## Fractal limit sets and Schottky groups

**Goal:**

The Möbius maps are transformations of the plane sending circles and
straight lines to circles and straight lines, and preserving the angles
between curves. Schottky groups are some particular groups consisting of
Möbius transformations, and constructed by pairing couples of circles.
Whenever such a group is given, there is an associated subset of the
plane, called its "limit set". These sets have lots of symmetries
related to the Shottky group, and are very often beautiful fractal sets.

The goal of this project is to be able to draw and explore these
beautiful subsets of the plane. It will be also a nice opportunity to
understand Möbius transformations and their link with 2x2 matrices, as
well as some topics around groups.

**Supervisor:**
Miguel Acosta

**Difficulty level:**
Any.

**Tools:**

- Any programming language or any Computer Algebra System such as sagemath
- Image processing software

**Literature:**

- Mumford D, Series C, Wright D, Gonick L. Indra's pearls: the vision of
Felix Klein. Cambridge, UK: Cambridge university press; 2002. xix+395.