# Non-Crystallographic Symmetry in Packing Spaces
## 3 Examples of Permutation Groups on Packing Spaces
### Quaternion Group \(D_{4}\)  
If the basis of the complex space are selected vectors:  
\(0=(1,\,\mathrm{i}),\,1=(-1,-\mathrm{i}),\,2=(\,\mathrm{i},\,1),\,3=(-\mathrm{i}, \,-1),\,4=(1,-\mathrm{i}),\,5=(-1,\,\mathrm{i}),\,6=(-\mathrm{i},\,1),\) and \(7=(\,\mathrm{i},\,-1)\)  
as it is well known in mathematics, they will match the operational set, forming a two-dimensional representation:  
\(e\,=\,(0)(1)(2)(3)(4)(5)(6)(7)\,=\, \begin{pmatrix}1&0\\ 0&1\end{pmatrix} \); \(-e=(0\,1)(2\,3)(4\,5)(6\,7)=\, \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} \)  
\(i\,=\,(0\,2\,1\,3)(4\,7\,5\,6)=\, \begin{pmatrix}i&0\\ 0&-i\end{pmatrix} \); \(-i=(0\,3\,1\,2)(4\,6\,5\,7)=\, \begin{pmatrix}-i&0\\ 0&i\end{pmatrix} \)  
\(j\,=\,(0\,4\,1\,5)(2\,6\,3\,7)=\, \begin{pmatrix}0&-i\\ -i&0\end{pmatrix} \); \(-j=(0\,5\,1\,4)(2\,7\,3\,6)=\, \begin{pmatrix}0&i\\ i&0\end{pmatrix} \)  
\(k\,=\,(0\,6\,1\,7)(2\,5\,3\,4)=\, \begin{pmatrix}0&-1\\ 1&0\end{pmatrix} \); \(-k=(0\,7\,1\,6)(2\,4\,3\,5)=\, \begin{pmatrix}0&1\\ -1&0\end{pmatrix} \)  
Expressing the elements of representation set in terms of substitution, we have eight operations:  
\(\mathrm{g}(0)=1=e=(0\,1\,2\,3\,4\,5\,6\,7)=(0)(1)(2)(3)(4)(5)(6)(7)\)  
\(\mathrm{g}(1)=I=(\,2\,3\,1\,0\,7\,6\,4\,5)=(0213)(4756)\)  
\(\mathrm{g}(2)=j=(\,4\,5\,6\,7\,1\,0\,3\,2)=(0415)(2637)\)  
\(\mathrm{g}(3)=k=(6\,7\,5\,4\,2 The symbols in the rows and columns of Table 7 for each transformation of the group are replaced by the number corresponding to this transformation of permutation operation. We will consider each row of Table 7 as a new operation of substitution. It is easy to verify that the resulting new operating set will generate a group isomorphic to the quaternion group. Table 8, derived from elements of rows 5, will have the properties of the original Table 7.  
We apply the same approach to the resulting Cayley Table 8 again. However, despite the changed permutation operation workflow, the table view remains the same (Table 9). Therefore, all operating sets obtained from the original group of quaternions are isomorphic, and any of them can be used to represent corresponding structures in the packing space. Obviously, when we visualize the quaternion group, the order of the appropriate packing space should be a chosen multiple of the order of the permutations subgroup, that is, \(N=8\)k. In this case, such as in Example 1.1, every cycle in an operation of the permutation subgroup selects in PS identical \begin{table}
7 & 7 & 6 & 1 & 0 & 3 & 2 & 5 & 4 \\ \end{tabular}
\end{table} 
Table 7: Cayley table for the quaternion group. \begin{table}
5 & 5 & 0 & 7 & 2 & 1  
(13) (46) (57) (Figure 12a) and six two-colored transformation operations, dividing space into three pairs of equivalent structures. Thus, there are only five independent structures that correspond to the five classes of quaternions:  
C\({}_{0}\) (Figure 10a) = {g(0)}, C\({}_{1}\) (Figure 12a) = {g(2)}, C\({}_{2}\) (Figure 11a, Figure 13a) = {g(1),g(3)}  
C\({}_{3}\) (Figure 10b, Figure12b) = {g(4),g(6)}, C\({}_{4}\) (Figure 11b, Figure 13b) = {g(5),g(7)}  
and consequently, the group has five irreducible representations. From a geometrical point of view, the quaternion group splits PS P81\({}_{1}\) into heterolayers (see, for example, Figures 7-11) and P42\({}_{1}\) into regions in the form of polyomino. There is a \"degeneration\" of permutation transformations in PS P42\({}_{1}\): (1 6 3 4 5 2 7 0), (3 2 7 6 1 0 5), (5 4 1 0 7 6 3 2 4), (7 0 5 2 3 4 1 6), which gives the same result. Thus, the structure of the packing space affects the number of independent partitions. The transition between structures that are determined by one fixed operation of the permutation group, belonging to different packing spaces of the same order \(N\), is discrete due to changes in the angular parameters of the sublattices. More information will be shown below in Example 1.6. Therefore, we can assume that the lattice is deformed in such a transition, and matrices of packing space function as a discrete deformation transformation (DDT).