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”
The notion of the informational measure of symmetry is introduced according to: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>s</mi><mi>...
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| Autores principales: | , , , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/3fe85a9c12b845ca999f44a9b6eddc73 |
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| Sumario: | The notion of the informational measure of symmetry is introduced according to: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msubsup><mstyle mathsize="70%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>G</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mi>l</mi><mi>n</mi><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>G</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>G</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the probability of appearance of the symmetry operation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> within the given 2D pattern. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is interpreted as an <i>averaged</i> uncertainty in the presence of symmetry elements from the group <i>G</i> in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>D</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo></mrow></semantics></math></inline-formula> 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>s</mi><mi>y</mi><mi>m</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. 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. |
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