The majority of Tangible User Interfaces (TUIs) consist of rigid objects that are either held in the hands, or arranged relative to each other on a horizontal or vertical surface. In this paper we consider the design space of TUIs that can be created by moving parts of an assembly relative to each other – creating articulated tangible interfaces. An analytic approach to this design space allows us to identify the potential applications and trade-offs of TUIs that include mechanically articulated parts.