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On the root mean square quantitative chirality and quantitative symmetry measures

Abstract : The properties of the root mean square chiral index of a d-dimensional set of n points, previously investigated for planar sets, are examined for spatial sets. The properties of the root mean squares direct symmetry index, defined as the normalized minimized sum of the n squared distances between the vertices of the d-set and the permuted d-set, are compared to the properties of the chiral index. Some most dissymetric figures are analytically computed. They differ from the most chiral figures, but the most dissymetric 3-tuples and the most chiral 3-tuples have a common remarkable geometric property: the squared lengths of the sides are each equal to three times a squared distance vertex to the mean point.
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Michel Petitjean. On the root mean square quantitative chirality and quantitative symmetry measures. Journal of Mathematical Physics, American Institute of Physics (AIP), 1999, 40 (9), pp.4587-4595. ⟨10.1063/1.532988⟩. ⟨hal-02122820⟩

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