The category of mixed Tate motives (iterated extensions of the pure Tate motives $Q(-n)$ for all integers $n$ is well understood from an abstract point of view but the structure of its periods is still very mysterious. This includes open questions on special values of Dedekind zeta functions, and irrationality questions for higher regulators. It is therefore an important task to understand certain extensions in mixed Tate motives as explicitly and geometrically as possible. In this talk I will survey different aspects of this problem, and explain in particular a construction of « small » motives whose periods are odd zeta values.