{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T13:35:03Z","timestamp":1767706503467,"version":"build-2065373602"},"reference-count":67,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2009,8,20]],"date-time":"2009-08-20T00:00:00Z","timestamp":1250726400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is \u03b1 and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values \u03b1&gt;0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in \u03b1=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with \u03b10.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When \u03b1&gt;2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with \u03b1 until the Poisson-Voronoi limit is reached for \u03b1&gt;2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces.<\/jats:p>","DOI":"10.3390\/sym1010021","type":"journal-article","created":{"date-parts":[[2009,8,20]],"date-time":"2009-08-20T16:36:45Z","timestamp":1250786205000},"page":"21-54","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Symmetry-Break in Voronoi Tessellations"],"prefix":"10.3390","volume":"1","author":[{"given":"Valerio","family":"Lucarini","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK"},{"name":"Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading RG6 6BB, UK"},{"name":"Department of Physics, University of Bologna, Viale Berti Pichat 6\/2, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2009,8,20]]},"reference":[{"key":"ref_1","first-page":"97","article-title":"Nouvelles applications des param\u00e8tres continus \u00e0 la th\u00e9orie des formes quadratiques. Premier M\u00e9moire: Sur quelques propri\u00e9t\u00e9es des formes quadritiques positives parfaites","volume":"133","author":"Voronoi","year":"1907","journal-title":"J. Reine Angew. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1515\/crll.1908.134.198","article-title":"Nouvelles Applications des Parametres Continus a la Theorie des Formes Quadratiques. Duesieme Memoire: Recherches sur les Paralleloderes Primitifs","volume":"134","author":"Voronoi","year":"1908","journal-title":"J. Reine Angew. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1239\/aap\/1013540236","article-title":"Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces","volume":"32","author":"Isokawa","year":"2000","journal-title":"Adv. Appl. Probl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Okabe, A., Boots, B., Sugihara, K., and Chiu, S. N. (2000). Spatial Tessellations - Concepts and Applications of Voronoi Diagrams, Wiley. [2nd ed.].","DOI":"10.1002\/9780470317013"},{"key":"ref_5","unstructured":"Sortais, M., Hermann, S., and Wolisz, A. (, January April). Analytical Investigation of Intersection-Based Range-Free Localization Information Gain. Proceedings of the European Wireless 2007, Paris, France."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1016\/0022-2836(75)90148-5","article-title":"Volume occupation, environment and. accessibility in proteins. The problem of the protein surface","volume":"96","author":"Finney","year":"1975","journal-title":"J. Mol. Biol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF00642300","article-title":"Particles, space and time","volume":"244","author":"Icke","year":"1996","journal-title":"Astrophys. Space Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1139\/x96-214","article-title":"Voronoi tessellation methods to delineate harvest units for spatial forest planning","volume":"27","author":"Barrett","year":"1997","journal-title":"Can. J. For. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1002\/(SICI)1096-987X(19970715)18:9<1113::AID-JCC1>3.0.CO;2-U","article-title":"Voronoi cell: New method for allocation of space among atoms: Elimination of avoidable errors in calculation of atomic volume and density","volume":"18","author":"Goede","year":"1997","journal-title":"J. Comp. Chem."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"L101","DOI":"10.1080\/13642818608240647","article-title":"On the distribution of cell areas in a Voronoi network","volume":"53","author":"Weaire","year":"1986","journal-title":"Phil. Mag. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1103\/PhysRevLett.82.105","article-title":"Cell Crystals: Kelvin\u2019s Polyhedra in Block Copolymer Melts","volume":"82","author":"Dotera","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8270","DOI":"10.1103\/PhysRevB.34.8270","article-title":"Local atomic environments in periodic and aperiodic Al-Mn alloys","volume":"34","author":"Bennett","year":"1986","journal-title":"Phys. Rev. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/S0167-6636(03)00062-0","article-title":"Unit cells for micromechanical analyses of particle-reinforced composites","volume":"36","author":"Li","year":"2004","journal-title":"Mechanics of Materials"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3532","DOI":"10.1103\/PhysRevLett.85.3532","article-title":"Voronoi tessellation reveals the condensed matter character of folded proteins","volume":"85","author":"Soyer","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","unstructured":"Bassani, F., and Pastori-Parravicini, G. (1975). Electronic States and Optical Transitions in Solids, Pergamon."},{"key":"ref_16","unstructured":"Ashcroft, N. W., and Mermin, N. D. (1976). Solid State Physics, Saunders."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"8552","DOI":"10.1103\/PhysRevB.47.8552","article-title":"Statistics of Voronoi polyhedra in a model silicon glass","volume":"47","author":"Tsumuraya","year":"1993","journal-title":"Phys. Rev. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"051202","DOI":"10.1103\/PhysRevE.72.051202","article-title":"Structure analysis methods for crystalline solids and supercooled liquids","volume":"72","author":"Yu","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"50404(R)","DOI":"10.1103\/PhysRevE.75.050404","article-title":"Statistical mechanics of the glass transition as revealed by a Voronoi tessellation","volume":"75","author":"Hentschel","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3181","DOI":"10.1103\/PhysRevB.62.3181","article-title":"Voronoi-Delaunay analysis of normal modes in a simple model glass","volume":"62","author":"Luchnikov","year":"2000","journal-title":"Phys. Rev. