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”.

Bormashenko, Edward; Legchenkova, Irina; Frenkel, Mark; Shvalb, Nir; Shoval, Shraga

doi:10.20944/preprints202109.0076.v1

Cited by 4 publications

(1 citation statement)

References 13 publications

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“…The Voronoi entropy gives information on the orderliness of the set of points. Bormashenko et al [11] compared the CSM and the Voronoi entropy specifically for a Penrose tiling and pointed out that these symmetry measures are not objective.…”

Section: Introduction 1motivationmentioning

confidence: 99%

Symmetry Measure of Truss Structures with Disturbed Higher-order Symmetries

Módis,

Kovács

2024

Period. Polytech. Civil Eng.

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This paper covers aspects of quantifying the symmetry of two- and three-dimensional elastic bar-and-joint structures. The concept of symmetry as a quantitative property instead of a binary question of 'yes' or 'no' is widely accepted and thoroughly investigated, for example, in molecular physics but also in engineering sciences, mainly in chemical engineering. Similarly to most of the articles written on this topic, our method is also based on the comparison of specific metrics of the analyzed structure and a reference one, i.e., which possesses the desired (perfect) symmetry. The deviation of the analyzed (imperfect) structure from the reference structure is quantified by one scalar. The novelty in our approach is that we consider not just the relative position of the nodes but also the normal stiffness of the truss members, even for structures with higher-order, i.e., polyhedral symmetries. For both geometric and material properties to be accounted for, the eigenvalues of the stiffness matrix were chosen as metrics. The difficulty lies in finding the reference structure which will be carried out based on energy principles.

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Section: Introduction 1motivationmentioning

confidence: 99%

Symmetry Measure of Truss Structures with Disturbed Higher-order Symmetries

Módis,

Kovács

2024

Period. Polytech. Civil Eng.

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Shannon (Information) Measures of Symmetry for 1D and 2D Shapes and Patterns

et al. 2022

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In this paper, informational (Shannon) measures of symmetry are introduced and analyzed for patterns built of 1D and 2D shapes. The informational measure of symmetry Hsym(G) characterizes the averaged uncertainty in the presence of symmetry elements from group G in a given pattern, whereas the Shannon-like measure of symmetry Ωsym(G) quantifies the averaged uncertainty of the appearance of shapes possessing a total of n elements of symmetry belonging to group G in a given pattern. Hsym(G1)=Ωsym(G1)=0 for the patterns built of irregular, non-symmetric shapes, where G1 is the identity element of the symmetry group. Both informational measures of symmetry are intensive parameters of the pattern and do not depend on the number of shapes, their size, and the entire area of the pattern. They are also insensitive to the long-range order (translational symmetry) inherent for the pattern. Additionally, informational measures of symmetry of fractal patterns are addressed, the mixed patterns including curves and shapes are considered, the time evolution of Shannon measures of symmetry are examined, the close-packed and dispersed 2D patterns are analyzed, and an application of the suggested measures of symmetry for the analysis of the chemical reaction is demonstrated.

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Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings

Bormashenko

Legchenkova

Frenkel

et al. 2023

Entropy

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We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.

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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”.

Cited by 4 publications

References 13 publications

Symmetry Measure of Truss Structures with Disturbed Higher-order Symmetries

Symmetry Measure of Truss Structures with Disturbed Higher-order Symmetries

Shannon (Information) Measures of Symmetry for 1D and 2D Shapes and Patterns

Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings

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