Shape-changing robots adapt their own morphology to address a wider range of functions or environments than is possible with a fixed or rigid structure. Akin to biological organisms, the ability to alter shape or configuration emerges from the underlying mechanical structure, materials or control methods. Soft robots, for instance, employ malleable materials to adapt to their environment, modular robots assemble multiple units into various three-dimensional configurations and insect-like swarm robots interact in large numbers to fulfil tasks. However, the promise of broad functional versatility in shape-changing robots has so far been constrained by the practical implications of either increasing the degree of morphological flexibility or addressing specific applications. Here we report a method for creating robotic systems that realizes both sides of this trade-off through the introduction of physical polygon meshing. By abstracting functional three-dimensional structures, collections of shape-changing robotic modules can recreate diverse three-dimensional shapes and dynamically control the resulting morphology. We demonstrate this approach by developing a system of polygon robots that change their own shape, attach to each other, communicate and reconfigure to form functional and articulated structures. Applying the system to three distinct application areas of robotics involving user interaction, locomotion and manipulation, our work demonstrates how physical polygon meshing provides a new framework for more versatile intelligent machines.|Robots that can change their shape offer flexible functionality. A modular robotic platform is shown that implements physical polygon meshing, by combining triangles with sides of adjustable lengths, allowing flexible three-dimensional shape configurations.