Periodic graph (crystallography)

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 [1]), 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.

Modern atomic theory traces back to Johannes Kepler and his work on geometric packing problems.

[2] 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 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).

There is also a thread of interest in the very-large-scale integration (VLSI) community for using these crystal nets as circuit designs.

Uninodality corresponds with isogonality in geometry and vertex-transitivity in graph theory, and produces examples objective structures.

For example, in a crystal net, it is presumed that edges do not “collide” in the sense that when treating them as line segments, they do not intersect.

[9] (mathematicians [10] use the term ``harmonic realiaztions" instead of ``crystal nets in equilibrium positions" because the positions are characterized by the discrete Laplace equation; they also introduced the notion of standard realizations which are special harmonic realizations characterized by a certain minimal principle as well;see [11]).

[15] This effort, in what Omar Yaghi described as reticular chemistry is proceeding on several fronts, from the theoretical[16] to synthesizing highly porous crystals.

Synthesis of a specific zeolite de novo from a novel crystal net design remains one of the major goals of contemporary research.

[citation needed] Control is more tractable if the constituents are molecular building blocks, i.e., stable molecules that can be readily induced to assemble in accordance with geometric restrictions.