# On the origin of crystallinity: a lower bound for the regularity radius of Delone sets.

@article{Baburin2018OnTO, title={On the origin of crystallinity: a lower bound for the regularity radius of Delone sets.}, author={Igor A. Baburin and Mikhail Bouniaev and Nikolay P. Dolbilin and Nikolay Erokhovets and Alexey Garber and Sergey V. Krivovichev and Egon Schulte}, journal={Acta crystallographica. Section A, Foundations and advances}, year={2018}, volume={74 Pt 6}, pages={ 616-629 } }

The mathematical conditions for the origin of long-range order or crystallinity in ideal crystals are one of the very fundamental problems of modern crystallography. It is widely believed that the (global) regularity of crystals is a consequence of `local order', in particular the repetition of local fragments, but the exact mathematical theory of this phenomenon is poorly known. In particular, most mathematical models for quasicrystals, for example Penrose tiling, have repetitive local…

## 11 Citations

The Local Theory for Regular Systems in the Context of t-Bonded Sets

- MathematicsSymmetry
- 2018

The recent results of the local theory for a wider class of point sets compared with the Delone sets are reviewed and might provide new insight into the case for which the atomic structure of matter is a Delone set of a “microporous” character, i.e., a set that contains relatively large cavities free from points of the set.

Bounding the regularity radius for regular crystals.

- MathematicsActa crystallographica. Section A, Foundations and advances
- 2019

It is stated that the Delone set X is a crystal if and only if the number of different local patterns of X of radius stays bounded if tends to infinity.

Local Groups in Delone Sets

- MathematicsLecture Notes in Computational Science and Engineering
- 2021

In the paper, we prove that in an arbitrary Delone set $X$ in $3D$ space, the subset $X_6$ of all points from $X$ at which local groups have axes of the order not greater than 6 is also a Delone set.…

Algorithms for continuous metrics on periodic crystals

- Mathematics, Computer Science
- 2022

The proofs develop new tools such as a boundary-tolerant metric applicable to other data ambiguity problems and a polynomial-time algorithm distinguishing all periodic point sets up to isometry in any dimension.

Soft Packings, Nested Clusters, and Condensed Matter

- Mathematics
- 2020

An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer k ≥ 3, there exists a normal…

On the Regularity Radius of Delone Sets in ${\mathbb {R}}^3$

- MathematicsDiscret. Comput. Geom.
- 2021

It is shown that if all $10R$-clusters of a Delone set X with parameters $(r,R)$ are equivalent, then $X$ is a regular system.

Embedding parallelohedra into primitive cubic networks and structural automata description.

- Computer ScienceActa crystallographica. Section A, Foundations and advances
- 2020

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…

A Critical Contact in Molecular Crystals

- Chemistry, PhysicsMoscow University Chemistry Bulletin
- 2019

Abstract—The definitions of the critical contact and critical energy coordination number (CECN) of a molecular crystal are introduced. Some characteristics of the critical contact in the diagram of…

Coordination Numbers and Topology of Crystalline Hydrocarbons

- ChemistryMoscow University Chemistry Bulletin
- 2020

Different types of coordination numbers in molecular crystals are discussed. Monosystem crystal structures of hydrocarbons are used as an example to study the relation between parameter R of a…

A Fuzzy Set of Generating Contacts in a Molecular Agglomerate

- Computer ScienceMoscow University Chemistry Bulletin
- 2019

It is shown that, from the viewpoint of the theory of fuzzy sets, a set of generating (key) contacts in any molecular agglomerate can be considered as a fuzzy set. Sets of generating contacts are…

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