Self-similar carpets over finite fields

In [4] an informal algorithm ’to display interesting numeric patterns’ is described without any proof. We generalize this algorithm over arbitrary finite fields of characteristic p and we prove that it really generates self-similar carperts, provided that they contain at least one zero in the first (p+1)/2 lines. For the fields Fp the generalized algorithm produces p− 1 different self-similar carpets. These self-similar carpets are classified according to their arithmetic and their groups of symmetry. All this phenomena can be also interpreted in the framework of the aperiodic tilings. A.M.S.-Classification: 11A07, 28A80.