Languages Associated with Crystallographic Symmetry

One way to depict a crystallographic structure is by a periodic (di)graph, i.e., a graph whose group of automorphisms has a translational subgroup of finite index acting freely on the structure. We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages 𝒟𝒞ℒd, d = 0, 1, 2, …, within the intersection classes of deterministic context-free languages. We introduce a class of counter machines that accept these languages, where the machines with d counters recognize the class 𝒟𝒞ℒd An intersection of d languages in 𝒟𝒞ℒ1 defines 𝒟𝒞ℒd. We prove that there is a one-to-one correspondence between sets of walks starting and ending in the same unit of a d-dimensional periodic (di)graph and the class of languages in 𝒟𝒞ℒd. The proof uses the following result: given a digraph Δ and a group G, there is a unique digraph Γ such that G ≤ Aut Γ, G acts freely on the structure, and Γ/G ≅ Δ.

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“…We end with a few concluding remarks. A preliminary extended abstract of this paper appeared in Jonoska et al (2014).…”

Section: Introductionmentioning

confidence: 99%

Counter machines and crystallographic structures

2015

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Traversal Languages Capturing Isomorphism Classes of Sierpiński Gaskets

Jonoska

KrajăźEvski

McColm

2016

Unconventional Computation and Natural Computation

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Automatically generated periodic graphs

McColm

2015

Zeitschrift Für Kristallographie - Crystalline Materials

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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.

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Languages Associated with Crystallographic Symmetry

Cited by 3 publications

References 18 publications

Counter machines and crystallographic structures

Counter machines and crystallographic structures

Traversal Languages Capturing Isomorphism Classes of Sierpiński Gaskets

Automatically generated periodic graphs

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