# Non-Crystallographic Symmetry in Packing Spaces
## 2 Basic Concepts and Definitions
The number of cycles determines the number of independent nodes in the cell of the sublattice, which is the index of a sublattice. From the geometry of the nodal lines in the space \(4\times 4\) (Figure 5) and Table 2, the operations of g(2), g(8) and g(10) are distinguished by one cell with an index of 8. Simple division of these cells into two independent parts leaves only two of them (g(10) and g(8)), each of which will be characterized by dividing the index, equal to 4. The remaining operations are divided into each subgroup to the equivalent of a pair, the number of which is equal to 5. Thus, the number of independent nodal areas is 7 for lattices with the index \(N=4\), which coincides with the sum of the divisors of the number 4, as shown in the simple brute force of previously published works [13]. Figure 4: A series of parallel nodal lines. Packing space (PS) or partition space is a lattice in which each node is assigned weight (symbol or color) so that any set of lattice nodes with the same weight makes the same (up to shift) sublattice of translations of the original lattice. The columns of the vectors' coordinates (based on the original lattice) of one of the sublattice bases of this lattice form an integral matrix:  
\[Y=\left[ \begin{array}{ccc}x_{1}&x_{2}&x_{3}\\ 0&y_{2}&y_{3}\\ 0&0&z_{3}\end{array} \right]\]  
where \(0\leq x_{2}<x_{1}\), \(0\leq x_{3}<x_{1}\), \(0\leq y_{3}<y_{2}\), and \(z_{3}>0\). Matrix \(Y\) is a matrix of three-dimensional packing space. The order of packing space coincides with the index of the sublattice and is the multiplication of the diagonal elements of the packing matrix \(N=x_{1}y_{2}z_{3}\). For a two-dimensional lattice in the matrix, only numbers \(x_{1}\), \(x_{2}\), and \(y_{2}\) will remain. The order of the packing space in this case is \(N=x_{1}y_{2}\). For each node of the packing space, we construct a partition of the Dirichlet domains by dividing the interstitial distances by the median plane, and we obtain a partition of space into minimum areas that do not contain nodes, except one. These are cubes in three dimensions, and squares in the two dimensions (Figure 6). In solid state physics, such lattice domains are called Wigner-Seitz cells. In [25], the question of the structural organization of the complex compound with carbamide on the basis of the division of space into Dirichlet-polihedras was considered. \begin{table}
\begin{tabular}{l l Let the size of the square coincide with the size of cells that make up the \(N\)-minoes of arbitrary shape. Then, we can address the problem of investing polyomino into the fundamental domain of two-dimensional packing space. If this can be conducted, the selected polyomino will tile space. For example, we choose a polyomino of 5 cells in the form of a cross and put its middle cell with any of the packing spaces of P51\({}_{2}\) (Figure 7). We assume the following statement (criterion of partition): if all the cells with the given orientation of the polyomino coincide with cells that have different weights in the packing space, the chosen polyomino breaks space. This conclusion is general and answers the question: Which polyomino (policube) or its assembly may become a fundamental domain of space partition? Figure 6: Definition of the packing space. PS P 51\({}_{3}\). Figure 7: PS P51\({}_{2}\). One of the cells, corresponding to the range of PS, is selected by vectors. The criterion of the space partition is tested by the polyomino shape—the “cross”. The center of the figure coincides with the “weight 3” of PS. The formalization of the basic concepts of the \(n\)-dimensional packing space discrete model, policube and the partition criterion are given in [21]. For the space presented in Figure 6, we have \(x_{1}=5\), \(x_{2}=3\), and \(y_{2}=1\). The two-dimensional packing space, presented in Figure 3, is denoted as P5 \(1_{3}\).