sage-flatsurf: Flat Surfaces in SageMath

sage-flatsurf is a Python package for working with flat surfaces in SageMath.

We aim for sage-flatsurf 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.

Installation

The preferred way to install software should be to use your package manager (e.g. \(apt-get\) on Debian or Ubuntu, \(pacman\) on Arch Linux, \(brew\) on MacOS, etc). However, as of this writing, sage-flatsurf has not been picked up by any of the major distributions yet.

We therefore recommend to install sage-flatsurf with the Mamba package manager.

Detailed installation instructions:

• Install with Mamba and Upgrade an existing installation with Mamba

• Install into an existing SageMath source build

• Install with pip

If you are planning to work on the sage-flatsurf source code, you can also build sage-flatsurf from source. For this, please have a look at our Developer’s Guide.

A Tour of sage-flatsurf

Demos of some of the capabilities of sage-flatsurf:

• A Tour of the flatsurf Suite
• Defining Surfaces
• Graphics Configuration
• The GL(2,R) Action, the Veech Group, Delaunay Decomposition
• Relative Period Deformations
• Working with Saddle Connections
• Exploring Orbit Closures
• Siegel-Veech Constants
• Straight-Line Flow
• Notes from the Warwick EPSRC Symposium on “Computation in geometric topology”
• Boshernitzan’s Conjectures

These examples can also be explored interactively by clicking . The interactive session might take a moment to start. Once ready, press Shift + Enter to execute the cells, starting from the top.

Module Reference

The sage-flatsurf source code is split into two packages: graphical which contains plotting logic, and geometry containing everything else. The links below lead to the documentation for the modules that make up these packages with more usage examples and the source code for all the classes and functions implemented in sage-flatsurf. (Note that you can also access this documentation from an interactive SageMath session with ? and ??.

• The flatsurf.cache Module
• The flatsurf.features Module
• The flatsurf.geometry Package
• The flatsurf.geometry.categories Package
• The flatsurf.geometry.pyflatsurf Package
• The flatsurf.graphical Package