Collective assembly and shape formation of modular cubes under uniform global magnetic control

Achieving coordinated collective behaviors in modular robotic systems under a single global input remains challenging, particularly for forming and manipulating structures beyond planar configurations. Here, we introduce magnetic modular cubes with face-specific magnetic encoding across all six faces and investigate their assembly and locomotion under uniform global magnetic field control. Two cube types, each with a 2 mm edge length and embedded with ten Neodymium micro-magnets, are designed with complementary face-specific encoding that enables multimodal locomotion, selective 2D assembly, and 3D assembly through vertical stacking. To coordinate multiple module groups, we introduce a multiplex workspace that exploits boundary-mediated symmetry breaking to generate differentiated manipulation and assembly behaviors without local sensing or agent-specific control. Using a triaxial Helmholtz coil system, we demonstrate parallel 2D assembly, structure-level pivot walking with per-cycle displacement approaching half the structure length, controlled in-plane rotation, rolling locomotion reaching approximately three body lengths per cycle, and tilted pivot walking for vertical stacking of assembled structures. Dipole interaction modeling and quantitative locomotion analysis relate these behaviors to actuation geometry and magnetic interactions. These results establish a framework for coordinated multimodal assembly and locomotion of modular structures under global magnetic control with applications in programmable matter and collective microrobotic systems.