Nikolai P. Dolbilin

Periodic Delone Tilings

Given a periodic point set in 3-dimensional Euclidean space, an algorithm is described for computing the corresponding Delone tiling (and its Delaney symbol). Examples of applications in tiling theory and crystallography are discussed.

Affine equivalent classes of parallelohedra

In the paper the affine equivalence relation in the set of parallelohedra is studied. One proves the uniqueness theorem for a wide class of d-dimensional parallelohedra. From here it follows that for every d (≥2) the space of affine equivalent classes of d-dimensional primitive parallelohedra has dimension d(d+1)/2−1.

The countability of a tiling family and the periodicity of a tiling

If a given finite protoset, together with a given finite matching rule, gives rise to at most countably many different tilings ofd-dimensional space, then at least one of them is periodic.

Geometrical Problems Related to Crystals, Fullerenes, and Nanoparticle Structure

This paper focuses on three groups of geometrical problems, closely related to material sciences in general and particularly to crystal/quasicrystal structures along with their formations and fullerenes. Some new results in mathematics are presented and discussed, for example, in section one, new estimates of minimum radius of local identity that guarantee that a Delone set is a point regular set. New results related to locally rigid packings are discussed in section two. One of the goals of the paper is to establish some internal (mathematically) and external (applications to material science) connections between research agendas of various studies in geometry and material sciences.

On the optimality of functionals over triangulations of Delaunay sets

In this short paper, we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation, for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations.

Regular and Multi-regular t-bonded Systems

The concept of t-bonded sets was briefly introduced by the second author in 1976 under the name of dconnected sets, though it has not received due consideration. This concept is a generalization of the concept of Delone (r,R)-systems. In light of the developments in the local theory for crystals that occurred since 1976 and demands in chemistry and crystallography, we believe the local theory for t-bonded sets deserves to be developed to describe materials whose atomic structures is multi-regular “microporous” point set. For a better description of such “microporous” structures it is worthwhile to take into consideration a parameter that represents atomic bonds within the matter. The overarching goal of this paper is to prove that analogous local conditions that guarantee that a Delone set is a regular (or multi-regular) system also guarantee that a t-bonded set is a regular (or multi-regular) t-bonded system.

Two finiteness theorems for periodic tilings of d-dimensional euclidean space

Consider the d-dimensional Euclidean space . Two main results are presented: First, for any N ∈ ℕ, the number of types of periodic equivariant tilings (T, Γ) that have precisely N orbits of (2,4, 6,...)-flags with respect to the symmetry group Γ, is finite. Second, for any N ∈ ℕ, the number of types of convex, periodic equivariant tilings (T, Γ) that have precisely N orbits of tiles with respect to the symmetry group T, is finite. The former result (and some generalizations) is proved combinatorially, using Delaney symbols, whereas the proof of the latter result is based on both geometric arguments and Delaney symbols.

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

The main goal of the local theory for crystals developed in the last quarter of the 20th Century by a geometry group of Delone (Delaunay) at the Steklov Mathematical Institute is to find and prove the correct statements rigorously explaining why the crystalline structure follows from the pair-wise identity of local arrangements around each atom. Originally, the local theory for regular and multiregular systems was developed with the assumption that all point sets under consideration are ( r , R ) -systems or, in other words, Delone sets of type ( r , R ) in d-dimensional Euclidean space. In this paper, we will review the recent results of the local theory for a wider class of point sets compared with the Delone sets. We call them t-bonded sets. This theory, in particular, 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.

Two Groups of Geometrical Problems Related to Study of Fullerenes & Crystals

The paper focuses on two groups of geometrical problems closely related to formations of crystal/quasi-crystal structures, and fullerenes. In section one we discuss a minimum radius of local identity that guarantee that a Delone set is a point regular set, and prove the local theorem for crystals. New results related to locally rigid packings are discussed in section two. One of the goals of the paper is to establish some internal (mathematically), and external (applications to material science), connections between research agendas of various studies in material sciences, and two classes of geometric problem related to Delone sets and packings.

Об оптимальности функционалов на триангуляциях множеств Делоне@@@Optimality of functionals on triangulations of Delaunay sets

В заметке мы рассматриваем плотности функционалов на равномерно ограниченных триангуляциях с заданным множеством вершин. Доказано, что если функционал достигает минимума на триангуляциях Делоне для конечных множеств, то минимум плотности достигается и на бесконечной триангуляции Делоне. Множеством Делоне называется множество X ⊂ E, для которого существуют положительные числа r и R такие, что для любого открытого d-шара B r и замкнутого шара BR радиусов r и R соответственно выполнены неравенства |B r ∩ X| 6 1 и |BR∩X| > 1, где |Y | означает мощность множества Y . В статье будем рассматривать множества Делоне в общем положении, в том смысле, что никакое подмножество такого множества, состоящее из d+ 2 точек, не лежит на одной (d− 1)-сфере. Будем рассматривать семейство симплициальных разбиений (триангуляций) T пространства E с фиксированным множеством вершин X, где X есть множество Делоне. Если для триангуляции T существует положительное число q = q(T ) такое, что для любого d-симплекса S ∈ T радиус R(S) его описанного шара не превосходит q : R(S) 6 q, то будем называть триангуляцию T равномерно ограниченной и обозначать семейство всех таких триангуляций через Θ(X). Множества Делоне были введены Б.Н. Делоне (1924 г.) под названием (r,R)-система. Он доказал (см., например, работу [1]), что для множества Делоне X существует и притом единственная триангуляция Делоне DT (X), множество вершин которой совпадает с X. Эта триангуляция является равномерно ограниченной с q = R, т. е. DT (X) ∈ Θ(X). Для любого множества Делоне можно построить триангуляцию, которая не является равномерно ограниченной. Вместе с тем остается открытой проблема (Л. Данцер и М. Бошерницан): верно ли, что для любого множества Делоне на плоскости существует не содержащий точек из этого множества треугольник сколь угодно большой площади? Рассмотрим функционал F , определенный на d-симплексах. Мы будем считать, что F непрерывно зависит от параметров симплекса (например, его ребер). Если X – конечное множество и T – триангуляция выпуклой оболочки conv(X) с множеством вершин в X, можно определить функционал на конечной триангуляции T : F (T ) = ∑