# Non-Crystallographic Symmetry in Packing Spaces
## 3 Examples of Permutation Groups on Packing Spaces
### An Example of a Real Structure with Non-Crystallographic Permutation Group Symmetry  
For the partition of three-dimensional structures, you can use the methodology for constructing molecular domains of Voronoi-Dirichlet by means of Fisher and Koch (see for example, reference [25]). However, as the analysis of three-dimensional partitions of a periodic lattice is hard to perceive, we choose the real structure of [(C\({}_{2}\)H\({}_{5}\))\({}^{4}\)N\({}^{+}\)]\({}_{2}\cdot\)C\({}_{4}\)O\({}_{4}\)\({}^{2-}\cdot\)2(C\({}_{2}\)H\({}_{5}\))\({}_{4}\)N + HCO\({}_{3}\)\({}^{-}\)4(NH\({}_{2}\))\({}_{2}\)CO-6H\({}_{2}\)O [27], based on a molecular layer (Figure 22). We use a qualitative approach: select the layer of the two types of dimer: 2(HCO\({}_{3}\))\({}^{-}\), 2 CO(NH\({}_{2}\))\({}_{2}\) and two pairs of water molecules per unit cell, bound by hydrogen bonds. The result is a three-colored partition \"map\" into non-overlapping substructures (Figure 22b).  
Figure 21: Coordination (magic) numbers of growth structure (Figure 19).  
Figure 22: The molecular layer **(a)** in the structure of the compound [(C\({}_{2}\)H\({}_{5}\))\({}^{4}\)N\({}^{+}\)]\({}_{2}\cdot\)C\({}_{4}\)O\({}_{4}\)\({}^{2-}\cdot\)2(C\({}_{2}\)H\({}_{5}\))\({}_{4}\)N + HCO\({}_{3}\)\({}^{-}\)4(NH\({}_{2}\))\({}_{2}\)CO-6H\({}_{2}\)O, “map” of its partition; **(b)** into disjoint regions and **(c)** the discrete layer structure model.  
Without changing the coordination environment of each region, a polyomino-like presentation of this structure's layer (Figure 22c) is obtained for the packing space P31 1\({}_{6}\). The sizes of cells are selected in such way that the order of the cell of packing space is a prime number \(N=31\). The number of independent types of lattices in this case is minimal, which is important when iterating the options in the prediction structures task. The image of the PS P31\({}_{6}\) structure, rotated to locate the yellow-red layer consisting of water molecules horizontally, is presented in Figure 23. The form of layered structure growth is presented in [16].  
In this example, the formed four-colored operation of permutation is written as follows:  
\(\mathrm{G(1)=(1\ 2\ 3\ 28\ 10\ 11\ 5\ 8\ 14\ 15\ 9\ 18\ 6\ 12\ 17\ 16\ 22\ 23\ 25\ 13\ 19\ 27\ 21\ 24\ 7\ 26\ 20\ 4\ 29\ 30\ 0)}\)  
\(=(0\ 1\ 2\ 3\ 28\ 29\ 30)\ (4\ 10\ 9\ 15\ 16\ 22\ 21\ 27\ )\ (5\ 11\ 18\ 25\ 26\ 20\ 19\ 13\ 12\ 6)\ (7\ 8\ 14\ 17\ 23\ 24).  
In this article, we do not give a Cayley matrix of the cyclic group (G (1))\({}^{m}\), which has an order greater than \(2^{N}\). In terms of group properties, it is interesting that when the order of \(m=p\), where \(p\) is a prime number \(>\)10, the structure is invariant. Rearrangements occur within the number of cycles, and we use this fact to calculate the statistical probability and statistical entropy:  
\(\mathrm{W_{T}=31!/6!7!8!10!=10^{13}}\); \(\mathrm{\sigma=lnW_{T}=13 ln10=29}\).  
The formula of Boltzmann's entropy of the structure is \(\mathrm{S=klnW_{T}=41.3\ \times 10^{-23}\ (J/K)}\).  
In other orders of m, the number of cycles (color set in the PS P31 1\({}_{6}\) partition) changes, as well as their structure and entropy.  
Keeping the original structure unchanged, that is, without changing the operation G(1), we consider gradual deformation of the structure when there is a transition from one packing space with \(N=31\)k to another, the number of which, as for the primes, is equal to \(N+1=32\) (Figure 24).  
The first and last PS P31 1\({}_{0}\) and P31 1\({}_{30}\) are not completely presented in Figure 24. It is easily observed from the slope of the layers in the partition that the deformation transformations of the lattice shift result in similar pairs of partitions associated with each other as \"left\" and \"right\" forms. These phase structures are obtained by a mirror image.  
Figure 23: A discrete model of the structure layer [27] in PS P31 1\({}_{6}\).