{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T08:52:30Z","timestamp":1778662350208,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T00:00:00Z","timestamp":1672617600000},"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>We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.<\/jats:p>","DOI":"10.3390\/e25010092","type":"journal-article","created":{"date-parts":[[2023,1,3]],"date-time":"2023-01-03T02:51:39Z","timestamp":1672714299000},"page":"92","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings"],"prefix":"10.3390","volume":"25","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":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3025-6130","authenticated-orcid":false,"given":"Mark","family":"Frenkel","sequence":"additional","affiliation":[{"name":"Chemical Engineering Department, Engineering Faculty, Ariel University, P.O.B. 3, Ariel 407000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8246-3727","authenticated-orcid":false,"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":[{"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.B. 3, Ariel 407000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,2]]},"reference":[{"key":"ref_1","unstructured":"Reinhardt, K. (1918). \u00dcber die Zerlegung der Ebene in Polygone. [Ph.D. Thesis, Universitat Frankfurt]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s10711-017-0270-9","article-title":"Convex pentagons that admit i-block transitive tilings","volume":"194","author":"Mann","year":"2018","journal-title":"Geom. Dedicata"},{"key":"ref_3","unstructured":"Heesch, H. (1935). Aufbau der Ebene aus Kongruenten Bereichen, Vandenhoeck Ruprecht."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1080\/00029890.1968.11971075","article-title":"On paving the plane","volume":"75","author":"Kershner","year":"1968","journal-title":"Am. Math. Mon."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1080\/17513472.2017.1399680","article-title":"Marjorie Rice (16 February 1923\u20132 July 2017)","volume":"12","author":"Schattschneider","year":"2018","journal-title":"J. Math. Arts"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1080\/17513472.2018.1453740","article-title":"Marjorie Rice and the MAA tiling","volume":"12","author":"Schattschneider","year":"2018","journal-title":"J. Math. Arts"},{"key":"ref_7","unstructured":"Rao, M. (2017). Exhaustive search of convex pentagons which tile the plane. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/BF03024384","article-title":"Pentaplexity A Class of Non-Periodic Tilings of the Plane","volume":"2","author":"Penrose","year":"1979","journal-title":"Math. Intell."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1088\/0034-4885\/69\/2\/R03","article-title":"The role of aperiodic order in science and technology","volume":"69","author":"Macia","year":"2005","journal-title":"Rep. Prog. Phys."},{"key":"ref_10","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_11","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_12","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_13","doi-asserted-by":"crossref","first-page":"054109","DOI":"10.1103\/PhysRevB.81.054109","article-title":"Strain-induced isosymmetric phase transition in BiFeO3","volume":"81","author":"Hatt","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7551","DOI":"10.1103\/PhysRevB.57.7551","article-title":"High-pressure isosymmetric phase transition in orthorhombic lead fluoride","volume":"57","author":"Haines","year":"1998","journal-title":"Phys. Rev. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1021\/acs.jpcb.9b07313","article-title":"An Isosymmetric High-Pressure Phase Transition in \u03b1-Glycylglycine: A Combined Experimental and Theoretical Study","volume":"124","author":"Clarke","year":"2020","journal-title":"J. Phys. Chem. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"17448","DOI":"10.1021\/acs.jpcc.1c04659","article-title":"Pressure-Driven Symmetry-Preserving Phase Transitions in Co(IO3)2","volume":"125","author":"Liang","year":"2021","journal-title":"J. Phys. Chem. C"},{"key":"ref_17","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. Re-cherches sur les parall\u00e9lo\u00e8dres primitifs","volume":"134","author":"Voronoi","year":"1908","journal-title":"J. Reine Angew. Math."},{"key":"ref_18","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_19","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_20","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":"Phil. Mag. Lett."},{"key":"ref_21","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_22","doi-asserted-by":"crossref","unstructured":"Lopez-Sauceda, J., von B\u00fclow, P., Ortega-Laurel, C., Perez-Martinez, F., Miranda-Perkins, K., and Gonz\u00e1lez, J.G.C. (2022). Entropy as a Geometrical Source of Information in Biological Organizations. Entropy, 24.","DOI":"10.3390\/e24101390"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","unstructured":"Bormashenko, E., Legchenkova, I., Frenkel, M., Shvalb, N., and Shoval, S. (2021). Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the \u201cVoronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling. Symmetry, 13.","DOI":"10.20944\/preprints202109.0076.v1"},{"key":"ref_25","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_26","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_27","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1107\/S0108768195001728","article-title":"Isosymmetric structural phase transitions: Phenomenology and examples","volume":"51","author":"Christy","year":"1995","journal-title":"Acta Cryst. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"041309","DOI":"10.1103\/PhysRevE.79.041309","article-title":"Optimal packings of superballs","volume":"79","author":"Jiao","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"248104","DOI":"10.1103\/PhysRevLett.110.248104","article-title":"Phyllotaxis, Pushed Pattern-Forming Fronts and Optimal Packing","volume":"110","author":"Pennybacker","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s004540010071","article-title":"The Honeycomb Conjecture","volume":"25","author":"Hales","year":"2001","journal-title":"Discret. Comput. Geom."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1090\/noti838","article-title":"Isoperimetric pentagonal tilings","volume":"59","author":"Chung","year":"2012","journal-title":"Not. Am. Math. Soc."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"040601","DOI":"10.1103\/PhysRevLett.94.040601","article-title":"Detection and Characterization of Structural Changes in the Hard-Disk Fluid under Freezing and Melting Conditions","volume":"94","author":"Moucka","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"258001","DOI":"10.1103\/PhysRevLett.96.258001","article-title":"Crystallization of a Quasi-Two-Dimensional Granular Fluid","volume":"96","author":"Reis","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"7827","DOI":"10.1021\/acs.langmuir.8b01411","article-title":"Pattern Formation in Binary Colloidal Assemblies: Hidden Symmetries in a Kaleidoscope of Structures","volume":"34","author":"Lotito","year":"2018","journal-title":"Langmuir"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"102252","DOI":"10.1016\/j.cis.2020.102252","article-title":"Pattern detection in colloidal assembly: A mosaic of analysis techniques","volume":"284","author":"Lotito","year":"2020","journal-title":"Adv. Colloid Interface Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/92\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:56:35Z","timestamp":1760118995000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/92"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,2]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010092"],"URL":"https:\/\/doi.org\/10.3390\/e25010092","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202212.0012.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,2]]}}}