## 2 Results and Discussion
### Information-Theoretic Measures of Symmetry of the Patterns Built of 1D Objects  
First, consider the IMS and the VSMS of the patterns comprising only 1D objects (lines) lying on a plane. It should be noted that the symmetry group of an object depends not only on the object itself but also on the space in which we view it. For example, the symmetry group of a line segment in R1 is of order 2, the symmetry group of a line segment considered as a set of points in R2 is of order 4, and the symmetry group of a line segments viewed as a set of points in R3 is of infinite order. Hereinafter, we assume that the considered objects (lines) are treated as sets of points in R2 space. Let us start from the pattern built of \(p\) irregular lines, such as depicted in Figure 1. The analysis should necessarily start from the establishment of the symmetry group of the pattern. In this case, we recognize that the single element of symmetry for the shapes constituting the pattern is the onefold rotational symmetry, reduced to the rotation of the lines by the angle \(\varphi_{1}=k\frac{2\pi}{1}=2\pi\), denoted \(G_{1}\), which is an identity element of the symmetry group. Let us first calculate \(H_{sym}\) for the pattern, shown in Figure 1a. In this case, \(N_{G}=p\); thus, \(P(G_{1})=1\), and consequently, we calculate \(H_{sym}(G_{1})=-P(G_{1})lnP(G_{1})=-1\times ln1=0\). Now, we establish \(\Omega_{sym}(G)=-\sum_{n=1}^{l}P_{n}(n(G))lnP_{n}(n(G))\). It is easily seen that, for the irregular shapes possessing a single element of symmetry, \(P_{n}(n=1)=1\); hence, \(\Omega_{sym}(G_{1})=0\). The same conclusion, i.e., \(H_{sym}(G_{1})=\Omega_{sym}(G_{1})=0\), holds for the pattern built from non-identical, irregular, non-symmetrical lines. It should be emphasized that the same conclusion, i.e., \(H_{sym}(G_{1})=\Omega_{sym}(G_{1})=0\), is true for the regular pattern, comprising irregular, non-symmetrical lines, such as the pattern shown in Figure 1b, demonstrating a pattern characterized by the translational symmetry. Thus, we concluded that both of the introduced measures of symmetry are insensitive to the long-range order. How should the obtained result be interpreted? \(H_{sym}(G_{1})=0\) means that an averaged uncertainty to reveal symmetry operation \(G_{1}\) within the pattern is zero; indeed, all of the 1D shapes depicted in Figure 1 possess this element of symmetry; in turn, \(\Omega_{sym}(G_{1})=0\) means an average uncertainty for finding the shape demonstrating the single element of symmetry is zero within the pattern; indeed, every shape in the pattern has only one element of symmetry, namely \(G_{1}\) (see Ref. [28]). It should be emphasized that neither Voronoi entropy nor continuous measure of symmetry could be defined and calculated for the patterns presented in Figure 1a,b. Now, consider the patterns composed of straight line segments, shown in Figure 2a,b, and seen as sets of points in R2 space. The symmetry group of the straight line segments is built of four elements, namely the unity element, which is the onefold rotational symmetry, which is the rotation by the angle \(\varphi_{1}=k\frac{2\pi}{1}=2\pi\) (denoted \(G_{1}\)); the twofold rotational symmetry by the angle \(\varphi_{1}=k\frac{2\pi}{2}=\pi\), denoted \(G_{2}\); and the mirror axes, denoted \(G_{3}\) and \(G_{4}\). Let us calculate \(H_{sym}\). For the pattern, built of the \(p\) segments, we obtain \(P(G_{1})=P(G_{2})=P(G_{3})=P(G_{4})=\frac{1}{4p}=\frac{1}{4}\); thus, \(H_{sym}=-4\frac{1}{4}ln\frac{1}{4}=1.39\). Figure 1: Pattern built of the 1D irregular (non-symmetrical) lines. (a) Random pattern; (b) regular pattern demonstrating the translational symmetry. Figure 2: _Cont._  
Now, let us establish \(\Omega_{sym}(G)=-\sum_{n=1}^{l}P_{n}(n(G)))lnP_{n}(n(G))\). In Figure 2a,b, all of the shapes in the patterns have four aforementioned elements of symmetry. Thus, \(P_{4}(n=4)=1\), and hence, \(\Omega_{sym}=0\).