The study of Teichmüller curves has since its first beginnings with results of Masur and Veech in the 1990’s become a very active field today. It has been advanced by major impetus of Fields medalists as Yoccoz, Kontsevich, McMullen, Okounkov, Avila and Mirzakhani. The central objects in our project are origamis, also called square-tiled surfaces, which are given by a few combinatorial data. They define an important class of Teichmüller curves which can be studied with algorithmic approaches. Supported by computer experiments we examine geometric aspects (boundary points and systolic geometry), arithmetic aspects (congruence properties of the Veech groups) and algebraic aspects (action of the affine group on homology).