Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 20 September 2021 / Approved: 20 September 2021 / Online: 20 September 2021 (16:39:57 CEST)

How to cite: Bormashenko, E.; Legchenkova, I.; Frenkel, M.; Shvalb, N.; Shoval, S. Shannon (Information) Measures of Symmetry for 1D and 2D Shapes and Patterns. Preprints 2021, 2021090347 (doi: 10.20944/preprints202109.0347.v1). Bormashenko, E.; Legchenkova, I.; Frenkel, M.; Shvalb, N.; Shoval, S. Shannon (Information) Measures of Symmetry for 1D and 2D Shapes and Patterns. Preprints 2021, 2021090347 (doi: 10.20944/preprints202109.0347.v1).

Abstract

Informational (Shannon) measures of symmetry are introduced and analyzed for the patterns built of 1D and 2D shapes. The informational measure of symmetry Hsym (G) characterizes the an averaged uncertainty in the presence of symmetry elements from the group G in a given pattern; whereas the Shannon-like measure of symmetry Ωsym (G) quantifies averaged uncertainty of appearance of shapes possessing in total 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. Both of informational measures of symmetry are intensive parameters of the pattern and do not depend on the number of shapes, their size and area of the pattern. They are also insensitive to the long-range order inherent for the pattern. Informational measures of symmetry of fractal patterns are addressed. The mixed patterns including curves and shapes are considered. Time evolution of the Shannon measures of symmetry is treated. The close-packed and dispersed 2D patterns are analyzed.

Keywords

informational measure of symmetry; 1D shapes; 2D shapes; fractal patterns; time evolution; symmetry; pattern

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