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8115","DOI":"10.1103\/PhysRevB.39.8115","article-title":"Pseudospherical integration scheme for electronic-structure calculations","volume":"39","author":"Averill","year":"1989","journal-title":"Phys. Rev. B"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"007281","DOI":"10.1103\/PhysRevB.57.7281","article-title":"Consistent methodology for calculating surface and interface energies","volume":"57","author":"Rapcewicz","year":"1998","journal-title":"Phys. Rev. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"025301","DOI":"10.1103\/PhysRevE.73.025301","article-title":"Hexagonal convection patterns in atomistically simulated fluids","volume":"73","author":"Rapaport","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_24","first-page":"269","article-title":"Geophysical parameterization and parameter structure identification using natural neighbors in groundwater inverse problems","volume":"308","author":"Sun","year":"2004","journal-title":"J. Hydrology"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"D13103","DOI":"10.1029\/2006JD008360","article-title":"Does the Danube exist? Versions of reality given by various regional climate models and climatological data sets","volume":"112","author":"Lucarini","year":"2007","journal-title":"J. Geophys. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"D09107","DOI":"10.1029\/2007JD009167","article-title":"Hydrological Cycle in the Danube basin in present-day and XXII century simulations by IPCCAR4 global climate models","volume":"113","author":"Lucarini","year":"2008","journal-title":"J. Geophys. Res."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Thiessen, A. H., and Alter, J. C. (1911). Climatological Data for July, 1911: District No. 10, Great Basin. Monthly Weather Review, 1082\u20131089.","DOI":"10.1175\/1520-0493(1911)39<1082b:PAFLA>2.0.CO;2"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1093\/comjnl\/24.2.162","article-title":"Computing Dirichlet tessellations","volume":"24","author":"Bowyer","year":"1981","journal-title":"Computer J."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1093\/comjnl\/24.2.167","article-title":"Computing the n-dimensional tessellation with application to Voronoi polytopes","volume":"24","author":"Watson","year":"1981","journal-title":"Computer J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0021-9991(83)90087-6","article-title":"A new algorithm for three-dimensional Voronoi tessellation","volume":"51","author":"Tanemura","year":"1983","journal-title":"J. Compu. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1145\/235815.235821","article-title":"The Quickhull Algorithm for Convex Hulls","volume":"22","author":"Barber","year":"1996","journal-title":"ACM TOMS"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"W11502","DOI":"10.1029\/2005WR004365","article-title":"Automated Thiessen polygon generation","volume":"42","author":"Han","year":"2006","journal-title":"Water Resour. Res."},{"key":"ref_33","first-page":"270","article-title":"Interface area, edge length, and number of vertices in crystal aggregates with random nucleation: Phillips Research Reports","volume":"8","author":"Meijering","year":"1953","journal-title":"Philips Res. Rep."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/0550-3213(82)90222-X","article-title":"Random lattice field theory: General formulation","volume":"202","author":"Christ","year":"1982","journal-title":"Nuclear Physics B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0550-3213(84)90147-0","article-title":"Random geometry and the statistics of two-dimensional cells","volume":"235","author":"Drouffe","year":"1984","journal-title":"Nucl. Phys. B"},{"key":"ref_36","unstructured":"Harding, E. F., and Kendall, D. G. (1974). Stochastic Geometry, Wiley."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"37","DOI":"10.2307\/1427197","article-title":"Random tessellations in Rd","volume":"21","year":"1989","journal-title":"Adv. Appl. Prob."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1239\/aap\/1059486817","article-title":"Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson Voronoi tessellation and a Poisson line process","volume":"35","author":"Calka","year":"2003","journal-title":"Adv. Appl. Probab."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"7227","DOI":"10.1088\/0305-4470\/39\/23\/004","article-title":"Planar Voronoi cells: the violation of Aboav\u2019s law explained","volume":"39","author":"Hilhorst","year":"2006","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_40","unstructured":"Finch, S. R. (2005). Mathematical Constants, Cambridge University Press. unpublished. Available on http:\/\/algo.inria.fr\/csolve\/vi.pdf. Addendum to Finch S. R."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1007\/BF02733102","article-title":"Proof of David Kendall\u2019s conjecture concerning the shape of large random polygons","volume":"33","author":"Kovalenko","year":"1997","journal-title":"Cybernetics and Systems Analysis"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s00454-007-1340-9","article-title":"Typical cells in Poisson hyperplane tessellations","volume":"38","author":"Hug","year":"2007","journal-title":"Discr. Comput. Geom."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1214\/aop\/1079021474","article-title":"The limit shape of the zero cell in a stationary Poisson hyperplane tessellation","volume":"32","author":"Hug","year":"2004","journal-title":"Ann. Probab."