Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

Symmetry

Mark Frenkel¹  Irina Legchenkova¹  Edward Bormashenko¹  Shraga Shoval²  Nir Shvalb³ 

[1] Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel;Department of Industrial Engineering and Management, Faculty of Engineering, Ariel University, P.O.B. 3, Ariel 407000, Israel;Department of Mechanical Engineering & Mechatronics, Faculty of Engineering, Ariel University, P.O.B. 3, Ariel 407000, Israel;

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

DOI  :  10.3390/sym13091659

来源: DOAJ

【 摘 要 】

A continuous measure of symmetry and the Voronoi entropy of 2D patterns representing Voronoi diagrams emerging from the Penrose tiling were calculated. A given Penrose tiling gives rise to a diversity of the Voronoi diagrams when the centers, vertices, and the centers of the edges of the Penrose rhombs are taken as the seed points (or nuclei). Voronoi diagrams keep the initial symmetry group of the Penrose tiling. We demonstrate that the continuous symmetry measure and the Voronoi entropy of the studied sets of points, generated by the Penrose tiling, do not necessarily correlate. Voronoi diagrams emerging from the centers of the edges of the Penrose rhombs, considered nuclei, deny the hypothesis that the continuous measure of symmetry and the Voronoi entropy are always correlated. The Voronoi entropy of this kind of tiling built of asymmetric convex quadrangles equals zero, whereas the continuous measure of symmetry of this pattern is high. Voronoi diagrams generate new types of Penrose tiling, which are different from the classical Penrose tessellation.

【 授权许可】

Unknown