Before revision: Periodic structures such as solid crystalline materials or textiles were previously studied as isolated structures without taking into account the continuity of their configuration spaces.
After revision: Periodic structures such as solid crystalline materials or textiles were previously classified in discrete and coarse ways that depend on manual choices or are unstable under perturbations. Since crystal structures are determined in a rigid form, their finest natural equivalence is defined by rigid motion or isometry, which preserves inter-point distances. Due to atomic vibrations, isometry classes of periodic point sets form a continuous space whose geometry and topology were unknown.