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/BF01049719","article-title":"Properties of a three-dimensional Poisson-Voronoi tessellation: a Monte Carlo study","volume":"67","author":"Kumar","year":"1992","journal-title":"Journal of Statistical Physics"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1080\/00949658008810370","article-title":"Monte Carlo estimates of the distributions of the random polygons of the Voronoi tessellation with respect to a Poisson process","volume":"10","author":"Hinde","year":"1980","journal-title":"Journal of Statistical Computation and Simulation"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2765","DOI":"10.1080\/01418610010032364","article-title":"The geometrical properties of irregular two-dimensional Voronoi tessellations","volume":"81","author":"Zhu","year":"2001","journal-title":"Philosophical Magazine A"},{"key":"ref_47","first-page":"221","article-title":"Statistical distributions of Poisson-Voronoi cells in two and three Dimensions","volume":"18","author":"Tanemura","year":"2003","journal-title":"Forma"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Hilhorst, H. J. (2005). Asymptotic statistics of the n-sided planar Poisson\u2013Voronoi cell: I. Exact results. J. Stat. Mech., P09005.","DOI":"10.1088\/1742-5468\/2005\/09\/P09005"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"074502","DOI":"10.1063\/1.2000233","article-title":"Voronoi neighbor statistics of hard-disks and hard-spheres","volume":"123","author":"Kumar","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1109\/TIT.1982.1056492","article-title":"The Hexagon Theorem","volume":"28","author":"Newman","year":"1982","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.1016\/j.camwa.2004.12.008","article-title":"The Optimal Centroidal Voronoi Tessellations and the Gersho\u2019s Conjecture in the Three Dimensional Space","volume":"49","author":"Du","year":"2005","journal-title":"Comput. Math. Appl."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1016\/j.physa.2006.02.018","article-title":"A Gibbs point field model for the spatial pattern of coronary capillaries","volume":"369","author":"Karch","year":"2006","journal-title":"Physica A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1103\/PhysRevA.15.319","article-title":"Hydrodynamic fluctuations at the convective instability","volume":"15","author":"Swift","year":"1977","journal-title":"Phys. Rev. A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.4007\/annals.2005.162.1065","article-title":"A Proof of the Kepler Conjecture","volume":"162","author":"Hales","year":"2005","journal-title":"Ann. Math."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1080\/09500839408241577","article-title":"A Counter-Example to Kelvin\u2019s Conjecture on Minimal Surfaces","volume":"69","author":"Weaire","year":"1994","journal-title":"Philos. Mag. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Gabbrielli, R. (2009). A new counter-example to Kelvin\u2019s conjecture on minimal surfaces. Phil. Mag. Lett., 89.","DOI":"10.1080\/09500830903022651"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1109\/TVCG.2007.70429","article-title":"Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice","volume":"14","author":"Entezari","year":"2008","journal-title":"IEEE T. Vis. Comput. Gr."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1209\/epl\/i1998-00224-x","article-title":"Statistics of Voronoi cells of slightly perturbed face-centered cubic and hexagonal close-packed lattices","volume":"42","author":"Troadec","year":"1998","journal-title":"Europhy. Lett."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1007\/s10955-007-9475-x","article-title":"From Symmetry Breaking to Poisson Point Process in 2D Voronoi Tessellations: the Generic Nature of Hexagons","volume":"130","author":"Lucarini","year":"2008","journal-title":"J. Stat. Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s10955-008-9668-y","article-title":"Three-Dimensional Random Voronoi Tessellations: From Cubic Crystal Lattices to Poisson Point Processes","volume":"134","author":"Lucarini","year":"2009","journal-title":"J. Stat. Phys."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1002\/ar.1090380305","article-title":"The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of Cucumis","volume":"38","author":"Lewis","year":"1928","journal-title":"Anat. Rec."},{"key":"ref_62","first-page":"241","article-title":"The solidification of metals from the liquid state","volume":"22","author":"Desch","year":"1919","journal-title":"J. Inst. Metals"},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Finch, S. R. (2003). Mathematical Constants, Cambridge University Press.","DOI":"10.1017\/CBO9780511550447"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1038\/nmat1849","article-title":"The rise of grapheme","volume":"6","author":"Geim","year":"2007","journal-title":"Nature Materials"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Hilhorst, H. J. (Heuristic theory for many-faced d-dimensional Poisson-Voronoi cells, 2009). Heuristic theory for many-faced d-dimensional Poisson-Voronoi cells.","DOI":"10.1088\/1742-5468\/2009\/08\/P08003"},{"key":"ref_66","unstructured":"Dodson, C. T. J. (2008). On the entropy flows to disorder."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Coles, S. G. (2001). An Introduction to Statistical Modeling of Extreme Values, Springer.","DOI":"10.1007\/978-1-4471-3675-0"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/1\/1\/21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:10:59Z","timestamp":1760220659000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/1\/1\/21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,8,20]]},"references-count":67,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,9]]}},"alternative-id":["sym1010021"],"URL":"https:\/\/doi.org\/10.3390\/sym1010021","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2009,8,20]]}}}