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Embedding parallelohedra into primitive cubic networks and structural automata description.

Mikhail M Bouniaev1, Sergey V Krivovichev2

  • 1School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, One University Boulevard, Brownsville, TX 78520, USA.

Acta Crystallographica. Section A, Foundations and Advances
|October 30, 2020
PubMed
Summary

This study develops an algorithmic model for crystal growth by embedding 3D parallelohedra into a primitive cubic network (pcu net). It reveals limitations for the rhombic dodecahedron, necessitating a 4D embedding, and introduces automata for modeling crystal structures.

Keywords:
crystalline structuresdeterministic finite automataparallelohedraprimitive cubic netsstructural automata

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Area of Science:

  • Crystallography
  • Computational Geometry
  • Materials Science

Background:

  • Crystallization processes are fundamental to materials science.
  • Developing algorithmic models for crystal growth is an ongoing challenge.
  • Understanding crystal complexity requires robust mathematical frameworks.

Purpose of the Study:

  • To develop an algorithmic model for crystallization.
  • To measure crystal complexity using embeddings into a primitive cubic network (pcu net).
  • To investigate the embeddability of 3D parallelohedra into the 3D pcu net.

Main Methods:

  • Constructing embeddings of 3D parallelohedra into the 3D primitive cubic network (pcu net).
  • Proving the existence or non-existence of embeddings for various parallelohedra.
  • Developing deterministic finite automata to model crystalline structures.

Main Results:

  • Any parallelohedron P, except the rhombic dodecahedron, can be embedded into the 3D pcu net.
  • The rhombic dodecahedron cannot be embedded into the 3D pcu net but can be embedded into a 4D pcu net.
  • The number of ways to embed a parallelohedron is determined.
  • Deterministic finite automata were developed for modeling crystal growth.

Conclusions:

  • The study provides a method for algorithmic modeling of crystal growth.
  • The complexity of crystals can be analyzed through parallelohedron embeddings.
  • The findings offer insights into the topological constraints of crystal structures.