## 2 Results and Discussion  
### Alternative Definitions of the Informational Measure of Symmetry  
Consider a 2D pattern built of 1D and/or 2D shapes or lines, demonstrating a number of symmetry elements (rotational symmetry, centers of symmetry, axes of symmetry, etc.), denoted as \(G_{i}\), \(i=1,2\ldots k\), where \(k\) is a number of non-identical symmetry operations. Pattern is understood as a 2D arrangement made from repeated lines or shapes on a surface. Element \(G_{i}\) forms the symmetry group of the shape \(G\) (which should be clearly distinguished from the symmetry group of the entire pattern). Thus, the informational measure of symmetry of the pattern (abbreviated to IMS) is defined in a Shannon-like form as follows:  
\[H_{sym}(G)=-\sum_{i=1}^{k}P_{i}(G_{i})lnP_{i}(G_{i}), \tag{2}\]  
where \(P_{i}(G_{i})\) is the probability of appearance of the symmetry operation \(G_{i}\) within the shapes (lines) constituting the pattern, defined as follows:  
\[P_{i}(G_{i})=\frac{m(G_{i})}{N_{G}}\leq 1, \tag{3}\]  
where \(N_{G}=\sum_{i=1}^{k}m(G_{i})\) is the total number of symmetry elements (operations) appearing in the 1D or 2D shapes, recognized in a given pattern, and \(m(G_{i})\) is the number of the same symmetry elements (operations) \(G_{i}\), calculated for a given pattern. The normalization condition given by Equation (4) takes place:  
\[\sum_{i=1}^{k}P_{i}(G_{i})=1. \tag{4}\]  
Let us explain in detail the meaning of the introduced Shannon measure of symmetry \  
Multiplying \(-lnP_{i}(G_{i})\) by \(P_{i}(G_{i})\), and summing over all \(i\), we obtain an average uncertainty about the presence of symmetry elements \(G_{i}\) within the given pattern. When \(P_{i}(G_{i})=0\), we are certain that element \(G_{i}\) is not met within the pattern (numerous examples of calculation of \(H_{sym}(G)\) are supplied below).  
Alternatively, the symmetry of the pattern may be quantified with the parameter \(\Omega_{sym}(G)\), defined according to Equation (5):  
\[\Omega_{sym}(G)=-\sum_{n=1}^{l}P_{n}(n(G))lnP_{n}(n(G)), \tag{5}\]  
where \(n\) denotes the _total_ number of elements of symmetry recognized in a shape; \(n\) changes from unity to the maximal number of elements of symmetry inherent for the shapes appearing in the pattern and denoted \"\(l\)\". \(P_{n}(n(G))\) is the probability of finding the shape possessing \(n\) elements of symmetry in total belonging to group \(G\) within the pattern, defined as follows:  
\[P_{n}(n(G))=\frac{r(n(G))}{N_{S}}, \tag{6}\]  
where \(r(n(G))\) is the number of shapes possessing \(n\) elements of symmetry in total and \(N_{S}\) is the total number of shapes in the given pattern. The normalization condition given by Equation (7) takes place:  
\[\sum_{n=1}^{l}P_{n}(n(G))=1 \tag{7}\]  
The Shannon-shaped measure \(\Omega_{sym}(G)\) is interpreted, in turn, as an average across the pattern uncertainty of finding the shape possessing in total \(n\) elements of symmetry within a given pattern [27; 28]. The definition of the Shannon measure of symmetry \(\Omega_{sym}(G)\) provided by Equations (5)-(7) resembles the definition of the Voronoi entropy, provided by Equation (1) with the following difference: instead of the number of polygon edges appearing in the definition of the Voronoi entropy, in our definition, we exploit the number of elements of symmetry \(n\) in the shape; thus, we label this measure as the Voronoi-Shannon measure of symmetry (VSMS). The definition of VSMS provided by Equations (5)-(7) latently implies that a more symmetrical shape is characterized by the larger number of symmetry elements. We illustrate the difference between the information-theoretic measure of symmetry (IMS) and the Voronoi-Shannon measure of symmetry with several examples. The paper is organized as follows: the measures of symmetry introduced by Equations (2)-(7) are calculated for the patterns built of (i) 1D shapes (curves); (ii) 2D shapes, namely triangles and ellipses; and (iii) mixed patterns comprised of 1D and 2D shapes.