Self-similar carpets over finite fields

In [Dann E. Passoja, Akhlesh Lakhtakia, Carpets and rugs: An exercise in numbers, Leonardo 25 (1) (1992) 69–71] an informal algorithm ‘to display interesting numeric patterns’ is described without any proof. We generalize this algorithm over arbitrary finite fields FqFq of characteristic pp and we prove that it always generates some self-similar sets. For the prime fields FpFp the generalized algorithm produces p−1p−1 different self-similar sets. These sets are classified according to their arithmetic and their groups of symmetry.