By investigating the canonical commutation rules for gravitating quantized particles in a (2 + 1)-dimensional world, it is found that these particles live on a spacetime lattice. The spacetime lattice points can be characterized by three integers. Various representations are possible, the details depending on the topology chosen for energy - momentum space. We find that an 0264-9381/13/5/018/img1 topology yields a physically most interesting lattice within which first quantization of Dirac particles is possible. An 0264-9381/13/5/018/img2 topology also gives a lattice, but does not allow first quantized particles.