [--upgrade]

where TARBALL_NAME has to be replaced with the full path to your tarball. Under Windows, it should be a Cygwin path and will look something like /cygdrive/c/Users/You/Downloads/

Then you should be able to use the following within sage:

sage: import flatsurf.geometry.similarity_surface_generators as sfg
sage: T = sfg.translation_surfaces.regular_octagon()
sage: T
Translation surface built from 1 polygon
sage: T.stratum()

To uninstall the package, you can do:

$ sage -pip uninstall flatsurf

Run the tests

Running the tests of a specific file or directory is done by running:

$ sage -t --force-lib ARG

where ARG is either a directory or file. In particular, to test all the files in the module just do:

$ sage -t --force-lib flatsurf

Tests on the master branch are automatically run through Travis-CI.

Related projects

There are several related projects

  • surface-dynamics (SageMath module): more focused on dynamics (interval exchanges)
  • veerer (Python module): specific to handle specific triangulation of half-translation surfaces
  • flatsurf: (C++ library with Python interface) computing GL(2,R)-orbit closure of translation surfaces

Primary Contributors

  • Vincent Delecroix (Bordeaux)
  • Pat Hooper (City College of New York and CUNY Graduate Center)

We welcome others to contribute.


  • This software project was created during a thematic semester at ICERM.
  • Hooper's contribution to the project has been supported by the National Science Foundation under Grant Number DMS-1500965. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
  • Delecroix's contribution to the project has been supported by OpenDreamKit, Horizon 2020 European Research Infrastructures project #676541.


Flat surfaces in Sage




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