Embedding parallelohedra into primitive cubic networks and structural automata description

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Embedding parallelohedra into primitive cubic networks and structural automata description. / Bouniaev, Mikhail M.; Krivovichev, Sergey V.

In: Acta Crystallographica Section A: Foundations and Advances, Vol. 76, 01.11.2020, p. 698-712.

Research output: Contribution to journal › Article › peer-review

Author

Bouniaev, Mikhail M. ; Krivovichev, Sergey V. / Embedding parallelohedra into primitive cubic networks and structural automata description. In: Acta Crystallographica Section A: Foundations and Advances. 2020 ; Vol. 76. pp. 698-712.

BibTeX

@article{ef1dbcb0fb2f47b49ca85cb3e859071d,

title = "Embedding parallelohedra into primitive cubic networks and structural automata description",

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. ",

keywords = "crystalline structures, deterministic finite automata, parallelohedra, primitive cubic nets, structural automata",

author = "Bouniaev, {Mikhail M.} and Krivovichev, {Sergey V.}",

note = "Funding Information: SVK thanks the Russian Science support (grant 19-17-00038). Publisher Copyright: {\textcopyright} 2020 International Union of Crystallography. All rights reserved. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.",

year = "2020",

month = nov,

day = "1",

doi = "10.1107/S2053273320011663",

language = "English",

volume = "76",

pages = "698--712",

journal = "Acta Crystallographica Section A: Foundations and Advances",

issn = "0108-7673",

publisher = "International Union of Crystallography",

}

RIS

TY - JOUR

T1 - Embedding parallelohedra into primitive cubic networks and structural automata description

AU - Bouniaev, Mikhail M.

AU - Krivovichev, Sergey V.

N1 - Funding Information: SVK thanks the Russian Science support (grant 19-17-00038). Publisher Copyright: © 2020 International Union of Crystallography. All rights reserved. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - 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.

AB - 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.

KW - crystalline structures

KW - deterministic finite automata

KW - parallelohedra

KW - primitive cubic nets

KW - structural automata

UR - http://www.scopus.com/inward/record.url?scp=85094936297&partnerID=8YFLogxK

U2 - 10.1107/S2053273320011663

DO - 10.1107/S2053273320011663

M3 - Article

C2 - 33125353

AN - SCOPUS:85094936297

VL - 76

SP - 698

EP - 712

JO - Acta Crystallographica Section A: Foundations and Advances

JF - Acta Crystallographica Section A: Foundations and Advances

SN - 0108-7673

ER -

ID: 75306647