Generalization of the continuous symmetry measure: the symmetry of vectors, matrices, operators and functions

2009 ◽  
Vol 11 (42) ◽  
pp. 9653 ◽  
Author(s):  
Chaim Dryzun ◽  
David Avnir
Keyword(s):  
Continuous Symmetry ◽  
Symmetry Measure

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

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1271
Author(s):  
Jiri Neustupa ◽  
Yvonne Nemcova
Keyword(s):  
Morphometric Analysis ◽  
Outer Surface ◽  
Honeycomb Structure ◽  
Small Scale ◽  
Continuous Symmetry ◽  
Caco3 Content ◽  
Symmetry Measure ◽  
Perfect Symmetry ◽  
Lateral Margins ◽  
The Individual

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.

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

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1659
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel ◽  
Nir Shvalb ◽  
Shraga Shoval
Keyword(s):  
Symmetry Group ◽  
Voronoi Diagrams ◽  
Continuous Measure ◽  
Penrose Tiling ◽  
Continuous Symmetry ◽  
Measure Of Symmetry ◽  
Symmetry Measure ◽  
Seed Points

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.

Continuous Symmetry Measure vs Voronoi Entropy of Droplet Clusters

2021 ◽  
Vol 125 (4) ◽  
pp. 2431-2436
Author(s):  
Mark Frenkel ◽  
Alexander A. Fedorets ◽  
Leonid A. Dombrovsky ◽  
Michael Nosonovsky ◽  
Irina Legchenkova ◽  
...  
Keyword(s):  
Continuous Symmetry ◽  
Symmetry Measure

scholarly journals Symmetry aspects in the macroscopic dynamics of magnetorheological gels and general liquid crystalline magnetic elastomers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Harald Pleiner ◽  
Helmut R. Brand
Keyword(s):  
Electric Fields ◽  
Time Reversal ◽  
Liquid Crystalline ◽  
Second Law Of Thermodynamics ◽  
Time Reversal Symmetry ◽  
Local Conservation ◽  
Continuous Symmetry ◽  
Polar Order ◽  
Preferred Direction ◽  
Symmetry Properties

Abstract We investigate theoretically the macroscopic dynamics of various types of ordered magnetic fluid, gel, and elastomeric phases. We take a symmetry point of view and emphasize its importance for a macroscopic description. The interactions and couplings among the relevant variables are based on their individual symmetry behavior, irrespective of the detailed nature of the microscopic interactions involved. Concerning the variables we discriminate between conserved variables related to a local conservation law, symmetry variables describing a (spontaneously) broken continuous symmetry (e.g., due to a preferred direction) and slowly relaxing ones that arise from special conditions of the system are considered. Among the relevant symmetries, we consider the behavior under spatial rotations (e.g., discriminating scalars, vectors or tensors), under spatial inversion (discriminating e.g., polar and axial vectors), and under time reversal symmetry (discriminating e.g., velocities from polarizations, or electric fields from magnetic ones). Those symmetries are crucial not only to find the possible cross-couplings correctly but also to get a description of the macroscopic dynamics that is compatible with thermodynamics. In particular, time reversal symmetry is decisive to get the second law of thermodynamics right. We discuss (conventional quadrupolar) nematic order, polar order, active polar order, as well as ferromagnetic order and tetrahedral (octupolar) order. In a second step, we show some of the consequences of the symmetry properties for the various systems that we have worked on within the SPP1681, including magnetic nematic (and cholesteric) elastomers, ferromagnetic nematics (also with tetrahedral order), ferromagnetic elastomers with tetrahedral order, gels and elastomers with polar or active polar order, and finally magnetorheological fluids and gels in a one- and two-fluid description.

Phase Transitions and Continuous Symmetry Breaking

1976 ◽  
Vol 36 (14) ◽  
pp. 804-806 ◽  
Author(s):  
J. Fröhlich ◽  
B. Simon ◽  
T. Spencer
Keyword(s):  
Phase Transitions ◽  
Symmetry Breaking ◽  
Continuous Symmetry

scholarly journals 300 nm bandwidth adiabatic SOI polarization splitter-rotators exploiting continuous symmetry breaking

2015 ◽  
Vol 23 (15) ◽  
pp. 19261 ◽  
Author(s):  
Luciano Socci ◽  
Vito Sorianello ◽  
Marco Romagnoli
Keyword(s):  
Symmetry Breaking ◽  
Polarization Splitter ◽  
Continuous Symmetry
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