Geometric interpretation of Bravais lattices

The study presents the results of the analysis of two-dimensional hexagonal Bravais lattices in regular packings of arbitrary congruent geometric objects on a plane. For computer modeling, three geo-metric objects of different shapes were used in the study. The relationships between four types of congruent representations in a regular packing of geometric objects on a plane are shown. The transformations of rotation of objects by 1800, transformations of mirror symmetry, as well as a joint trans-formation of mirror symmetry and rotation of objects by 1800 are considered. The use of cognitive visualization allowed us to identify two common properties of regular packings of arbitrary congruent geometric objects on a plane. The first property is related to the fact that any object of a dense regular packing of similarly oriented objects has a tangency with six surrounding objects, forming a honeycomb structure. The analysis of the pole hexagon led to the heuristic judgment that the basis of two-dimensional hexagonal Bravais lattices of regular packings similarly oriented are affine-regular hexagons. The possibility of forming regular packings in which the geometric objects of each row have different types of congruence is shown.