Automatically generated periodic graphs

Abstract

A crystal structure may be described by a periodic graph, so the automatic generation of periodic graphs satisfying given criteria should prove useful in understanding crystal structure and developing a theory for crystal design. We present an algorithm for constructing periodic graphs, based on a discrete variant of “turtle geometry,” which we formalize using isometries represented as affine transformations. This algorithm is given information about the putative orbits (kinds) of vertices and edges and produces a finite graph which is then collapsed into a unit cell of a periodic graph. We verify that this algorithm will generate any periodic graph.

Realizations of crystal nets. I. (Generalized) derived graphs

“…The motivation from Abelson & deSessa (1980) as applied by McColm et al (2011) and McColm (2012, 2015 is that a 'turtle', 3 whose standard position on the plane at the origin is oriented so that it sees a coordinate system on the plane with respect to the standard basis ðe 1 ; e 2 Þ, is moved as follows. Imagine that our voltage graph had voltages and weights consisting of isometries of R 2 , and that we wanted the turtle to move around on the resulting derived graph which we embed in the plane using voltages starting from (v 0 , !…”

“…Using these data, we generate one of the crystal nets of the associated generalized derived graph by starting at one vertex and generating vertices and edges, one at a time, until we reach an identifiable unit cell. This process is a formalization of the algorithms described by McColm et al (2011) and McColm (2012McColm ( , 2015.…”

“…We then adapt the construction of Potoc ˇnik & Toledo (2021) to the derivation of graphs realized in Euclidean space. But as that construction produces very large graphs of many components, and all we want is one component, we adapt the algorithm of McColm (2015) to obtain (realizations of) the desired crystal nets. As an example, we generate graphs for the diamond structure [denoted dia by O' Keeffe et al (2008)].…”