We introduce a model for a robot hand based on the theory of expansions in non-integer bases. Self-similarity of configurations and an arbitrarily large number of fingers (including the opposable thumb) and phalanxes are the main features. Binary controls rule the dynamics of the hand, in particular the extension and the rotation of each phalanx. We investigate the reachable set of this hand and we introduce a grasping model. Joint work with P. Loreti.