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

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> characterizes the averaged uncertainty in the presence of symmetry elements from group <i>G</i> in a given pattern, whereas the Shannon-like measure of symmetry <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Ω</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> quantifies the averaged uncertainty of the appearance of shapes possessing a total of <i>n</i> elements of symmetry belonging to group <i>G</i> in a given pattern. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi mathvariant="sans-serif">Ω</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> for the patterns built of irregular, non-symmetric shapes, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> 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.