On quantitative assessment of chirality: right-sided and left-sided geometric objects

Capa

Citar

Texto integral

Acesso aberto Acesso aberto
Acesso é fechado Acesso está concedido
Acesso é fechado Acesso é pago ou somente para assinantes

Resumo

Two methods for quantitatively assessing the chirality of a set are considered, the first of which uses the calculation of the area of their symmetric difference of two sets as a measure of the discrepancy between them, and the second uses the Hausdorff distance between them. It is shown that these methods, generally speaking, do not provide a correct quantitative estimate for a fairly wide class of sets, such as bounded Borel sets. Using the example of flat triangles and convex quadrangles, the problem of dividing geometric objects into right-handed and left-handed is considered. For triangles, level lines of two versions of the chirality measure were calculated on the plane of the angular parameters. For a spatial spiral, the values of two versions of the chirality index are found, based respectively on the calculation of the mixed product of vectors and the Hausdorff distance between two sets.

Texto integral

Acesso é fechado

Sobre autores

Yu. Kriksin

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences

Autor responsável pela correspondência
Email: kriksin@imamod.ru
Rússia, Moscow

V. Tishkin

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences

Email: v.f.tishkin@mail.ru

Corresponding Member of the RAS

Rússia, Moscow

Bibliografia

  1. Guye P.-A. Influence de la constitution chimique des dérivés du carbone sur le sens el les variations de leur pouvoir rotatoire // Compt. Rend. (Paris) 1890. V. 110. P. 714–716. http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3066&I=766&M=tdm
  2. Gilat G. Chiral coefficient-a measure of the amount of structural chirality // J. Phys. A Math. Gen. 1989. V. 22. P. L545–L550. https://doi.org/10.1088/0305-4470/22/13/003
  3. Gilat G. On quantifying chirality – Obstacles and problems towards unification // J. Math. Chem. 1994. V. 15. P. 197–205. https://doi.org/10.1007/BF01277559
  4. Zimpel Z. A geometrical approach to the degree of chirality and asymmetry // J. Math. Chem. 1993. V. 14. P. 451–465. https://doi.org/10.1007/bf01164481
  5. Zabrodsky H., Avnir D. Continuous Symmetry Measures. 4. Chirality // J. Am. Chem. Soc. 1995. V. 117. P. 462–473. https://doi.org/10.1021/ja00106a053
  6. Petitjean M. About second kind continuous chirality measures. 1. Planar sets // J. Math. Chem. 1997. V. 22. P. 185–201. https://doi.org/10.1023/A:1019132116175
  7. Petitjean M. Chirality and Symmetry Measures: A Transdisciplinary Review // Entropy 2003. V. 5. № 3. P. 271–312. https://doi.org/10.3390/e5030271
  8. Petitjean M. Chirality in metric spaces // Optim Lett. 2020. V. 14. P. 329–338. https://doi.org/10.1007/s11590-017-1189-7
  9. Dryzun C. Avnir D. Chirality Measures for Vectors, Matrices, Operators and Functions // ChemPhysChem. 2011. V. 12. P. 197–205. https://dx.doi.org/10.1002/cphc.201000715
  10. Mezey P.G. Chirality Measures and Graph Representations // Coputers Math. Applic. 1997. V. 34. № 11. P. 105–112. https://doi.org/10.1016/S0898-1221(97)00224-1
  11. Buda A.B., Auf der Heyde T.P.E., Mislow K. On Quantifying Chirality // Angewandte Chemie. 1992. V. 31. № 8. P. 989–1007. https://doi.org/10.1002/anie.199209891
  12. Buda A.B., Mislow K.A. Hausdorff chirality measure // J. Am. Chem. Soc. 1992. V. 114. № 15. P. 6006–6012. https://doi.org/10.1021/ja00041a016
  13. Buda A.B., Mislow K. On Geometric Measures of Chirality // J. Mol. Struct. 1991. V. 232. P. 1–12. https://doi.org/10.1016/0166-1280(91)85239-4
  14. Fowler P.W. Quatification of chirality: attempting the impossible // Symmetry: Culture and Science. 2005. V. 16. № 4. P. 321-334. https://symmetry.hu/oldsite/content/fowler-05-4.pdf
  15. Rassat A., Fowler P.W. Any Scalene Triangle Is the Most Chiral Triangle // Helvetica Chimica Acta. 2003. V. 86. P. 1728–1740. https://doi.org/10.1002/hlca.200390143
  16. Osipov M.A., Pickup B.T., Fehervari M., Dunmur D.A. Chirality measure and chiral order parameter for a two-dimensional system // Molecular Physics 1998. V. 94. № 2. P. 283-287. https://dx.doi.org/10.1080/002689798168150
  17. Kriksin Y.A., Potemkin I.I., Khalatur P.G. Chirality in Self-Assembling Rod-Coil Copolymers: Macroscopic Homochirality Versus Local Chirality // Polymer Science, Series C. 2018. V. 60. Suppl. 1. P. S135–S147. https://dx.doi.org/10.1134/S1811238218020133
  18. Kitaigorodskii A.I. Organic Chemical Crystallography, NY: Consultants Bureau, 1961. 541 p.

Arquivos suplementares

Arquivos suplementares
Ação
1. JATS XML
Baixar
2. Fig. 1. Examples of chiral sets: a) isosceles triangle ABC (AC = BC) and an isolated point O on the extension OA of side AB; b) the left half of the isosceles triangle ABC (AC = BC) (shaded in gray) consists of points with rational coordinates, and the right unshaded half contains all the points included in it.

Baixar (11KB)
Metadados
3. Fig. 2. Triangle ABC and its reflection, triangle A1B1C1 (the set that forms the symmetric difference of these triangles is colored gray): a) sides AB and A1B1 lie on the same line, and triangle A1B1C1 is the reflection of triangle ABC relative to the perpendicular to side AB, passing through the geometric center O of both triangles; b) the reflected triangle A1B1C1 is rotated around point O counterclockwise by an angle φ.

Baixar (11KB)
Metadados
4. Fig. 3. The region of admissible values of the triangle’s angular parameters α > 0, β > 0, α + β < π. Light subregions “1”, “3” and “5” correspond to positive values, and dark subregions “2”, “4” and “6” to negative values of the triangle’s chirality index.

Baixar (9KB)
Metadados
5. Fig. 4. Level lines of numerical values of centered measures (indices) of chirality in the region (7): a) according to Kitaygorodskii: 1 − boundary of the region (7); 2 − ; 3 − ; 4 − ; 5 − ; 6 − ; 7 − ; “ʘ” denotes the point of global maximum , , ; b) according to Hausdorff: 1 − boundary of the region (7); 2 − ; 3 − ; 4 − ; 5 − ; “ʘ” denotes the point of global maximum , , .

Baixar (30KB)
Metadados
6. Fig. 5. Parameterization of a convex quadrilateral using four independent positive angular parameters: , , , .

Baixar (10KB)
Metadados
7. Fig. 6. Dependences of different versions of the chirality indices of the spatial helix (10) on the dimensionless parameter λ = h/r: 1 − ; 2 − ; 3 − ; 4 − (dashed line).

Baixar (12KB)
Metadados

Declaração de direitos autorais © Russian Academy of Sciences, 2024