Flat surfaces in Sage
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This is a software package (Sage module) for working with flat surfaces in Sage. The flatsurf documentation is available http://www.labri.fr/perso/vdelecro/flatsurf/latest/

We aim for this module 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 program can generate images of flat surfaces, plot straight-line trajectories, deform surfaces through the SL(2,R) action, and compute Delaunay decompositions. Sage is used to perform exact arithmetic.

This software is open source (see the file LICENSE). 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 the package, please contact the authors.

There is also a related surface_dynamics package. The code in this repository is currently independent of the package but the aim is to get them merged.

Installing the dependency

Our software depends on the surface_dynamics package. The module is distributed on PyPI. To install it, you just need to run the following command:

$ sage -pip install surface_dynamics [--user]

The --user option is optional and allows to install the module in your user space (and does not require administrator rights).

Installing the module

To install sage-flatsurf, run the following command:

$ sage -pip install git+https://github.com/videlec/sage-flatsurf [--user] [--upgrade]

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

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

: $ 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.

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.