On the root mean square quantitative chirality and quantitative symmetry measures

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 spat ial sets. The properties of the root mean squares direct symmetry index, de fined as the normalized minimized sum of the n squared distances between th e vertices of the d-set and the permuted d-set, are compared to the propert ies of the chiral index. Some most dissymetric figures are analytically com puted. They differ from the most chiral figures, but the most dissymetric 3 -tuples and the most chiral 3-tuples have a common remarkable geometric pro perty: the squared lengths of the sides are each equal to three times a squ ared distance vertex to the mean point. (C) 1999 American Institute of Phys ics. [S0022-2488(99)01009-9].