Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”.

Version 1 : Received: 2 September 2021 / Approved: 3 September 2021 / Online: 3 September 2021 (21:36:08 CEST)

How to cite: Bormashenko, E.; Legchenkova, I.; Frenkel, M.; Shvalb, N.; Shoval, S. Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”.. Preprints 2021, 2021090076 (doi: 10.20944/preprints202109.0076.v1). Bormashenko, E.; Legchenkova, I.; Frenkel, M.; Shvalb, N.; Shoval, S. Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”.. Preprints 2021, 2021090076 (doi: 10.20944/preprints202109.0076.v1).

Abstract

The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.

Keywords

symmetry; informational measure; penrose tiling; Voronoi entropy; continuous symmetry measure; ordering

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