Artificial Intelligence (AI) supports many of our everyday activities and decisions. However, personalized algorithmic recommendations often produce adverse experiences due to a lack of awareness, control, or transparency. While research has directed solutions on graphical user interfaces (GUIs), there are no explorations of Tangible User Interfaces (TUIs) to improve the experience with such systems, despite the valid existing academic arguments in favor of this exploration. Therefore, centering on transparency and control, we analyzed how 18 users of movie recommender systems perceived four different TUIs using individual co-design sessions and post-interview questionnaires. Through thematic analysis, we identified seven design considerations while designing TUIs to interact with algorithmic movie recommender systems: (1) Distinctions between TUIs and GUIs; (2) TUIs replacing predominant interfaces; (3) Preference for single-device TUIs; (4) The relevance of granular control for TUIs; (5) Apparent transparency limitations of TUIs; (6) TUIs and algorithmic social computing; and (7) Overview of specific design choices, including advantages and disadvantages of soft, hard, rounded, cubic, and humanoid interfaces. These findings inspired Recffy: the first functional TUI designed to enhance awareness and control in personalized movie recommendations. Based on this study, we propose the concept of Tangible Algorithms: TUIs dedicated to enhancing the interaction of algorithmic systems and their profiling processes or decisions in a specific context. Furthermore, we describe the relevance of tangible algorithms and design guidelines to promote them in diverse AI contexts. Finally, we invite the HCI and CSCW community to continue exploring tangible algorithms to address the interaction with algorithmic systems, including the collaborative and social computing dynamics they can promote in diverse AI contexts.