Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

A continuous measure of symmetry and the Voronoi entropy of 2D patterns representing Voronoi diagrams emerging from the Penrose tiling were calculated. A given Penrose tiling gives rise to a diversity of the Voronoi diagrams when the centers, vertices, and the centers of the edges of the Penrose rhombs are taken as the seed points (or nuclei). Voronoi diagrams keep the initial symmetry group of the Penrose tiling. We demonstrate that the continuous symmetry measure and the Voronoi entropy of the studied sets of points, generated by the Penrose tiling, do not necessarily correlate. Voronoi diagrams emerging from the centers of the edges of the Penrose rhombs, considered nuclei, deny the hypothesis that the continuous measure of symmetry and the Voronoi entropy are always correlated. The Voronoi entropy of this kind of tiling built of asymmetric convex quadrangles equals zero, whereas the continuous measure of symmetry of this pattern is high. Voronoi diagrams generate new types of Penrose tiling, which are different from the classical Penrose tessellation.

Morphometric Analysis of Surface Utricles in Halimeda tuna (Bryopsidales, Ulvophyceae) Reveals Variation in Their Size and Symmetry within Individual Segments

Calcifying marine green algae of genus Halimeda have siphonous thalli composed of repeated segments. Their outer surface is formed by laterally appressed peripheral utricles which often form a honeycomb structure, typically with varying degrees of asymmetry in the individual polygons. This study is focused on a morphometric analysis of the size and symmetry of these polygons in Mediterranean H. tuna. Asymmetry of surface utricles is studied using a continuous symmetry measure quantifying the deviation of polygons from perfect symmetry. In addition, the segment shapes are also captured by geometric morphometrics and compared to the utricle parameters. The area of surface utricles is proved to be strongly related to their position on segments, where utricles near the segment bases are considerably smaller than those located near the apical and lateral margins. Interestingly, this gradient is most pronounced in relatively large reniform segments. The polygons are most symmetric in the central parts of segments, with asymmetry uniformly increasing towards the segment margins. Mean utricle asymmetry is found to be unrelated to segment shapes. Systematic differences in utricle size across different positions might be related to morphogenetic patterns of segment development, and may also indicate possible small-scale variations in CaCO3 content within segments.

Planar Robot Grasping Planning Based on Symmetry

One of the main challenges in robotic manipulation is to decide how to grasp objects. Symmetry has great significance for characterizing planar shapes for recognition, handling etc. Most of the existing methods use an approach suited for a class of shapes. In the proposed method the axes of symmetry are simultaneously searched using two approaches: as eigenvectors of the moment of inertia covariance matrix of the object and as directions that minimize a symmetry index of the contour of the object in polar coordinates. The best candidate is selected by a symmetry measure in Cartesian coordinates. The method was tested on various shapes, and then implemented in a robotic system with artificial vision. It proved to be efficient even in the case of slightly asymmetric shapes. The proposed method proved to be well suited for the symmetric objects usually grasped by the robots. The implementation must be adjusted for different classes of objects and lightning conditions using two threshold values.