In geometric crystallography, there are 32 well-known point crystallographic groups, or A. V. Gadolin’s 32 crystal classes, which make up a complete list of symmetry groups of crystal shapes whose internal structure is subordinate to one of the 230 Fedorov groups existing in ℝ^3 . In 2022, the author constructed two point crystal structures located in ℝ^3 whose possible external shapes have the symmetry groups D_8h and D_12h , respectively. However, the internal structure of the crystal was not taken into account in the considerations of these groups. The central result of the author’s 2022 paper is as follows: if a possible external shape of an ideal crystal has an ordinary rotation of non-crystallographic order n , then either n=8 or n=12 and in this case the external shape is a right prism of finite height. But only after the paper was published did the author notice that the proof of this result was incomplete, although the result itself is correct. The present paper provides a complete proof of this result without relying on the 2022 text.

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.

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.

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.

Faceting with a `filling'. An ideal crystal structure consists of finitely many equal and parallel translational point lattices. In R-3 it extends unboundedly in all directions. We distinguish in it a finite part situated in a closed convex polyhedron every face of which contains nodes of a translational point lattice involved in the structure not belonging to the same straight line. Such a polyhedron is called a possible faceting of the ideal crystal structure. There are 32 well-known crystal classes, or 32 crystallographic point groups. Among them is the symmetry group of the possible faceting calculated taking account of the nodes of the ideal crystal structure belonging to it. A cyclic subgroup Cn of the symmetry group of any possible faceting has order n. 4 or n = 6. Faceting without `filling'. In this paper we construct two crystal structures in which there are crystal polyhedra whose symmetry groups, calculated without taking account of the nodes of the crystal structure belonging to it, have rotation axes of orders n = 8 and n = 12. In both cases, the crystal polyhedron is a right prism of finite height. Without taking account of the internal structure, a possible faceting of a crystal structure in three-dimensional Euclidean space cannot have an axes of rotation of order n satisfying 6 < n < infinity. The proposed constructions are accompanied by a detailed analysis of ideal crystal structures, as well as Delone sets S of type (r, R) in R2 and R3. In particular, we produce an expanded proof of one of the theorems stated in 2010 at an international conference dedicated to the 120th anniversary of B. N. Delone.

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 $$2R$$ .

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.

Rational and irrational rotations for the set of rational directions in the flat point lattice are considered. It is proved that in the case of rational rotations an order of noncrystallographic turn can be only 8 or 12. The set of rational 2 2 directions in the rectangular point lattice with metric quadratic form x +λ2 y and arbitrary its centering has irrational rotation if and only if the number λ2 is rational.

We construct a closed orientable polyhedral surface of arbitrary genus that is embedded in three-dimensional Euclidean space and admits a one-parameter bending under which all its handles bend. This surface admits no other bendings. We also construct a flexible closed nonorientable polyhedral surface of arbitrary genus such that all its handles and Möbius strips bend during its bending.

On 19 December 2013 the prominent Russian geometer and doctor of the physical and mathematical sciences Nikolai Petrovich Dolbilin observed his 70th birthday. Dolbilin was born in the city of Simferopol. His parents, Petr Andrianovich and Raisa Petrovna were civil engineers, and the family often moved. Nikolai graduated from high school in 1960 in the town of Angarsk, in Irkutsk Province. In 1965 he graduated from the Faculty of Mathematics and Mechanics at Moscow State University. In March 1966 he enrolled in the graduate school of the Steklov Mathematical Institute of the USSR Academy of Sciences. His advisor was the famous mathematician Boris Nikolaevich Delauney, head of the institute’s Department of Geometry and corresponding member of the Academy of Sciences. Dolbilin defended his Ph.D. thesis “On regular Dirichlet tessellations of a sphere” in June 1972, although he had worked as a researcher in the Department of Geometry from March 1969. Since then the Steklov Institute has become a part of Dolbilin’s life, and he has become an integral part of the institute, a relationship that has never been interrupted. In his early works Dolbilin studied geometric properties of lattices. In the very first paper “A new theory of lattice coverings of n-dimensional space by equal balls”, written in conjunction with Delauney, S. S. Ryshkov and M. I. Shtogrin, he obtained an interesting and important result in the geometric theory of positive quadratic forms. In the mid-1970s he started to work on local conditions on crystallographic structures. His first papers on this new subject were “A local test for the regularity of a system of points” (coauthored with Delauney, R.V. Galiulin, and Shtogrin) and “Combinatorial and metric theory of planigons” (coauthored with Delaunay and Shtogrin). Subsequently Dolbilin developed the results of these works into his well-known local theory of periodic structures. Among the most important results of this theory are necessary and sufficient local conditions on a polyhedral tessellation of space for the tessellation to be crystallographic (the corresponding theorem

In 1976 R. Connelly [1] constructed a flexible immersion of a polyhedral sphere in R, and then in 1977 a flexible embedding [2], described in detail in [3]. There ([3], Remark 5) the embedding was further generalized to closed orientable surfaces of arbitrary genus. However, in the flexing of this surface only a part of it homeomorphic to a disk is actually flexed, while all its handles remain passive. In this paper we construct a flexible embedding in R of a closed orientable polyhedral surface of arbitrary genus in which every handle is active. In 1995 Aleksandrov [4] constructed a flexible torus (see Fig. 1 and [5]). This torus (a frame) is not embedded and not immersed in R. It has two saddle vertices. We transform the torus into a flexible handle, and attach this handle to a flexible surface.