## Independent Projects

### Visualizing SubRiemanniann Billiards

**Mentor(s):** Lucas Dahinden, Lutz Hofmann, and Maksim Schreck**Student(s) participating:** Jakob Niessner**Details:** All the details are contained in this pdf.

### Planimeters and Bicycle Tires

**Mentor(s):** Lucas Dahinden**Student(s) participating:** Aaron Osburg**Details:** Planimeters are devices used to measure the area of a region in the plane by tracing around its boundary. As such, they are a mechanical manifestation of Green’s Theorem. The goal of this project is to build models of the polar and of the hatchet planimeter and understand the mathematics behind them. The hatchet planimeter is connected to bicycle tire tracks and the project could further explore a conjecture by Menzin about their geometry using this planimeter.

### Laminations on Hyperbolic Surfaces in the Universal Cover

**Mentor(s):** Mareike Pfeil**Student(s) participating:** Christian Alber**Details:** A lamination on a hyperbolic surface is a collection of simple disjoint geodesics whose union is closed. Laminations are very helpful when studying surfaces. For instance, they can be used to deform hyperbolic surfaces and to define coordinates on Teichmüller space. The goal of this project is to visualize lifts of laminations on hyperbolic surfaces in the Poincaré disk model for the hyperbolic plane. **Prerequisites**: A first course in differential geometry, basic algebra, basic hyperbolic geometry will be needed, but can be learned during the project.

### Geometric Constructions with the MIRA

**Mentor(s):** Denis Vogel**Student(s) participating:** Silas Gramlich**Details:** The MIRA is a semi-transparent reflective device used for constructing geometric objects. The goal of this project is to build such a device and understand the mathematics behind it.

## SFB Funded Projects

### Symplectic Billiards

**Mentor**: Peter Albers**Student(s) participating**: Fabian Lander, Jannik Westermann**Details**: The goal of this project is to simulate symplectic billiards on non-convex polygons, and to search for periodic orbits in particular.

### Hyperbolic Billiards

**Mentor**: Dia Taha**Student(s) participating**: Christian Alber, Mara-Eliana Popescu**Details**: The project aims to investigate the statistics of hyperbolic billiards.

## Proseminar/Seminar Projects

### Ping Pong Game in the Poincaré Disk

**Mentors**: Brice Loustau, Anja Randecker, Ricardo Waibel and Mathias Häberle.**Students participating**: …**Details**: The goal of this project is to implement a game inspired by the classic Ping Pong game, but in a hyperbolic setting. The game takes place in the Poincaré disc, only a finite area (the hyperbolic equivalent of a rectangle) is available to the ball. Players can move their racket along a fixed geodesic (a generalization of the concept of a straight line), and the ball obeys these different laws for its movement as well. Through the game players can then develop an intuitive understanding of hyperbolic geometry.**GitHub repository**: link

### Delta-Slim Triangles in the Poincaré Disk

**Mentors**: Brice Loustau, Anja Randecker, Ricardo Waibel, and Mathias Häberle.**Students participating**: …**Details: **The delta-slim property of a triangle is defined as follows: take two sides of the triangle and draw a delta neighbourhood around them. Then check whether the third side lies completely in the union of the neighbourhoods. If this is true for all choices of two sides, then the triangle is called *delta-slim*. The project takes an experimental approach to firstly finding the minimum delta for a given hyperbolic triangle on the Poincaré disc, and secondly, to determine a minimum delta for all possible hyperbolic triangles on the Poincaré disc, i.e., a delta, for which all hyperbolic triangles on the Poincaré disc are delta-slim.**GitHub repository**: link

### 3D Prints of Tilings in the Poincaré Disk

**Mentors**: Brice Loustau, Anja Randecker, Ricardo Waibel, and Mathias Häberle.**Students participating**: …**Details**: Inspired by M.C. Escher’s works (especially Circle Limit I-IV), this project looks at tilings of the Poincaré disc and recreates their mathematical aspects as models using 3D printing. On top of that, the HEGL logo is adapted to a 3D model, whose components include geodesics and horocycles in the Poincaré disc.**GitHub repository**: link