# Non-Crystallographic Symmetry in Packing Spaces
## 2 Basic Concepts and Definitions  
In crystallography, figures built on the basis of joining squares are called polyomino, on the basis of triangles are called \"polyiamonds\" and on the basis of hexagon are called polyhexes (see for example, [11]). The shape of the square and cube has been chosen to describe the space partition for simplicity of algorithms and software partition implementation and continuity of these figures in the transition from two- to three-dimensional, and then to \(n\)-dimensional space [21]. Three-dimensional analogs of the triangle (tetrahedron) and hexagon (hexagonal bipyramid or prism) do not give simple partitions. The figures compiled from the \(N\)-unicellular-squares or cubes in \(\mathrm{E}^{3}\) so that each square (or cube) is adjacent to at least one neighbor that has a common side, are called polyominoes [11] (3D-polyomino, or policube). Figure 6 shows an example of figures \"polyomino-cross\" with \(N=5\), which splits the space, that is, fills it without voids or overlays. It is evident that any \(N\)-cellular polyomino can be fully presented on the plane in the area of the square containing the \(N\times N\) cells, each of which coincides with the size of the cells polyomino (Figure 4, where \(N=6\)). Centers \(N^{2}\) elementary cell of the square form \(N^{2}\) nodes of the lattice translations of all the space. If the basic directions of space are used to select the horizontal and vertical lines, the nodes in this basis will have integer values (Figure 4). The beginning of the reference in the chosen basis accept node is located in the upper left hand corner of the square. In the general case of not-primitive sublattice, this lattice, which has \(N\) nodes, has a fundamental area of \(N\) of the corresponding cells polyomino. The number of \(N\) in this case is called the index of a sublattice or the order. Red (Figure 4) highlights a nodal direction, determined by the translation of \(0\to 8\) that will be permutated in the cycles:  
\( \begin{array}{l}\mbox{(0 8 16 18 26 34)(1 9 17 19 27 35)(2 10 12 20 28 30)(3 11 13 21 29 31)(4 6 14 22 24 32)}\\ \mbox{(5 7 15 23 25 33).}\end{array} \)  
This is a permutation of highlights in the lattice of the series of parallel nodal lines (Figure 4). Cell sublattice defines the coordinates of nodes: (6,0) and (2,1), but the fundamental area of the cell sublattice contains \(N=6\) nodes, or 6 cells polyomino. The coordinates of the selected nodes can be represented in the form of vector-matrix columns:  
\[ \begin{bmatrix}6&2\\ 0&1\end{bmatrix} \]  
and all the information about the selected cell sublattice is recorded as follows: PS-P 6 \(1_{2}\), where the designation PS is a brief record of the \"Pack Space\", and \(P\)-plane. To partition, the policubes will use the symbol PS S, where S is the space--\"Space\". As sublattice in two-dimensional space is determined by two independent directions coming from the origin, the result was the following theorem:  
**Theorem 1. ** A beam translational independent nodal line conducted through the origin and every node of the square \(N\times N\) is a subgroup \"\(N^{2}\)\" of the permutations of the order of \(N^{2}\). By forming a group of some of the directions in the lattice, it is simple, with the help of operations of multiplication, to find all of the subgroup \"\(N^{2}\)\" and its table Cayley. The result of this calculation for a simple case of \(N=4\) is presented in Table 1.  
The splitting of the elements of the group on the equivalence classes that are defined as independent lines, and irreducible representations of the group \"\(N^{2}\)\", which have the same color, is presented in the Table 2. The number of directions is 16. The group is divided into 5 subgroups of 4 elements each, including the identity transformation. In each subgroup, operations are divided into cycles. Each cycle determines the number of equivalent nodes that form the sublattice translations.