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© 2002, 2003 Michel Petitjean

petitjean@itodys.jussieu.fr

The quantitative mathematics of geometric chirality (handedness), symmetry, and probability theory are presented. Geometric chirality differs from physical interactions between light and matter - chiroptical phenomena like optical rotation or circular dichroism - in which optical chirality (requiring only absence of a symmetry point of inversion) can occur absent geometric chirality (requiring absence of a point, planes, and improper axes of symmetry). Petitjean has published an exhaustive technical review of quantiatitve symmetry measure, available as a 330K pdf file.

(Return to the parity Eötvös experiment.)

This definition of chirality embraces the classical and intuitive ideas that both direct
and indirect symmetries are dichotomous properties of a Euclidean set. A set * is*
or

A two-dimensional square's four vertices in 3-space form an achiral set.
Perturbation of the vertices' coordinates can create a chiral figure. A
tetrahedron is the limiting three-dimensional array, again achiral. What
criterion identifies these four points as an achiral or chiral figure?
Any 4-tuple of points may be viewed as a perturbed square or a perturbed
regular tetrahedron. Are all 4-tuples of points both achiral and chiral? No!
Chirality is a * continuous* measure: a set is

A quantitative chiral index is valid for various situations in Euclidean space:

- Different points can have identical coordinates (weighted points). The weights can be any non-negative real value.
- The number of points is not limited. "Continuous" geometric domains and homogeneous solids are included.
- Chirality is sensitive to dimensionality. A scalene triangle is chiral in a plane but achiral in three-space. The chirality measure should be defined for any number of spatial dimensions.

The chiral index should be independent of selection of a particular reflection plane to generate the mirror image, and insensitive to any isometric transformation of the distribution. Coincidence of a distribution with its inverted image is tested by translation and rotation of the inverted image to best superpose the two distributions, then measurement of how similar they are.

Measuring chirality is one case of measuring the similarity of two classes of equivalence of distributions: the set of all translated and rotated images of the original distribution, and the set of all its translated and rotated and inverted images. Any probability metric measuring the distance between two distributions measures this similarity. Quantitative chirality measure is related to the theory of probability metrics [1] and the Monge-Kantorovitch transportation problem [2] for which the cost of transporting one mass distribution to another one must be minimized.

What is a "continuous" measure of chirality? Two "close" distributions are expected to have "close" chiral indices. When a sequence of distributions converges to a limit, the associated sequence of chiral indices should converge to the chiral index of the limiting distribution. Any theory dealing with a "continuous" measure of chirality should specify which kind of convergence is considered in the space of the chiral entities.

Consider chlorofluoromethane C-H-H-Cl-F. It is an achiral structure provided the first hydrogen of the original molecule is allowed to match the second hydrogen of the reflected molecule, and the second hydrogen of the original molecule is allowed to match the first one of the reflected molecule.

Consider a physical model wherein the molecule is a mixture of a negative charge distribution, a positive charge distribution, and a mass distribution. The degree of chirality of each distribution could be evaluated separately without weights (i.e., the ratios of the physical units), although the mixture has its own unique chirality content. This example repeats the previous example. When a color is located at N points, the distribution is concentrated on these N points. Alternatively... all the space has a specified color, but most points have a null weight and are invisible.

The formal model is a colored mixture [3]. It is a mixture of d-dimensional random vectors, each of them attached to a color. The complete space is the product of Euclidean space and the color space.

Color may be viewed as a supplementary value on a (d+1)^{th} pseudo axis
in a space having a numeric component and a non-numeric component. Probability metrics
operating on the space of function distributions are inadequate. When the physical
model includes the probability law of a random variable taking simultaneously
numerical and non-numerical values, metrics operating on the space of probability laws
are preferred. Extending the Wasserstein distance to colored mixtures provides a metric
sensitive to colors [3]. This is why the associated chiral index has been retained.

