Local groups in Delone sets in the Euclidean space.

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.

Tilings by hexagonal prisms and embeddings into primitive cubic networks.

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.

Version of zones and zigzag structure in icosahedral fullerenes and icosadeltahedra.

A circuit of faces in a polyhedron is called a zone if each face is attached to its two neighbors by opposite edges. (For odd-sized faces, each edge has a left and a right opposite partner.) Zones are called alternating if, when odd faces (if any) are encountered, left and right opposite edges are chosen alternately.

Pentaheptite modifications of the graphite sheet.

Pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossover from cubic-close-packed to hexagonal-close-packed packings in 3D. Symmetries realizable for these pentaheptite layers are three strip groups (periodic in one dimension), p1a1, p112, and p111, and five Fedorov groups (periodic in two dimensions), cmm, pgg, pg, p2, and p1.