## 2 Results and Discussion
### Information-Theoretic Measures of Symmetry of Patterns Built of 2D Shapes  
Next, we address the patterns composed of 2D shapes. Let us start from the completely disordered pattern built of \(p\) irregular shapes depicted in Figure 4a. In this case, we recognize for all of the non-symmetrical shapes constituting the pattern the single element of symmetry, namely the rotation \(\varphi_{1}=k\frac{2\pi}{1}=2\pi\); thus, \(N_{G}=p;P(G_{1})=1\); and consequently, \(H_{sym}=-\sum_{i=1}^{1}P(G_{i})lnP(G_{i})=-\sum_{i=1}^{1}ln(1)=0\). It is easily seen that \(\Omega_{sym}(G)=0\) takes place for the same pattern. Let us analyze the 2D pattern comprising \(p\) identical equilateral triangles depicted in Figure 4b. The symmetry group of the equilateral triangle is the dihedral symmetry group, usually labeled \(D_{3}\). In the case of the equilateral triangles shown in Figure 4b, we have \(N_{g}=6p\) elements of symmetry, which are \(3p\) symmetry axes and \(3p\) rotations. The IMS calculated with Equation (2) equals \(H_{sym}(D_{3})=1.792\). On the other hand, all of the shapes constituting the pattern shown in Figure 4b have the same number of symmetry elements; thus, \(\Omega_{sym}(D_{3})=0\). Figure 3: Mixed patterns comprising 1D objects are depicted: (**a**) the pattern built from segments of cubic parabola and arcs of a circle, and (**b**) the pattern composed of irregular (non-symmetrical) shapes and arcs of a circle. Consider the fractal Sierpinski gasket built of equilateral triangles, shown in Figure 4c [33,34]. It is easily seen that \(H_{sym}(G)=1.792\); \(\Omega_{sym}=0\) for any scaling level of its fractal structure. Thus, we conclude that informational measures of symmetry are invariant with respect to the scaling of the Sierpinski gasket. Obviously, the conclusion holds for the Sierpinski gasket built of squares. The general problem of the calculation of the informational measures of symmetry of fractal structures deserves additional research. The pattern characterized by the informational measures of symmetry should not be necessarily close-packed, as shown in Figure 4d, representing a set of randomly dispersed equilateral triangles of various areas. In this case, \(H_{sym}(D_{3})=1.792\), \(\Omega(D_{3})=0\) takes place. Again, the introduced informational measures of symmetry are insensitive to the presence/absence of the long-range order in a given pattern. It is noteworthy that a pattern should not necessarily be built of polygons; it may comprise curvilinear shapes such as the ellipses, shown in Figure 5.  
Figure 4: (**a**) Pattern filled with different unsymmetrical polygons, demonstrating \(H_{sym}(G)=0\); \(\Omega_{sym}(G)=0\). (**b**) Two-dimensional pattern comprising identical equilateral triangles, demonstrating \(H_{sym}(G)=1.792\); \(\Omega_{sym}=0\). (**c**) Sierpinski gasket built of equilateral triangles. (**d**) Pattern composed of the randomly dispersed equilateral triangles of various areas. Let us start from the pattern depicted in Figure 5a comprising the dispersed random ellipses of various sizes. The group of symmetry of the ellipses includes four elements, namely the onefold rotational symmetry denoted \(G_{1}\), the twofold rotational symmetry denoted \(G_{2}\), and two distinguishable mirror axes denoted correspondingly as \(G_{3}\) and \(G_{4}\); in this case, \(P(G_{1})=P(G_{2})=P(G_{3})=P(G_{4})=\frac{1}{4}\). Thus, we calculate \(H_{sym}=-4\frac{1}{4}ln\frac{1}{4}=1.39\). All of the ellipses have in total four elements of symmetry; hence, \(\Omega_{sym}=0\). The same is true for the dispersed identical long-range ordered ellipses, shown in Figure 5b. The same conclusion holds for the close-packed ordered identical ellipses forming the patterns shown in Figure 5c,d. Figure 5c,d illustrate the idea that the set of the shapes should be clearly defined. Indeed, \(H_{sym}=1.39;\ \Omega_{sym}=0\) holds for the patterns built of the ellipses _only_.