symmetry-break in voronoi tessellations在泰森多边形法symmetry-break镶嵌.pdf

Symmetry 2009, 1, 21-54; doi:10.3390/sym1010021 OPEN ACCESS symmetry ISSN 2073-8994 /journal/symmetry Review Symmetry-Break in Voronoi Tessellations Valerio Lucarini 1,2,3 1 Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK; E-mail: v.lucarini@reading.ac.uk 2 Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading RG6 6BB, UK 3 Department of Physics, University of Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy Received: 4 July 2009, in revised form: 4 August 2009 / Accepted: 6 August 2009 / Published: 20 August 2009 Abstract: 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 α 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