Embedding parallelohedra into primitive cubic networks and structural automata description

Authors

Mikhail M. Bouniaev, The University of Texas Rio Grande Valley
Sergey V. Krivovichev

Document Type

Article

Publication Date

8-2020

Abstract

The main goal of the paper is to contribute to the agenda of developing an algorithmic model for crystallization and measuring the complexity of crystals by constructing embeddings of 3D parallelohedra into a primitive cubic network (pcu net). It is proved that any parallelohedron P as well as tiling by P, except the rhombic dodecahedron, can be embedded into the 3D pcu net. It is proved that for the rhombic dodecahedron embedding into the 3D pcu net does not exist; however, embedding into the 4D pcu net exists. The question of how many ways the embedding of a parallelohedron can be constructed is answered. For each parallelohedron, the deterministic finite automaton is developed which models the growth of the crystalline structure with the same combinatorial type as the given parallelohedron.

Comments

Copyright 2020 International Union of Crystallography

Recommended Citation

Bouniaev, M. M., and S. V. Krivovichev. 2020. “Embedding Parallelo­hedra into Primitive Cubic Networks and Structural Automata Description.” Acta Crystallographica Section A: Foundations and Advances 76 (6): 698–712. https://doi.org/10.1107/S2053273320011663.

First Page

698

Last Page

712

Publication Title

Acta Crystallographica Section A: Foundations and Advances

DOI

10.1107/S2053273320011663