To anycorresponds a domainT(x) ⊂⊂Y= ]0, 1[2. For some ε > 0, the domain Ω is occupied by a quasiperiodic structure which has the property that if an ε-neighborhood ofxis enlarged by the scale factor (1/ε) then it appears like aY-periodically perforated piece of material, with holes “slightly different” fromT(x). The torsion problem of this structure is studied. The homogenization procedure is completed, that is all the convergences which reveal the system which governs the limit phenomenon, when ε → 0, are proved. In the periodic case there are already two distinct approaches to this problem: [1] and [2], The present work is based on them and on the stepwise method [3] used for proving the homogenization of linear elliptic equations with quasiperiodic coefficients.