{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T07:49:19Z","timestamp":1779263359217,"version":"3.51.4"},"reference-count":41,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,8]],"date-time":"2022-06-08T00:00:00Z","timestamp":1654646400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.<\/jats:p>","DOI":"10.3390\/e24060802","type":"journal-article","created":{"date-parts":[[2022,6,10]],"date-time":"2022-06-10T00:22:39Z","timestamp":1654820559000},"page":"802","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6384-7570","authenticated-orcid":false,"given":"Irina","family":"Legchenkova","sequence":"first","affiliation":[{"name":"Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3025-6130","authenticated-orcid":false,"given":"Mark","family":"Frenkel","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nir","family":"Shvalb","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering & Mechatronics, Faculty of Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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. Box 3, Ariel 407000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Oleg V.","family":"Gendelman","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, Technion\u2014Israel Institute of Technology, P.O. Box 10, Haifa 3200003, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1356-2486","authenticated-orcid":false,"given":"Edward","family":"Bormashenko","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1103\/PhysRev.176.250","article-title":"Crystalline Order in Two Dimensions","volume":"176","author":"Mermin","year":"1968","journal-title":"Phys. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1103\/PhysRev.65.117","article-title":"Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition","volume":"65","author":"Onsager","year":"1944","journal-title":"Phys. Rev."},{"key":"ref_3","first-page":"493","article-title":"Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems","volume":"32","author":"Berezinskii","year":"1971","journal-title":"Sov. Phys. JETP"},{"key":"ref_4","first-page":"610","article-title":"Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group II, Quantum systems","volume":"34","author":"Berezinskii","year":"1972","journal-title":"Sov. Phys. JETP"},{"key":"ref_5","unstructured":"Hunt, B.R., Li, T.Y., Kennedy, J.A., and Nusse, H.E. (1976). A Two-dimensional Mapping with a Strange Attractor. The Theory of Chaotic Attractors, Springer."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","unstructured":"Kovalerchuk, B. (2018). Visual Knowledge Discovery and Machine Learning, Springer International Publishing.","DOI":"10.1007\/978-3-319-73040-0"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"062122","DOI":"10.1103\/PhysRevE.95.062122","article-title":"Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination","volume":"95","author":"Hu","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1038\/s41524-019-0221-0","article-title":"Recent advances and applications of machine learning in solid-state materials science","volume":"5","author":"Schmidt","year":"2019","journal-title":"NPJ Comput. Mater."},{"key":"ref_10","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_11","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_12","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1016\/j.jcis.2020.09.019","article-title":"Colloidal monolayers with cell-like tessellations via interface assisted evaporative assembly","volume":"583","author":"Mayarani","year":"2021","journal-title":"J. Colloid Interface Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"110191","DOI":"10.1016\/j.matdes.2021.110191","article-title":"Optimisation of rGO-enriched nanoceramics by combinatorial analysis","volume":"212","author":"Borodin","year":"2021","journal-title":"Mater. Des."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1038\/s41593-020-00729-w","article-title":"Mesophasic organization of GABAA receptors in hippocampal inhibitory synapses","volume":"23","author":"Liu","year":"2020","journal-title":"Nat. Neurosci."},{"key":"ref_15","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_16","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_17","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_18","doi-asserted-by":"crossref","first-page":"023527","DOI":"10.1103\/PhysRevD.105.023527","article-title":"Minkovski functionals and the nonlinear perturbation theory in the large-scale structure: Second-order effects","volume":"105","author":"Matsubara","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_19","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_20","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_21","doi-asserted-by":"crossref","first-page":"088002","DOI":"10.1103\/PhysRevLett.124.088002","article-title":"Multiple Particle Correlation Analysis of Many-Particle Systems: Formalism and Application to Active Matter","volume":"124","author":"Stroteich","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","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_23","doi-asserted-by":"crossref","first-page":"2429","DOI":"10.1080\/02664763.2017.1421917","article-title":"Testing symmetry based on empirical likelihood","volume":"45","author":"Zhang","year":"2018","journal-title":"J. Appl. Stat."