- X is a colored mixture of random vectors in R
^{d} - Y is a colored mixture distributed as an inverted image of X, and submitted to a rotation R and a translation t
- W is a joint distribution of X and Y such that the marginals of X and Y in the space of colors are fully correlated; the couple (X,Y) has a null probability when X and Y have different colors
- T is the inertia of X (or Y), and E denotes the mathematical expectation:
CHI = (d/4T) · Inf

_{{W,R,t}}E[(X-Y)^{t}·(X-Y)] - The chiral index depends only upon the probability law of X, and is insensitive to isometries.
- The normalizing coefficient (d/4T) is such that the chiral index is scale independant, and takes values onto the interval [0,1].
- The value 0 indicates that X is achiral.
- The optimal translation is known to be such that X and Y should have a null expectation.

CHI = (d/2) · (1 - [SupIn the unidimensional case, the chiral index depends only on the maximal correlation coefficient r between X and Y, Y being distributed as -X:_{{W,R}}(c_{1}+c_{2}+...+c_{d})]/T)

CHI = (1 - SupIf X and Y are identically distributed on the real line, the chiral index depends only on the minimal correlation r between X and Y:_{{W}}r)/2

CHI = (1 + InfConsider the uncolored situation in multidimensional space. X is now an ordinary random vector. The chiral index is a shape coefficient of the distribution. Most statisticians consider distributions such as the unidimensional gaussian to be symmetric. These are better described as achiral because achirality is related to indirect symmetry and because direct symmetry has no sense on the real line._{{W}}r )/2

The chiral index is a better asymmetry coefficient than the skewness because there are asymmetric distributions having a null skewness, just as for the symmetric (i.e. achiral) distributions. This problem does not occur with the chiral index, because its properties are induced from those of a probability metric.

The general colored mixture model is not universal. When the total mass of a system is infinite, such as for unbounded lattices, the model fails. Only sequences of finite lattices can be so considered. When the colored mixture model is adequate, the chiral index exists if and only if the inertia is finite and non-null - if the Cauchy distribution has no chiral index.

Null inertia occurs if and only if all the mass is concentrated at a single point. This trivial situation is not related to any achiral or chiral situation. It can occur either as the limit of a sequence of chiral or achiral distributions.

- The chiral index of an achiral set or an achiral colored mixture is null.
- The chiral index of the Bernoulli distribution with parameter p=1-q, equals 1-1/2p when p>1/2, and equals 1-1/2q when p<1/2. It is null when p=q=1/2.
- The chiral index of the colored Bernoulli distribution, such that the two weighted points have a different color, equals 1/2p when p>1/2, and equals 1/2q when p<1/2. It reaches the maximal value CHI=1 when p=q=1/2.
- The chiral index of a set of colored points in R
^{d}, such that no two of them have the same color, equals d·V_{d}/T, V_{d}being the smallest eigenvalue of the covariance matrix. These sets have the maximal chiral index CHI=1 when d=1, or when the covariance matrix is proportional to the identity matrix. This occurs when the points are vertices of a regular planar polygon, or are vertices of a full d-dimensional cube, regular d-octahedron, or regular d-simplex. - The chiral index of 3 uncolored equally weighted points on the real line
equals (1-a)
^{2}/4(1+a+a^{2}), where a is the ratio of lengths of adjacent segments. - The most chiral triangle, i.e. the most chiral set of 3 uncolored equally
weighted points in the plane, has a chiral index equal to 1-2/5
^{1/2}. This triangle pertains to a family of triangles having a particular geometrical property (see section VII-C in ref.[5]). - For the uncolored model and a fixed rotation, computing CHI is an instance
of the Monge-Kantorovitch transportation problem. Analytical results are available for
continuous unidimensional distributions having an invertible distribution function
(see chap.3 in ref. [2]). The chiral index of an exponential distribution equals
1-Pi
^{2}/12.

