# Regular Two-Dimensional Packing of Congruent Objects: Cognitive Analysis of Honeycomb Constructions
###### Abstract
Figure 4b-d shows regular packing of the three introduced congruent relationship types. Two-row packing of congruent GOs is described as double-lattice packing that combines two single-lattice packings, translated one into the other [9; 12]. The concept of dense double-lattice packing is introduced [13]. Honeycomb constructions (Figure 4b-d) determine regular packing of congruent GOs and have three parallel packing directions with identical translation vectors in each direction. The last fact determines the necessity of introducing the concept of triple-lattice packing. The analysis of Figure 4 leads us to the following heuristic judgement: \"Regular packings of congruent GOs have a honeycomb structure\". Figure 3: Regular packing of GOs (type_0): (a) Directions of packing; \(\overline{a_{1}}\), \(\overline{a_{2}}\), \(\overline{a_{3}}\)—translation vectors. The main objective of this research is to investigate the properties of regular packing of congruent objects on the plane. The main objects of the research were honeycomb constructions of regular packing of congruent GOs. Before proceeding to the analysis of honeycomb constructions of the regular packing of congruent objects, we will consider the methods of their construction. For accurate construction of regular packing of congruent GOs (Figure 3), the apparatus of functions of tight arrangements (FTA) and algorithms of formation of the FTA hodograph are used [7]. The FTA hodograph for packing two objects on the plane is the pole path of the movable object (Figure 5) with its plane-parallel movement around the fixed object in conditions of constant contact with each other. According to the Science Citation Index Expanded (Sci-EXPANDED), the number of annual publications on 'honeycomb' is constantly growing, now at several hundreds. Honeycomb structures have found widespread applications in various fields, including architecture, transportation, mechanical engineering, and chemical engineering. The geometry of a honeycomb minimizes the amount of material used for minimal weight and minimal cost. The mathematical structure of honeycomb was first formulated long before B.C. and later was called the honeycomb conjecture, which states \"any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling\". The conjecture was finally proven by [PERSON] [14]. No studies of the honeycomb conjecture in regular packings of congruent objects were found. Figure 4: Regular packing of congruent GOs:(**a**) type_0; (**b**) type_1; (**c**) type_2; (**d**) type_3. Approximate generation of FTA hodographs is possible using a suitable graphic editor. Depending on the type of congruent relation, we will distinguish the hodograph of type_0 (Figure 5a), hodograph of type_1 (Figure 5b), hodograph of type_2 (Figure 5c), and hodograph of type_3 (Figure 5d). The centers of all hodographs coincide with the pole of a fixed object. Development of a honeycomb construction (type_0) involves sequentially performing the following actions:  
1. The points of intersection of an arbitrary straight line passing through the pole of the object (Figure 6a) with the hodograph (type_0) are fixed as two poles of future objects and centers of future hodographs (type_0);
2. The points of intersection of the two new hodographs (type_0) with the already existing hodograph (type_0) are fixed as the four poles of future objects (Figure 6b);
3. The obtained six poles make it possible to form a honeycomb construction (Figure 6c). Figure 5: Generation of FTA hodographs of congruent GOs: (**a**) type_0; (**b**) type_1; (**c**) type_2; (**d**) type_3. 1—fixed object; 2—mobile object; 3—hodograph. Let us consider the development of honeycomb constructions of other types of congruent relations using the example of type_1:  
1. The points of intersection of an arbitrary straight line passing through the pole of the object (Figure 7a) with the hodograph (type_0) are fixed as two poles of future objects and centers of future hodographs (type_1);
2.