We describe a graphics processing unit (GPU)-based computational platform for six-dimensional configuration mapping, which is the description of the configuration space of rigid motions in terms of collision and contact constraints. The platform supports a wide range of computations in design and manufacturing, including three- and six-dimensional configuration space obstacle computations, Minkowski sums and differences, packaging problems, and sweep computations. We demonstrate dramatic performance improvements in the special case of configuration space operations that determine interference-free or containment-preserving configurations between moving solids. Our approach treats such operations as convolutions in the six-dimensional configuration space that are efficiently computed using the fast Fourier transform (FFT). The inherent parallelism of FFT algorithms facilitates a straightforward implementation of convolution on GPUs with existing and freely available libraries, making all such configuration space computations practical, and often interactive.