In crystallography, a periodic graph or crystal net is a three-dimensional periodic graph, i.e., a three-dimensional Euclidean graph whose vertices or nodes are points in three-dimensional Euclidean space, and whose edges (or bonds or spacers) are line segments connecting pairs of vertices, periodic in three linearly independent axial directions. There is usually an implicit assumption that the set of vertices are uniformly discrete, i.e., that there is a fixed minimum distance between any two vertices. The vertices may represent positions of atoms or complexes or clusters of atoms such as single-metal ions, molecular building blocks, or secondary building units, while each edge represents a chemical bond or a polymeric ligand. Although the notion of a periodic graph or crystal net is ultimately mathematical (actually a crystal net is nothing but a periodic realization of an abelian covering graph over a finite graph ), and is closely related to that of a Tessellation of space (or honeycomb) in the theory of polytopes and similar areas, much of the contemporary effort in the area is motivated by crystal engineering and prediction (design), including metal-organic frameworks (MOFs) and zeolites.
History A crystal net is an infinite molecular model of a crystal. Similar models existed in Antiquity, notably the atomic theory associated with Democritus, which was criticized by Aristotle because such a theory entails a vacuum, which Aristotle believed nature abhors. Modern atomic theory traces back to Johannes Kepler and his work on geometric packing problems. Until the twentieth century, graph-like models of crystals focused on the positions of the (atomic) components, and these pre-20th century models were the focus of two controversies in chemistry and materials science. The two controversies were (1) the controversy over Robert Boyle’s corpuscular theory of matter, which held that all material substances were composed of particles, and (2) the controversy over whether crystals were minerals or some kind of vegetative phenomenon. During the eighteenth century, Kepler, Nicolas Steno, René Just Haüy, and others gradually associated the packing of Boyle-type corpuscular units into arrays with the apparent emergence of polyhedral structures resembling crystals as a result. During the nineteenth century, there was considerably more work done on polyhedra and also of crystal structure, notably in the derivation of the Crystallographic groups based on the assumption that a crystal could be regarded as a regular array of unit cells. During the early twentieth century, the physics and chemistry community largely accepted Boyle's corpuscular theory of matter—by now called the atomic theory—and X-ray crystallography was used to determine the position of the atomic or molecular components within the unit cells (by the early twentieth century, unit cells were regarded as physically meaningful). However, despite the growing use of stick-and-ball molecular models, the use of graphical edges or line segments to represent chemical bonds in specific crystals have become popular more recently, and the publication of encouraged efforts to determine graphical structures of known crystals, to generate crystal nets of as yet unknown crystals, and to synthesize crystals of these novel crystal nets. The coincident expansion of interest in tilings and tessellations, especially those modeling quasicrystals, and the development of modern Nanotechnology, all facilitated by the dramatic increase in computational power, enabled the development of algorithms from computational geometry for the construction and analysis of crystal nets. Meanwhile, the ancient association between models of crystals and tessellations has expanded with Algebraic topology. There is also a thread of interest in the very-large-scale integration (VLSI) community for using these crystal nets as circuit designs.
Basic formulation A Euclidean graph in three-dimensional space is a pair (V, E), where V is a set of vertices (sometimes called points or nodes) and E is a set of edges (sometimes called bonds or spacers) where each edge joins two vertices. There is a tendency in the polyhedral and chemical literature to refer to geometric graphs as nets (contrast with polyhedral nets), and the nomenclature in the chemical literature differs from that of graph theory.
Symmetries and periodicity A symmetry of a Euclidean graph is an isometry of the underlying Euclidean space whose restriction to the graph is an automorphism; the symmetry group of the Euclidean graph is the group of its symmetries. A Euclidean graph in three-dimensional Euclidean space is periodic if there exist three linearly independent translations whose restrictions to the net are symmetries of the net. Often (and always, if one is dealing with a crystal net), the periodic net has finitely many orbits, and is thus uniformly discrete in that there exists a minimum distance between any two vertices. The result is a three-dimensional periodic graph as a geometric object. The resulting crystal net will induce a lattice of vectors so that given three vectors that generate the lattice, those three vectors will bound a unit cell, i.e. a parallelepiped which, placed anywhere in space, will enclose a fragment of the net that repeats in the directions of the three axes.
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