#### Tessellations of the plane and beyond

###### Autor: Fanny Kassel .

###### Tessellations of the plane have been used since antiquity as decorative patterns. With the development of modern mathematics, periodic tilings of R^n have been extensively studied in relation to crystallography; they include Euclidean tilings but also, more generally, tilings whose symmetry group consists of affine transformations. When the tiles are noncompact, the symmetry group may no longer be a group of translations up to finite index, and tilings with interesting nonsolvable symmetry groups have been constructed since the 1980s. We will discuss some of these developments, including recent joint work with J. Danciger and F. GuĂ©ritaud.