  Physical systems are frequently modeled as sets of points in space, each
representing the position of an atom, molecule, or mesoscale particle. As many
properties of such systems depend on the underlying ordering of their
constituent particles, understanding that structure is a primary objective of
condensed matter research. Although perfect crystals are fully described by a
set of translation and basis vectors, real-world materials are never perfect,
as thermal vibrations and defects introduce significant deviation from ideal
order. Meanwhile, liquids and glasses present yet more complexity. A complete
understanding of structure thus remains a central, open problem. Here we
propose a unified mathematical framework, based on the topology of the Voronoi
cell of a particle, for classifying local structure in ordered and disordered
systems that is powerful and practical. We explain the underlying reason why
this topological description of local structure is better suited for structural
analysis than continuous descriptions. We demonstrate the connection of this
approach to the behavior of physical systems and explore how crystalline
structure is compromised at elevated temperatures. We also illustrate potential
applications to identifying defects in plastically deformed polycrystals at
high temperatures, automating analysis of complex structures, and
characterizing general disordered systems.
