Embedding parallelo­hedra into primitive cubic networks and structural automata description

The main goal of the paper is to contribute to the agenda of developing an algorithmic model for crystallization and measuring the complexity of crystals by constructing embeddings of 3D parallelo­hedra into a primitive cubic network (pcu net). It is proved that any parallelo­hedron P as well as tiling by P, except the rhombic dodeca­hedron, can be embedded into the 3D pcu net. It is proved that for the rhombic dodeca­hedron embedding into the 3D pcu net does not exist; however, embedding into the 4D pcu net exists. The question of how many ways the embedding of a parallelo­hedron can be constructed is answered. For each parallelo­hedron, the deterministic finite automaton is developed which models the growth of the crystalline structure with the same combinatorial type as the given parallelo­hedron.