Computing the chiral index of a set of N equally weighted points, colored or not, is simple (see appendix 1 in ref. [7]). Computing CHI for the uncolored model is possible on a pocket calculator: compute the correlation coefficient between the set of values sorted in increasing order and the set sorted in decreasing order, add 1, then divide by 2. This correlation coefficient cannot be positive, and the chiral index has an upper bound of 1/2.

Consider N colored and weighted points in R^{d}. When the rotation is fixed
or when the optimal rotation is known, computing CHI is a linear programming
problem [4]. Also computing the rotation requires minimizing a linear function
under polynomial constraints: the linear constraints come from the joint density
and the non-linear constraints come from the rotation. When the joint density is
fixed or known, the optimal rotation is known in R^{2} [7], and in R^{3}
[5].

When the N colored points are equally weighted, computing the optimal joint density enumerates the permutations of the N points [6,7]. This is practically impossible above ten points with the same color. There are situations where the combinatoric can be drastically reduced. Chemists view the 3D molecular model of a conformer as a non-directed graph such that atoms are the vertices and edges are the chemical bonds. Atoms and bonds have their own colors. This graph induces constraints on the correspondences allowed between atoms. A carbon cannot match an oxygen, and a hydrogen of a methyl group cannot match a hydrogen of a hydroxy group. Among the N! correspondences (or permutations or joint densities), only those generated by the graph's automorphisms' enumeration are allowed [7]. The chiral index is thus easily computable for many heavy molecules [6]. As an example of reduction of the combinatoric, consider N equally weighted uncolored points, such that the N nodes of the graph define a single ring containing N edges. There are only 2N graph automorphisms, and the chiral index is computable even for large values of N.

The colored graph model is a generalization of the colored mixture model for which constraints on joint density are added to constraints coming from the colors themselves.

Freeware program QCM computes the chiral index of a conformer. It may be downloaded from the software page of ITODYS. Input of points' coordinates and colors (atom types) as a Hyperchem *.hin file allows a large lattice of points to be computed without regard to connectivity,

- if all points (atoms) are connected into a single graph, and
- if the points are not connected into a linear unbranched string.

A more general problem is quantitative evaluaiton of how different are the distributions of any two colored mixtures X and Y. The mixtures are assumed to share the same space of colors and, as previously, the couple (X,Y) offers a null probability to get different colors for X and Y. The similarity between the distributions is still expressed as a Wasserstein distance:

(InfWhen each of the two distributions is represented by a set of N colored points, the N colors being all different and the two sets of N colors being identical, there is a pairwise correspondence between the two sets of points. The distance is called the pure rotation Procrustes distance (a least squares method to superpose by rotation and translation two sets of N points pairwise associated [3]). The optimal translation is achieved by centering the sets. The optimal rotation is known in the plane (sect. 3 in ref.[7]) and in three-space (appendix of ref.[5]). When all colors are identical one obtains the pure rotation Procrustes algorithm without prefixed correspondence._{{W,R,t}}E[(X-Y)^{t}·(X-Y)])^{1/2}

Chirality is related to indirect symmetry. It has been quantified to evaluate how a set is similar to its inverted image, via an optimal superposition method. How can we quantify direct symmetry? How a set is similar to itself? For a finite number of equally weighted colored points, a direct symmetry index has been defined [5]. When the number of points increases toward infinity, the trivial optimal superposition of the distribution on itself is unavoidable. This problem may be reformulated as a local minimum problem, but this is much more difficult to solve.

A transdisciplinary review of methods dealing with chirality and symmetry measures has been published [8].

A relation between chirality and gravitation has been proposed, including a definitive experiment in existing apparatus using commercially available materials: the parity Eötvös experiment.

Rigorous mathematical description of chirality is new, complicated, subtle... and bears promise of great power and utility. If you have remarks, criticisms, suggestions, ideas; if you just wish to discuss chirality or symmetry, please email the mathematician, petitjean@itodys.jussieu.fr

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