%0 Book Section %1 statphys23_0862 %A Mazzoni, S. %A Giavazzi, F. %A Cerbino, R. %A Giglio, M. %A Vailati, A. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K diagrams formation pattern statphys23 topic-4 voronoi %T Mutual Voronoi diagrams in spoke pattern convection %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=862 %X In two-dimensional cellular patterns, the plane is divided into adjacent and non overlapping polygonal cells by a set of lines forming a reticulum. Cellular patterns are widespread in many mathematical, physical, biological or social processes. One of the features that is still cause of interest among researchers is the fact that many topological properties of cellular patterns are strongly universal and system-independent. The most ubiquitous topological feature is perhaps trivalency. That is, for the vast majority of cellular patterns, always three cell edges join at a vertex. For trivalent cellular patterns, the Euler equation states that the mean number of edges per cell must be equal to six. In this work we outline the peculiar topological properties of the pattern observed in spoke pattern convection. Spoke pattern convection is composed of upwelling and downwelling columnar flows erupting from thin boundary layers located across the confining plates. This peculiar form of convective transport is of interest in geophysics, for mantle convection is recognized to be akin to spoke pattern instability. If observed by means of optical shadowgraph technique, the convective flow takes the form of two reciprocal cellular patterns that have mean number of edges per cell and mean number of edges per vertex equal to four. The analysis of the topological properties of the spoke pattern shows that its peculiar fourfold sided domains and the fourfold coordination of vertices stem from the rather general condition of having two dual cellular patterns that possess the same topological properties. This is demonstrated by means of Euler Theorem. In addition, we show that each of the two cellular patterns can be well described by means of a Voronoi diagram up to a definite length scale. Quite remarkably, the two diagrams actually obey a Mutual Voronoi Relation (MVR), where each diagram is modeled by the Voronoi diagram generated by vertices of the other diagram. The MVR imposes symmetries for the position of the vertices of the two diagrams. That is, the cross correlation length between the position of the different species is narrowly distributed around a dominant value, while the distribution of distances among vertices of the same species shows a less pronounced correlation peak falling in correspondence of the first minimum of the cross correlation function. A similar metric behaviour is reported for molten salts, where an ion is surrounded by alternating shell of unlike and like ions, thus suggesting a qualitative similarity between the two systems. S.M. acknowledges partial support from the European Space Agency

@incollection{statphys23_0862, abstract = {In two-dimensional cellular patterns, the plane is divided into adjacent and non overlapping polygonal cells by a set of lines forming a reticulum. Cellular patterns are widespread in many mathematical, physical, biological or social processes. One of the features that is still cause of interest among researchers is the fact that many topological properties of cellular patterns are strongly universal and system-independent. The most ubiquitous topological feature is perhaps trivalency. That is, for the vast majority of cellular patterns, always three cell edges join at a vertex. For trivalent cellular patterns, the Euler equation states that the mean number of edges per cell must be equal to six. In this work we outline the peculiar topological properties of the pattern observed in spoke pattern convection. Spoke pattern convection is composed of upwelling and downwelling columnar flows erupting from thin boundary layers located across the confining plates. This peculiar form of convective transport is of interest in geophysics, for mantle convection is recognized to be akin to spoke pattern instability. If observed by means of optical shadowgraph technique, the convective flow takes the form of two reciprocal cellular patterns that have mean number of edges per cell and mean number of edges per vertex equal to four. The analysis of the topological properties of the spoke pattern shows that its peculiar fourfold sided domains and the fourfold coordination of vertices stem from the rather general condition of having two dual cellular patterns that possess the same topological properties. This is demonstrated by means of Euler Theorem. In addition, we show that each of the two cellular patterns can be well described by means of a Voronoi diagram up to a definite length scale. Quite remarkably, the two diagrams actually obey a Mutual Voronoi Relation (MVR), where each diagram is modeled by the Voronoi diagram generated by vertices of the other diagram. The MVR imposes symmetries for the position of the vertices of the two diagrams. That is, the cross correlation length between the position of the different species is narrowly distributed around a dominant value, while the distribution of distances among vertices of the same species shows a less pronounced correlation peak falling in correspondence of the first minimum of the cross correlation function. A similar metric behaviour is reported for molten salts, where an ion is surrounded by alternating shell of unlike and like ions, thus suggesting a qualitative similarity between the two systems. S.M. acknowledges partial support from the European Space Agency}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Mazzoni, S. and Giavazzi, F. and Cerbino, R. and Giglio, M. and Vailati, A.}, biburl = {https://www.bibsonomy.org/bibtex/27a844428888a38e52936a36b23354a54/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {348d4cfdeac032bbfe75c81e6f988402}, intrahash = {7a844428888a38e52936a36b23354a54}, keywords = {diagrams formation pattern statphys23 topic-4 voronoi}, month = {9-13 July}, timestamp = {2007-06-20T10:16:31.000+0200}, title = {Mutual Voronoi diagrams in spoke pattern convection}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=862}, year = 2007 }