One of the most popular self-similar fractal set is the Sierpinski carpet. To obtain it, we partition the unit square into 9 congruent copies and throw away the one in the middle. We repeat the same process for the remaining squares ad infinitum. The set we obtain is the Sierpinski carpet. The intersection of the Sierpinski carpet with a straight line is a fractal set itself. It is known that the size (Hausdorff dimension) of the intersection of the Sierpinski carpet and a straight line is different for different lines. However, for many (in some natural sense) lines the Hausdorff dimension of this intersection is equal to the Hausdorff dimension of the carpet (which is log 8/log 3) minus one. In this talk I will speak about our joint result with Anthony Manning concerning the exceptional behavior of the intersection of the Sierpinski carpet with lines of rational slopes.