We survey the results of the local theory of regular systems that are related to estimating the regularity radius. We also discuss recent results on local groups in arbitrary Delone sets in the plane and in the three-dimensional space, including new theorems and conjectures generalizing the classical statement on the absence of fivefold rotations in lattices.

Delone sets are discrete point sets $X$ in $\mathbb{R}^d$ characterized by parameters $(r,R)$, where (usually) $2r$ is the smallest inter-point distance of $X$, and $R$ is the radius of a largest ``empty ball" that can be inserted into the interstices of $X$. The regularity radius $\hat{\rho}_d$ is defined as the smallest positive number $\rho$ such that each Delone set with congruent clusters of radius $\rho$ is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our ``Weak Conjecture" states that $\hat{\rho}_{d}={{\rm O}(d^2\log d)}R$ as $d\rightarrow\infty$, independent of~$r$. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius $2r$ and those with full-dimensional sets of $d$-reachable points. We also offer support for the plausibility of a ``Strong Conjecture", stating that $\hat{\rho}_{d}={{\rm O}(d\log d)}R$ as $d\rightarrow\infty$, independent of $r$.

Дан обзор результатов локальной теории правильных систем, относящихся к оценке радиуса регулярности. Обсуждаются недавние результаты о локальных группах в произвольных множествах Делоне на плоскости и в трехмерном пространстве, в том числе новые теоремы и выдвинутые гипотезы, обобщающие классическое положение об отсутствии поворотов пятого порядка в решетках.

For an arbitrary convex polyhedral prism, a family of isometric embeddings of it is constructed that satisfy conditions similar to those that Pogorelov imposed on an isometry of a circular cylinder and called the ‘conditions of support on circles at the edges’. Bibliography: 4 titles.

Для произвольной выпуклой многогранной призмы строится семейство ее изометрических вложений, удовлетворяющих условиям, аналогичным тем, которые А. В. Погорелов накладывал на изометрию кругового цилиндра и называл условиями "опирания на окружности по краям". Библиография: 4 названия.

A Delone (Delaunay) set is a uniformly discrete and relatively dense set of points located in space, and is a natural mathematical model of the set of atomic positions of any solid, whether it is crystalline, quasi-crystalline or amorphous. A Delone set has two positive parameters: r is the packing radius and R is the covering radius. The value 2r can be interpreted as the minimum distance between points of the set. The covering radius R is the radius of the biggest `empty' ball, i.e. the radius of the biggest ball containing no points from the set. The central concept of this article is the so-called local group at a point of X which is defined as a group of a cluster (neighborhoods) around the point of radius 2R. This value 2R is notable because it is the minimum size of cluster that provides the finiteness of the cluster group at each point for any set X from the family of all Delone sets with the covering radius R. A few conjectures and theorems on the local groups for arbitrary Delone sets in the Euclidean plane and 3D space are discussed. Some of these statements significantly refine and generalize the famous Bravais theorem on the impossibility of fifth-order axes in 2D and 3D lattices. A complete proof is given that, in a Delone set X in the 3D Euclidean space, the subset \widetilde{X} of all points at which the local groups contain rotations of order at most 6 is also a Delone set with a certain covering radius \widetilde{R}, where \widetilde{R} < 3R and R is the covering radius for X.

It is proved that, in any Delone set on a Euclidean plane, a subset of points with a crystallographic local group, i.e., with local rotations of order n = 1, 2, 3, 4, or 6 , is also a Delone set. This result has a number of important implications for regular systems and crystalline structures. By the local group at a point of a set X, we mean the group of the cluster of radius 2R centered at this point, where R is the radius of a covering of the plane by equal disks with centers in X.

Излагаются новые результаты в локальной теории множеств Делоне, правильных систем и изогональных разбиений. Доказывается локальный критерий для изогональных разбиений евклидова пространства. Этот критерий применяется при исследовании $2R$-изометрических множеств Делоне, где $R$ - радиус покрытия для этих множеств. Установлено точное значение $\widehat {\rho }_2=4R$ радиуса регулярности для правильных систем на плоскости. Доказано, что в произвольном множестве Делоне на плоскости в любой ячейке разбиения Делоне имеется вершина, в которой локальная группа кристаллографическая. Следовательно, подмножество точек с локальной кристаллографической группой в множестве Делоне на плоскости само является множеством Делоне с радиусом покрытия, не превышающим $2R$.

We present new results in the local theory of Delone sets, regular systems, and isogonal tilings. In particular, we prove a local criterion for isogonal tilings of the Euclidean space. This criterion is then applied to the study of $$2R$$ -isometric Delone sets, where $$R$$ is the covering radius for these sets. For regular systems in the plane we establish the exact value $$\widehat{\rho}_2=4R$$ of the regularity radius. We prove that in any cell of the Delone tiling in an arbitrary Delone set in the plane, there is a vertex at which the local group is crystallographic. Hence, the subset of points with local crystallographic groups in a Delone set in the plane is itself a Delone set with covering radius at most $$2R$$ .

We complete the proof of the upper bound $${\hat{\rho }}_3\le 10R$$ for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all $$10R$$ -clusters of a Delone set X in $${\mathbb {R}}^3$$ with parameters (r, R) are equivalent, then X is regular (has a transitive symmetry group).

We prove that in an arbitrary Delone set X in the three-dimensional space, the subset X6 of all points from X at which the local group has no rotation axis of order larger than 6 is also a Delone set. Here, under the local group at pointx ∈ X we mean the symmetry group Sx(2R) of the cluster Cx(2R) of x with radius 2R, where R is the radius of the largest ball free of points of X (according to Delone’s empty sphere theory).

All possible combinatorial embeddings into primitive cubic networks of arbitrary tilings of 3D space by pairwise congruent and parallel regular hexagonal prisms are discussed and classified.

A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10 R -theorem. This theorem asserts that identity of neighborhoods within a radius 10 R of all points of a Delone set (in other words, an ( r, R )-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10 R -theorem. In the proof, we use some recent results of the author, which simplify the proof.

Данная статья посвящена 150-летию со дня рождения выдающегося российского математика Георгия Феодосьевича Вороного.