Shannon (Information) Measures of Symmetry for 1D and 2D Shapes and Patterns

Abstract

In this paper, informational (Shannon) measures of symmetry are introduced and analyzed for patterns built of 1D and 2D shapes. The informational measure of symmetry Hsym(G) characterizes the averaged uncertainty in the presence of symmetry elements from group G in a given pattern, whereas the Shannon-like measure of symmetry Ωsym(G) quantifies the averaged uncertainty of the appearance of shapes possessing a total of n elements of symmetry belonging to group G in a given pattern. Hsym(G1)=Ωsym(G1)=0 for the patterns built of irregular, non-symmetric shapes, where G1 is the identity element of the symmetry group. Both informational measures of symmetry are intensive parameters of the pattern and do not depend on the number of shapes, their size, and the entire area of the pattern. They are also insensitive to the long-range order (translational symmetry) inherent for the pattern. Additionally, informational measures of symmetry of fractal patterns are addressed, the mixed patterns including curves and shapes are considered, the time evolution of Shannon measures of symmetry are examined, the close-packed and dispersed 2D patterns are analyzed, and an application of the suggested measures of symmetry for the analysis of the chemical reaction is demonstrated.

Fibonacci Sequences, Symmetry and Ordering in Biological Patterns, Their Sources, Information Origin and the Landauer Principle

“…Alternative measures of ordering in biological patterns, such as Minkowski functionals were also introduced [79][80]. Shannon (informational) measure of symmetry was also suggested recently [81]. 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.…”

“…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 𝐺 𝑖 , i = 1,2 … 𝑘, where k is a number of non-identical symmetry operations [81]. Elements 𝐺 𝑖 form the symmetry group of the shape G (which should be clearly distinguished from the symmetry group of the entire pattern).…”

From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns

“…Thus, clearly distinguished 2D ordering emerged from the aforementioned mathematical procedure applied to the initially random set of points. We quantified the ordering with Voronoi entropy and the continuous measure of symmetry [ 24 , 25 , 26 , 27 , 28 , 29 ]. The Shannon (Voronoi) entropy and the continuous measure of symmetry of the addressed patterns demonstrated pronounced asymptotic behavior, while approaching the close asymptotic values with the increase in the number of iteration steps.…”

“…The details of the transformation of the non-symmetric triangle to the symmetric equilateral triangle and algorithm of the calculation of the continuous measure of symmetry are supplied in references [ 27 , 28 , 29 ].…”

“…The quantification of 2D ordering is crucial for understanding phase transitions [ 1 , 2 , 3 , 4 ], the characterization of attractors in non-linear systems [ 5 ], the treatment of images [ 6 ] and machine learning applied for study of physical systems [ 7 , 8 , 9 ]. Various measures and mathematical procedures were implemented for the quantification of ordering in 2D patterns, including Voronoi tessellations followed by the calculation of the information (Shannon) entropy of the distribution of the Voronoi polygons (which is also called the Voronoi entropy and abbreviated in the text as VE; VE below within the text denotes the Shannon entropy) [ 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 ], Minkovski functionals [ 18 , 19 , 20 ], method of correlation functions [ 21 , 22 , 23 ], and calculation of the recently introduced continuous and Shannon measures of symmetry [ 6 , 24 , 25 , 26 , 27 ]. It was demonstrated that the introduced measures of symmetry do not necessarily correlate [ 27 , 28 ].…”