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.

scholarly journals Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”.

2021 ◽  
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel ◽  
Nir Shvalb ◽  
Shraga Shoval
Keyword(s):  
Symmetry Operation ◽  
Continuous Measure ◽  
Penrose Tiling ◽  
Measure Of Symmetry ◽  
2D Pattern ◽  
Equilateral Triangles ◽  
The Given

The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.

scholarly journals Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2146
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel ◽  
Nir Shvalb ◽  
Shraga Shoval
Keyword(s):  
Symmetry Operation ◽  
Continuous Measure ◽  
Penrose Tiling ◽  
Measure Of Symmetry ◽  
2D Pattern ◽  
Equilateral Triangles ◽  
The Given ◽  
The Ideal

The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

scholarly journals The Numerical Measure of Symmetry for 3D Stick Creatures

2008 ◽  
Vol 14 (4) ◽  
pp. 425-443 ◽  
Author(s):  
Wojciech Jaśkowski ◽  
Maciej Komosinski
Keyword(s):  
Background Information ◽  
Continuous Measure ◽  
Mathematical Language ◽  
Evolutionary Processes ◽  
3D Objects ◽  
Measure Of Symmetry ◽  
Symmetry Measure ◽  
Prefer Symmetry

This work introduces a numerical, continuous measure of symmetry for 3D stick creatures and solid 3D objects. Background information about the property of symmetry is provided, and motivations for developing a symmetry measure are described. Three approaches are mentioned, and two of them are presented in detail using formal mathematical language. The best approach is used to sort a set of creatures according to their symmetry. Experiments with a mixed set of 84 individuals originating from both human design and evolution are performed to examine symmetry within these two sources, and to determine if human designers and evolutionary processes prefer symmetry or asymmetry.

MULTIFRACTAL CASCADE SYMMETRY MODEL FOR FULLY DEVELOPED TURBULENCE

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850070
Author(s):  
G. C. LAYEK ◽  
SUNITA
Keyword(s):  
Structure Function ◽  
Symmetry Group ◽  
Velocity Structure ◽  
Random Parameter ◽  
Continuous Symmetry ◽  
Space Filling ◽  
Support Set ◽  
Developed Turbulence ◽  
Symmetry Model ◽  
Velocity Structure Function

We report a symmetry model for turbulence intermittency. This is obtained by the compositions of continuous symmetry group transformations of statistical turbulent spectral equation at infinite Reynolds number limit. Flow evolution under group compositions yields velocity structure function exponents that depend on the dilation symmetry group parameter [Formula: see text] [Formula: see text] and a random parameter [Formula: see text]. The random parameter [Formula: see text] is associated with energy distribution. Since the correction to the space-filling Kolmogorov cascade is small, the value of [Formula: see text]. The asymptotic structures are filaments having dimension one, so [Formula: see text] is found to be related with [Formula: see text] by [Formula: see text]. The present model therefore depends only on [Formula: see text], and [Formula: see text] can be ascertained uniquely for [Formula: see text]. It is found that the velocity structure function exponents [Formula: see text], [Formula: see text] in present symmetry model agree well with the existing experimental, direct numerical simulation results and different phenomenological models for [Formula: see text]. For these values of [Formula: see text], the correction to Kolmogorov space-filling, universal [Formula: see text] law, belongs to the range [Formula: see text], and the fractal dimension for the support set lies in [Formula: see text].

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 Symmetry and Shannon Measure of Ordering: Paradoxes of Voronoi Tessellation

2019 ◽  
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel
Keyword(s):  
Voronoi Tessellation ◽  
Voronoi Diagrams ◽  
Random Patterns ◽  
Seed Points ◽  
Symmetric Patterns

Voronoi entropy for the random patterns and patterns demonstrating various elements of symmetry are calculated. The symmetric patterns are characterized by the values of the Voronoi entropy very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points, constituting the pattern. The extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables revealing of the elements of symmetry of the pattern.

scholarly journals Symmetry and Shannon Measure of Ordering: Paradoxes of Voronoi Tessellation

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 452 ◽  
Author(s):  
Bormashenko ◽  
Legchenkova ◽  
Frenkel
Keyword(s):  
Voronoi Tessellation ◽  
Voronoi Diagrams ◽  
Random Patterns ◽  
Seed Points ◽  
Symmetric Patterns

The Voronoi entropy for random patterns and patterns demonstrating various elements of symmetry was calculated. The symmetric patterns were characterized by the values of the Voronoi entropy being very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points constituting the pattern. Extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables the elements of symmetry of the patterns to be revealed.

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
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