We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.This is joint work with Pavlo Pylyavskyy.