{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T10:51:24Z","timestamp":1780051884514,"version":"3.53.1"},"reference-count":41,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,8]],"date-time":"2021-09-08T00:00:00Z","timestamp":1631059200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["19-19-00076"],"award-info":[{"award-number":["19-19-00076"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym13091659","type":"journal-article","created":{"date-parts":[[2021,9,8]],"date-time":"2021-09-08T21:28:45Z","timestamp":1631136525000},"page":"1659","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1356-2486","authenticated-orcid":false,"given":"Edward","family":"Bormashenko","sequence":"first","affiliation":[{"name":"Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6384-7570","authenticated-orcid":false,"given":"Irina","family":"Legchenkova","sequence":"additional","affiliation":[{"name":"Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Frenkel","sequence":"additional","affiliation":[{"name":"Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nir","family":"Shvalb","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering & Mechatronics, Faculty of Engineering, Ariel University, P.O.B. 3, Ariel 407000, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0582-4821","authenticated-orcid":false,"given":"Shraga","family":"Shoval","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering and Management, Faculty of Engineering, Ariel University, P.O.B. 3, Ariel 407000, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,8]]},"reference":[{"key":"ref_1","unstructured":"Lang, S. (1993). Algebra, Addison-Wesley. [3rd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1016\/j.brat.2003.07.001","article-title":"Symmetry, ordering and arranging compulsive behavior","volume":"42","author":"Radomsky","year":"2004","journal-title":"Behav. Res. Ther."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1038\/nphys2190","article-title":"Between order and chaos","volume":"8","author":"Crutchfield","year":"2012","journal-title":"Nature Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1515\/crll.1908.134.198","article-title":"Nouvelles applications des param\u00e8tres continus \u00e0 la th\u00e9orie des formes quadratiques. Deuxi\u00e8me m\u00e9moire. Recherches sur les parall\u00e9lo\u00e8dres primitifs","volume":"134","author":"Voronoi","year":"1908","journal-title":"Reine Angew. Math."},{"key":"ref_5","unstructured":"Descartes, R. (1644). Principia Philosophiae, Ludovicus Elzevirius."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2010.11.002","article-title":"Spatial networks","volume":"499","year":"2011","journal-title":"Phys. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1080\/00107518408210979","article-title":"Soap, cells and statistics\u2014random patterns in two dimensions","volume":"25","author":"Weaire","year":"1984","journal-title":"Contemp. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1080\/09500839.2019.1677957","article-title":"Geometric formulas of Lewis\u2019s law and Aboav-Weaire\u2019s law in two dimensions based on ellipse packing","volume":"99","author":"Xu","year":"2019","journal-title":"Philos Mag. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"156983","DOI":"10.1016\/j.jallcom.2020.156983","article-title":"Inconsistency of neighborhood based on Voronoi tessellation and Euclidean distance","volume":"854","author":"Wang","year":"2021","journal-title":"J. Alloys Compd."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1888","DOI":"10.1038\/s41598-017-02166-5","article-title":"Self-assembled levitating clusters of water droplets: Pattern-formation and stability","volume":"7","author":"Fedorets","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5599","DOI":"10.1021\/acs.jpclett.7b02657","article-title":"Small Levitating Ordered Droplet Clusters: Stability, Symmetry, and Voronoi Entropy","volume":"8","author":"Fedorets","year":"2017","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"866","DOI":"10.1016\/j.jcis.2020.10.053","article-title":"Quantification of ordering in active light driven colloids","volume":"586","author":"Frenkel","year":"2021","journal-title":"J. Colloid Interface Sci."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bormashenko, E., Frenkel, M., Vilk, A., Legchenkova, I., Fedorets, A.A., Aktaev, N.E., Dombrovsky, L.A., and Nosonovsky, M. (2018). Characterization of self-assembled 2D patterns with Voronoi Entropy. Entropy, 20.","DOI":"10.20944\/preprints201811.0535.v1"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"022720","DOI":"10.1103\/PhysRevE.88.022720","article-title":"Automatic sorting of point pattern sets using Minkowski functionals","volume":"88","author":"Parker","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"P12015","DOI":"10.1088\/1742-5468\/2008\/12\/P12015","article-title":"Utilizing Minkowski functionals for image analysis: A marching square algorithm","volume":"2008","author":"Mantz","year":"2008","journal-title":"J. Stat. Mech. Theor. Exp."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1002\/macp.200700552","article-title":"Mesoscopic patterning in evaporated polymer solutions: Poly (ethylene glycol) and room-temperature-vulcanized Polyorganosilanes\/-siloxanes Promote formation of honeycomb structures","volume":"209","author":"Bormashenko","year":"2008","journal-title":"Macromol. Chem. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Bormashenko, E. (2020). Entropy, Information, and Symmetry: Ordered is Symmetrical. Entropy, 22.","DOI":"10.3390\/e22020235"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bormashenko, E. (2020). Entropy, Information, and Symmetry; Ordered Is Symmetrical, II: System of Spins in the Magnetic Field. Entropy, 22.","DOI":"10.20944\/preprints202001.0215.v1"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"7843","DOI":"10.1021\/ja00046a033","article-title":"Continuous symmetry measures","volume":"114","author":"Zabrodsky","year":"1992","journal-title":"J. Am. Chem. Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"8278","DOI":"10.1021\/ja00071a042","article-title":"Continuous symmetry measures. 2. Symmetry groups and the tetrahedron","volume":"115","author":"Zabrodsky","year":"1993","journal-title":"J. Am. Chem. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1021\/ja00106a053","article-title":"Continuous Symmetry Measures. 4. Chirality","volume":"117","author":"Zabrodsky","year":"1995","journal-title":"J. Am. Chem. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5575","DOI":"10.1021\/ic9804925","article-title":"Continuous Symmetry Measures. 5. The Classical Polyhedra","volume":"37","author":"Pinsky","year":"1998","journal-title":"Inorg. Chem."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1154","DOI":"10.1109\/34.476508","article-title":"Symmetry as a continuous feature","volume":"17","author":"Zabrodsky","year":"1995","journal-title":"IEEE Trans. Pattern Anal. Mach. Intel."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2712","DOI":"10.1002\/jcc.20990","article-title":"Analytical methods for calculating Continuous Symmetry Measures and the Chirality Measure","volume":"29","author":"Pinsky","year":"2008","journal-title":"Comp. Chemistry"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1076","DOI":"10.1002\/ijch.201600082","article-title":"Adsorption-induced Symmetry Distortions in W@Au12 Nanoclusters, Leading to Enhanced Hyperpolarizabilities","volume":"56","author":"Sinai","year":"2016","journal-title":"Israel J. Chem."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3176","DOI":"10.1021\/cm0604817","article-title":"Distortions in Octahedrally Coordinated d0 Transition Metal Oxides:\u2009 A Continuous Symmetry Measures Approach","volume":"18","author":"Ok","year":"2006","journal-title":"Chem. Mater."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8367","DOI":"10.1038\/s41598-020-65097-8","article-title":"The near-symmetry of protein oligomers: NMR-derived structures","volume":"10","author":"Bonjack","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/S0009-2614(98)01101-4","article-title":"Continuous symmetry measures for electronic wavefunctions","volume":"297","author":"Grimme","year":"1998","journal-title":"Chem. Phys. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2431","DOI":"10.1021\/acs.jpcc.0c10384","article-title":"Continuous Symmetry Measure vs Voronoi Entropy of Droplet Clusters","volume":"125","author":"Frenkel","year":"2021","journal-title":"J. Phys. Chem. C"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"466","DOI":"10.3390\/sym2020466","article-title":"Fluctuating Asymmetry: Methods, Theory, and Applications","volume":"2","author":"Graham","year":"2010","journal-title":"Symmetry"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/1385-7258(81)90016-0","article-title":"Algebraic theory of Penrose\u2019s non-periodic tilings of the plane","volume":"84","year":"1981","journal-title":"Indag. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1951","DOI":"10.1103\/PhysRevLett.53.1951","article-title":"Metallic phase with long-range orientational order and no translational symmetry","volume":"53","author":"Shechtman","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"15961","DOI":"10.1038\/ncomms15961","article-title":"Imaging quasiperiodic electronic states in a synthetic Penrose tiling","volume":"8","author":"Collins","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1038\/316050a0","article-title":"Penrose tiling observed in a quasi-crystal","volume":"316","author":"Bursill","year":"1985","journal-title":"Nature"},{"key":"ref_35","unstructured":"Lanczos, C. (1986). The Variational Principles of Mechanics, Dover Publications Inc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1479","DOI":"10.1002\/chem.200400799","article-title":"The rich stereochemistry of eight-vertex polyhedra: A continuous shape measures study","volume":"11","author":"Casanova","year":"2005","journal-title":"Chem. Eur. J."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1016\/j.tet.2019.02.007","article-title":"Evaluating continuous chirality measure as a 3D descriptor in chemoinformatics applied to asymmetric catalysis","volume":"75","author":"Zahrt","year":"2019","journal-title":"Tetrahedron"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1186\/s13321-019-0360-9","article-title":"Improved algorithms for quantifying the near symmetry of proteins: Complete side chains analysis","volume":"11","author":"Alon","year":"2019","journal-title":"J. Cheminform."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/0031-3203(84)90064-5","article-title":"An optimal algorithm for constructing the weighted Voronoi diagram in the plane","volume":"17","author":"Aurenhammer","year":"1984","journal-title":"Pattern Recognit."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1111\/j.0033-0124.2004.05602007.x","article-title":"Polygon characterization with the multiplicatively weighted Voronoi diagram","volume":"56","author":"Mu","year":"2004","journal-title":"Prof. Geogr."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Bormashenko, E., Legchenkova, I., and Frenkel, M. (2019). Symmetry and Shannon Measure of Ordering. Entropy, 21.","DOI":"10.3390\/e21050452"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/9\/1659\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:59:04Z","timestamp":1760165944000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/9\/1659"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,8]]},"references-count":41,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["sym13091659"],"URL":"https:\/\/doi.org\/10.3390\/sym13091659","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,8]]}}}