This paper presents PolyHinge, a variable-form Tangible User Interface (TUI) that provides dynamic affordance and semantics and extends its malleability and mutability by enabling active shape-changing outputs. It is superior to traditional rigid TUIs in that it can provide multistate geometric transformations in the physical world. Highly diverse forms of transformations are allowed as PolyHinge is based upon extended structures of hinged dissections and connections. In addition, the concept of dynamic affordance and semantics is further developed within the context of shape-changing interfaces, as a functional and bidirectional interface applicable for both input and output. Meanwhile, a series of applications are developed, demonstrating how users can interact with PolyHinge on a tabletop, and how the dynamic affordance and semantics can aid users.