},{"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":"J. Comput. Chem."},{"key":"ref_25","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_26","doi-asserted-by":"crossref","unstructured":"Shalit, Y., and Tuvi-Arad, I. (2020). Side chain flexibility and the symmetry of protein homodimers. PLoS ONE, 15.","DOI":"10.1371\/journal.pone.0235863"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bormashenko, E., Legchenkova, I., Frenkel, M., Shvalb, N., and Shoval, S. (2022). Shannon (Information) Measures of Symmetry for 1D and 2D Shapes and Patterns. Appl. Sci., 12.","DOI":"10.20944\/preprints202109.0347.v1"},{"key":"ref_28","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_29","doi-asserted-by":"crossref","unstructured":"Bormashenko, E., Legchenkova, I., Frenkel, M., Shvalb, N., and Shoval, S. (2021). Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams. Symmetry, 13.","DOI":"10.3390\/sym13091659"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1016\/0032-3861(84)90076-4","article-title":"On the lamellar morphology of isotactic polypropylene spherulites","volume":"25","author":"Bassett","year":"1984","journal-title":"Polymer"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1081\/MB-120017116","article-title":"Polymer Spherulites: A Modern Assessment","volume":"42","author":"Bassett","year":"2003","journal-title":"J. Macromol. Sci. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.sedgeo.2017.12.024","article-title":"Origins of carbonate spherulites: Implications for Brazilian Aptian pre-salt reservoir","volume":"365","author":"Chafetz","year":"2018","journal-title":"Sediment. Geol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1130\/G36639.1","article-title":"Spherulites as in-situ recorders of thermal history in lava flows","volume":"43","author":"Befus","year":"2015","journal-title":"Geology"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.sedgeo.2018.07.002","article-title":"Thrombolites, spherulites and fibrous crusts (Holkerian, Purbeckian, Aptian): Context, fabrics and origins","volume":"374","author":"Kirkham","year":"2018","journal-title":"Sediment. Geol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1006\/jasc.1998.0322","article-title":"The Production and Preservation of Faecal Spherulites: Animals, Environment and Taphonomy","volume":"26","author":"Canti","year":"1999","journal-title":"J. Archaeol. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"9859","DOI":"10.1021\/ma1019686","article-title":"Conformational Ordering in Growing Spherulites of Isotactic Polypropylene","volume":"43","author":"Cong","year":"2010","journal-title":"Macromolecules"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3762","DOI":"10.1103\/PhysRevLett.76.3762","article-title":"Evidence for convective effects in breath figure formation on volatile fluid surfaces","volume":"76","author":"Limaye","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"73","DOI":"10.6028\/jres.064A.007","article-title":"Theory of Formation of Polymer Crystals with Folded Chains in Dilute Solution","volume":"64","author":"Lauritzen","year":"1960","journal-title":"J. Res. Natl. Bur. Stand."},{"key":"ref_39","first-page":"e10173","article-title":"Secondary nucleation in polymer crystallization: A kinetic view","volume":"4","author":"Zhang","year":"2021","journal-title":"Polym. Cryst."},{"key":"ref_40","first-page":"57","article-title":"Oxidation of Zr-based metallic glasses and nanocrystalline alloys","volume":"449","author":"Jastrow","year":"2007","journal-title":"Mater. Sci. Eng. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1016\/j.jnoncrysol.2005.01.049","article-title":"Structural evolution of Fe33Zr67 and Fe90Zr10 metallic glasses","volume":"351","author":"Liu","year":"2005","journal-title":"J. Non-Cryst. Solids"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/6\/802\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:26:11Z","timestamp":1760138771000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/6\/802"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,8]]},"references-count":41,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["e24060802"],"URL":"https:\/\/doi.org\/10.3390\/e24060802","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202204.0252.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,8]]}}}