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”

Symmetry

卷:13

Mark Frenkel¹  Irina Legchenkova¹  Edward Bormashenko¹  Shraga Shoval²  Nir Shvalb³ 

[1] Chemical Engineering Department, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel;

[2] Department of Industrial Engineering and Management, Faculty of Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel;

[3] Department of Mechanical Engineering & Mechatronics, Faculty of Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel;

关键词: symmetry;    informational measure;    Penrose tiling;    Voronoi entropy;    continuous symmetry measure;    ordering;   

DOI  :  10.3390/sym13112146

来源: DOAJ

【 摘 要 】

The notion of the informational measure of symmetry is introduced according to: $H_{sym}\left(G\right)=-{{\displaystyle \sum }}_{i=1}^{k}P\left(G_i\right)lnP\left(G_i\right)$, where $P\left(G_i\right)$ is the probability of appearance of the symmetry operation $G_i$ within the given 2D pattern. $H_{sym}\left(G\right)$ 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 $H_{sym}\left(D_3\right)=$ 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, $H_{sym}=0$. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

【 授权许可】

Unknown