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Voronoi cell analysis: The shapes of particle systems

Many physical systems can be studied as collections of particles embedded in space, evolving through deterministic evolution equations. Natural questions arise concerning how to characterize these arrangements - are they ordered or disordered? If they are ordered, how are they ordered and what kinds of defects do they possess? Originally introduced to study problems in pure mathematics, Voronoi tessellations have become a powerful and versatile tool for analyzing countless problems in pure and applied physics. In this paper we explain the basics of Voronoi tessellations and the shapes they produce, and describe how they can be used to study many physical systems.