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A useful robot is one that fulfils its intended function. In a factory setting, where robots have been used successfully for decades, this function is often singular and clearly defined. Similarly, the surroundings of the robot are mostly known, sterile, and unobstructed. Taking robotic systems out of these conditions and into the real world comes with numerous challenges that are non-existent in factory cages. We want personal robots to cope with the uncertain and dynamic environments we inhabit, while at the same time managing and solving diverse tasks. Reconfigurable robots aim to achieve this by changing shape and function to address a variety of applications, environments, and users. While reconfigurable robots carry a lot of promise, finding a balance between the system's adaptability, the extent to which it can alter shape and function, and the added complexity is difficult. Research efforts have largely focused on proof-of-concept studies with limited reconfigurability and application range, avoiding the increasingly overwhelming mechanical, computational, and electronic complexities. This thesis introduces a new paradigm to the world of reconfigurable robotics with an inherent adaptability through simplification of the underlying structure. Approximating physical structures through polygon abstractions, similar to computer graphics, such systems can assume a wide range of structural or functional three-dimensional shapes. Based on this paradigm, it also presents a new robotic platform combining the concepts behind both modular and origami robotics, as well as reconfigurable mechanisms and polygon meshing. In order to take advantage of this new paradigm, a diverse set of problems must be investigated, spanning multiple robotic disciplines. With an increasing degree of reconfigurability, both within a module and the overall system, the growing physical and mechatronic requirements need to be analysed and addressed accordingly. New reconfiguration algorithms and control strategies need to be developed to cope with the large, and constantly changing, number of degrees of freedom. These must then be synchronised and scaled appropriately, leveraging modularity at multiple levels, to accomplish diverse sets of tasks and functions. Addressing the challenges associated with this new robotic paradigm and proving its viability provides the context for this thesis. In a first phase it outlines the initial conception, studying scalability and applicability through the combination of modularity and origami robots with a first prototype and its use in multiple scenarios. It continues with the development and analysis of several building blocks of modular origami robots, both mechanical and algorithmic, analysing mechanisms for the coupling between modules and the reconfiguration process. In the second phase the proposed paradigm is elaborated into its full form, integrating and examining reconfigurable mechanisms and polygon meshing. The resulting morphological and functional flexibility is validated through the development and testing of a highly sophisticated modular robotic system. Individual modules can alter their own triangular shape, drive towards and attach to each other, and transform into functional three-dimensional configurations. The conceptual and physical systems developed and studied in this thesis answer some of the challenges posed by this new paradigm and underline the potential of reconfigurability in robotics.