DOI 10.1140/epje/i2010-10676-1 Regular Article Eur. Phys. J. E 33, 369–375 (2010) T HE EUROPEAN P HYSICAL JOURNAL E Morphometry and structure of natural random tilings A. Hoˇ cevar 1, a , S. El Shawish 1 , and P. Ziherl 1,2 1 Joˇ zef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia 2 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Received 23 April 2010 and Received in final form 4 August 2010 Published online: 25 November 2010 – c  EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2010 Abstract. A vast range of both living and inanimate planar cellular partitions obeys universal empirical laws describing their structure. To better understand this observation, we analyze the morphometric pa- rameters of a sizeable set of experimental data that includes animal and plant tissues, patterns in desiccated starch slurry, suprafroth in type-I superconductors, soap froths, and geological formations. We character- ize the tilings by the distributions of polygon reduced area, a scale-free measure of the roundedness of polygons. These distributions are fairly sharp and seem to belong to the same family. We show that the ex- perimental tilings can be mapped onto the model tilings of equal-area, equal-perimeter polygons obtained by numerical simulations. This suggests that the random two-dimensional patterns can be parametrized by their median reduced area alone. 1 Introduction Irrespective of their origin, many natural spatial parti- tions that can be regarded as polygonal tilings of a plane share the same structure [1]. Planar animal and plant tis- sues, magnetic froths, convection patterns, and geological formations [1–4] are all characterized by identical topo- logical correlations of neghboring polygons described by the Aboav-Weaire law [1]. Less universal are the empiri- cal and mutually exclusive Lewis [1,2] and Desch law [5] which state that the average polygon area and perimeter, respectively, are linear functions of the number of vertices: Many tilings obey the former and certain others are de- scribed by the latter. These seemingly simple rules apply to very diverse tilings across a range of length scales [1,6]. The various aspects of the observed topological and morphological universality of random two-dimensional cel- lular patterns have been interpreted in various ways. The purely topological models revolve around the number of vertices as the most apparent feature of polygons in a tiling [7] and use maximum-entropy arguments to re- produce the above laws [8–11]. Related concepts have been explored in statistical-mechanical theories based on polygon geometry [12]. In biological contexts the struc- ture of planar cellular partitions is often associated with cell division [13–15] although it can also be described by an entropy-maximizing, equilibrium model [16]. De- spite all these efforts, the link between the specific mi- croscopic mechanisms at work and the universal structure of the tilings remains poorly understood. In particular, it a e-mail: ana.hocevar@ijs.si is unclear which parameters characterize the tilings and whether their structure is determined kinetically or not. To provide an alternative insight into the common fea- tures of the tilings, we explore the relation between the structure of a tiling and its morphometry encoded in the distribution of the polygon reduced area a = 4πA L 2 , (1) where A is the polygon area and L is its perimeter 1 . The largest value a can adopt is 1, which corresponds to a circle, whereas for a regular polygon with n edges a =(π/n) cot(π/n)= a max (n). The reduced area of an irregular n-gon must be smaller than a max (n); the larger the shape anisotropy of a n-gon (quantified, e.g., by its aspect ratio) the smaller its reduced area. For any given reduced area a, one can construct polygons with n such that a max (n) >a. Apart from this constraint, the reduced area itself does not restrict the number of vertices in a polygonal shape. To the best of our knowledge, the distributions of the reduced area have barely been used to quantify the struc- ture of polygonal tilings. The only context where reduced 1 This parameter has been used in materials science [17, 18]. In other studies a is sometimes referred to as the circular- ity [19], the shape factor [20], the isoperimetric ratio or quo- tient [21] or the shape index [22]; but shape factor is also used for S = L/2 √ πA − 1=1/ √ a − 1 [23] or L 2 /4πA =1/a [24] and shape index has also been defined as πA/2d 2 where d is the cell diameter [25]. To avoid ambiguities, we use the phrase “reduced area” to state that a measures the area of polygons.