  The dilatancy angle describes the propensity of a granular medium to dilate
under an applied shear. Using a simple spin model (the ``tetris'' model) which
accounts for geometrical ``frustration'' effects, we study such a dilatancy
angle as a function of density. An exact mapping can be drawn with a directed
percolation process which proves that there exists a critical density $\rho_c$
above which the system expands and below which it contracts under shear. When
applied to packings constructed by a random deposition under gravity, the
dilatancy angle is shown to be strongly anisotropic, and it constitutes an
efficient tool to characterize the texture of the medium.
