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Poster Communications Year : 2018

## Définir et mesurer l'asymétrie

Michel Petitjean

#### Abstract

Asymmetry means the lack of symmetry while dissymmetry means the lack of some symmetry elements, usually the lack of indirect symmetry: in this sense, dissymetry means chiral. E.g., an object having a mirror symmetry is achiral. Although the concepts of symmetry and chirality are known since a long, they did not receive an unifying mathematical definition until recently [1,2], and chirality was erroneously thought to be definable only in oriented spaces since more than one century, when the terms chiral and chirality were introduced by Lord Kelvin. Then, numerous attempts to define quantitative measures of asymmetry were done since the end of the 19th century, particularly for chirality [3]. Direct symmetry, i.e. combinations of rotational symmetry and translational symmetry, was quantified in [4]. Indirect symmetry was quantified in [5]: the chiral index defined there is null if and only if the object has indirect symmetry. It can be applied either to molecular objects or to multivariate probability distributions. For these latter the chiral index is an asymmetry coefficient which offers a major advantage over the well known skewness of Karl Pearson and its multivariate analogs because the skewness can vanish for some classes of chiral distributions. References: [1] Petitjean M. A definition of symmetry. Symmetry Cult. Sci., 2007, 18[2-3], 99-119. https://hal.archives-ouvertes.fr/hal-01552499 [2] Petitjean M. Chirality in metric spaces. In memoriam Michel Deza. Optim. Letters, 2018. DOI 10.1007/s11590-017-1189-7 [3] Petitjean M. Chirality and symmetry measures: a transdisciplinary review. Entropy, 2003, 5[3], 271-312. [4] Petitjean M. On the root mean square quantitative chirality and quantitative symmetry measures. J. Math. Phys. 1999, 40[9], 4587-4595. [5] Petitjean M. Chiral mixtures. J. Math. Phys., 2002, 43[8], 4147-4157.

#### Domains

Group Theory [math.GR] Metric Geometry [math.MG]

### Dates and versions

hal-01778387 , version 1 (25-04-2018)

### Identifiers

• HAL Id : hal-01778387 , version 1

### Cite

Michel Petitjean. Defining and measuring asymmetry. First European Asymmetry Symposium (FEAS), Mar 2018, Nice, France. , 2018. ⟨hal-01778387⟩

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