## 2 Results and Discussion
### Information-Theoretic Measures of Symmetry of the Patterns Built of 1D Objects
Again, all of the measures of symmetry introduced coincide for the patterns presented in Figure 2a,b, and they are insensitive to the long-range order (translational symmetry) of segments, shown in Figure 2b. Consider the patterns built of the \(p\) arcs of a circle (or, perhaps, the symmetric segments of the parabola \(y=\alpha x^{2},\ \alpha=const\), shown in Figure 2c,d. The symmetry group of the shapes depicted in Figure 2c,d is built of two elements, namely the unity element, which is the onefold rotational symmetry (denoted \(G_{1}\)), and the mirror axis, denoted \(G_{2}\). For this kind of pattern, we obtain \((G_{1})=P(G_{2})=\frac{p}{2p}=\frac{1}{2}\); thus, \(H_{sym}=-2\frac{1}{2}ln\frac{1}{2}=0.69\). Consequently, in this case, \(\Omega_{sym}=0\) (due to \(P_{2}(n=2)=1\) and \(lnP_{2}=0\)); the long-range order, shown in Figure 2d plays no role in the calculation of both measures of symmetry. Now, consider the patterns built of the \(p\) symmetric segments of a cubic parabola \(y=\alpha x^{3},\ \alpha=const\)) shown in Figure 2e,f. The considerations akin to the aforementioned ones immediately yield \(H_{sym}=0.69\), \(\Omega_{sym}=0\). Thus, we come to the conclusion that the patterns depicted in Figure 2c-f are _equivalent_ from the point of view of the Shannon measures of symmetry introduced, namely, \(H_{sym}\) and \(\Omega_{sym}\). Moreover, the VSMS of the patterns depicted in Figures 1 and 2 is equal; however, the IMS of the patterns shown in Figures 1 and 2 are different. It should be emphasized that both of the Shannon and the Voronoi-Shannon information-theoretic measures of symmetry of the discussed patterns are _the intensive properties_ of the patterns and that they are independent of the area of the pattern or the density and size of the shapes, and in this sense, they are different from the  
Figure 2: Patterns built of shapes possessing two elements of symmetry: (**a**) random straight line segments, (**b**) ordered straight line segments, (**c**) random arcs of a circle, (**d**) ordered arcs of a circle, (**e**) random symmetrical segments of a cubic parabola, and (**f**) ordered symmetrical segments of a cubic parabola. The center of symmetry is shown by the blue circle. true thermodynamic entropy, which is an extensive parameter of the system. It should be emphasized that the long-range order does not influence either of the information-theoretic measures of symmetry. Consider now the mixed patterns, comprising 1D objects (lines), depicted in Figure 3.  
Let us start from the pattern comprising segments of a cubic parabola and arcs of a circle, shown in Figure 3a. Both shapes are characterized by the symmetry group containing two symmetry elements: the unity element, which is the onefold rotational symmetry and the twofold rotational symmetry in the case of segments of a cubic parabola, and the onefold rotational symmetry and the mirror axis in the case of arcs of a circle. Thus, \(H_{sym}=1.03\). Both shapes have two elements of symmetry; hence, \(\Omega_{sym}=0\). Now, consider the pattern built from \(p\) irregular, non-symmetrical curves and \(p\) arcs of a circle, presented in Figure 3b. We recognize two elements of symmetry in this pattern, namely the onefold rotational symmetry inherent for all of the shapes and denoted \(G_{1}\), and the mirror axis labeled \(G_{2}\) inherent for the arcs only. Thus, the entire number of the symmetry elements in the pattern is \(N_{G}=3p\) (see Equation (3)); consequently, we easily calculate \(P(G_{1})=\frac{2p}{3p}=\frac{2}{3};P(G_{2})=\frac{p}{3}=\frac{1}{3}\); and finally, we obtain \(H_{sym}=-\big{(}\frac{2}{3}ln\frac{2}{3}+\frac{1}{3}ln\frac{1}{3}\big{)}=0.54\). Let us now calculate \(\Omega_{sym}\) for the mixed pattern, shown in Figure 3b. For this pattern, we calculate (\(N_{s}=2p\)): \(P_{1}(n=1)=\frac{p}{2p}=\frac{1}{2}\); \(P_{2}(n=2)=\frac{p}{2p}=\frac{1}{2}\); thus, \(\Omega_{sym}=0.69\). Again, the Voronoi entropy could not be introduced for the patterns presented in Figure 3a,b.