A metamorphic robotic system is an aggregate of homogeneous robot units which can individually and selectively locomote in such a way as to change the global shape of the system. We introduce a mathematical framework for defining and analyzing general metamorphic robots. With this formal structure, combined with ideas from geometric group theory, we define a new type of configuration space for metamorphic robots—the state complex—which is especially adapted to parallelization. We present an algorithm for optimizing an input reconfiguration sequence with respect to elapsed time. A universal geometric property of state complexes—non-positive curvature—is the key to proving convergence to the globally timeoptimal solution obtainable from the initial path