An iterative seed finding algorithm quantifies Voronoiness

Cellular structures produced by natural processes are typically regarded as Voronoi tessellations, albeit without rigorous mathematical justification. Generating Voronoi tessellations from given seed points is well established, and there are complex algorithms that find the seed points given a Voronoi tessellation. Here, we introduce a simple iterative algorithm that finds seeds of Voronoi tessellations with arbitrary precision. The algorithm is robust against degenerate vertices and converges to a unique solution for non-Voronoi tessellations as well. Using this property of the algorithm, we introduce the notion of “Voronoiness”. We can quantify Voronoiness for each cell as a unitless number. The value starts at 1 (representing a perfect Voronoi cell) and decreases monotonically as the tessellation deviates from the ideal structure. We show that Voronoiness deviation arising from small linear transformations of the ideal honeycomb structure mimics the deviatoric stress of its mechanical counterpart. We propose that spatial quantification of Voronoiness could be beneficial for understanding cellular structures without assuming they are ideal Voronoi tessellations. As a case study, we generate spatial maps of Voronoiness of various insect wing venations. The resulting maps show that secondary venation patterns can have significant deviation from the ideal Voronoi tessellation.