This paper introduces the key concepts and problems of the new research area of Periodic Geometry and Topology for applications in Materials Science .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. The key new problem in Periodic Geometry is an isometry classification of periodic point sets. A required complete invariant should continuously change under point perturbations, because atoms always vibrate in real crystals. The main objects of Periodic Topology are embeddings of curves in a thickened plane that are invariant under lattice translations. Such periodic knots were classified in the past up to continuous deformations (isotopies) that keep a fixed lattice structure, hence are realized in a fixed thickened torus. The more practical equivalence is a periodic isotopyin a thickened plane without fixing a lattice basis. The paper states the first results in the new area and proposes further problems and directions .
This monograph introduces key concepts and problems in the new research area of Periodic Geometry and Topology for materials applications .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. The key new problem in Periodic Geometry is to unambiguously parameterize this space of isometry classes by continuous coordinates that allow a complete reconstruction of any crystal. The major part of this manuscript reviews the recently developed isometry invariants to resolve the above problem: (1) density functions computed from higher order Voronoi zones, (2) distance-based invariants that allow ultra-fast visualizations of huge crystal datasets, and (3) the complete invariant isoset (a DNA-type code) with a first continuous metric on all periodic crystals. The main goal of Periodic Topology is to classify textiles up to periodic isotopy, which is a continuous deformation of a thickened plane without a fixed lattice basis. This practical problem substantially differs from past research focused on links in a fixed thickened torus .
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an isometry classification of periodic point sets. A required complete invariant should continuously change under point perturbations, because atoms always vibrate in real crystals. The main objects of Periodic Topology are embeddings of curves in a thickened plane that are invariant under lattice translations. Such periodic knots were classified in the past up to continuous deformations (isotopies) that keep a fixed lattice structure, hence are realized in a fixed thickened torus. The more practical equivalence is a periodic isotopyin a
to unambiguously parameterize this space of isometry classes by continuous coordinates that allow a complete reconstruction of any crystal. The major part of this manuscript reviews the recently developed isometry invariants to resolve the above problem: (1) density functions computed from higher order Voronoi zones, (2) distance-based invariants that allow ultra-fast visualizations of huge crystal datasets, and (3) the complete invariant isoset (a DNA-type code) with a first continuous metric on all periodic crystals. The main goal of Periodic Topology is to classify textiles up to periodic isotopy, which is a continuous deformation of a
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The paper states the first results in the new area and proposes further problems and directions
This practical problem substantially differs from past research focused on links in a fixed thickened torus