Symmetry, Chirality, Symmetry Measures and Chirality Measures: General Definitions © Michel Petitjean, 2022 Author's professional address: INSERM ERL U1133 (BFA, CNRS UMR 8251), Université Paris 7 35 rue Hélène Brion, 75205 Paris Cedex 13, France.petitjean.chiral@gmail.com Formerly (2010-2018): MTi, INSERM UMR-S 973, Université Paris 7. Formerly (2007-2009): CEA/DSV/iBiTec-S/SB2SM (CNRS URA 2096), Saclay, France Formerly (1987-2006): ITODYS, CNRS UMR 7086, Université Paris 7.

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3-armed rhombic spirallohedron. Image file communicated the 23 January 2006 by Russell Towle, who discovered the spirallohedra and their space filling properties (Symmetry: Culture and Science, 2000,11[1-4],293-306).

Despite that symmetry is a concept known since millenaries, I could not find a general definition of symmetry in the literature, and the connection of group theory with symmetry, which is known since a long, did not seem deeply investigated. Furthermore, the role of metric spaces in a general mathematical definition of symmetry was neglected. It is why I produced my own definition of symmetry [1].
Objects are functions having their input argument taking values in a metric space. The space of the returned values can be any one. Objects are transformed via bijective isometries over this metric space. A symmetry is detected when an object is identical to one of its non trival transforms.
A rigorous presentation of the full mathematical definition of symmetry can be found in my paper, downloadable in PDF format (ref. [1]). It is aimed to cover a broad spectrum of situations, where several specific definitions were previously required: geometric figures (with or without colors), functions, matrices, graphs, probability laws, sequences of symbols (e.g., digits, words), etc. The metric space is not required to be Euclidean. Examples were explicited for graphs and tilings, at the occasion of an analysis of the local symmetry concept (ref. [2]).

Summary of basic definitions (for chirality and symmetry measures, see further):
1. An object is a function having its input argument in a metric space S.
(that can be extended to the case where the metric is not a true one)
2. An isometry is a distance preserving bijection of S onto S.
(the set of isometries is the group of distance preserving bijections of S onto S)
3. An object is symmetric when it is invariant under an isometry which is not the identity.
4. An isometry is direct when it can be expressed as a product of squared isometries.
An isometry which is not direct is an indirect isometry.
5. An object is achiral when it is invariant under an indirect isometry.
If no indirect isometry leaves an object invariant, it is chiral.
The historical definition of chirality of Lord Kelvin is based on the assumption that the space is orientable. It is an intuitive definition which is suitable for the Euclidean spaces, but which is sometimes not fully understood: see the case of the mirror paradox [3]. The orientability of the space is by no way required to define chirality: according to my own definition [4,5], chirality can be defined in any metric space, only with the help of group theory.
In the case of Euclidean spaces, this modern definition of chirality (published in open access in ref [5]), is equivalent to the historical one of Lord Kelvin. It is summarized on the slides of my communication at the Symmetry Festival 2021 (download: see ref. [6]; see also my poster presented at the First European Asymmetry Symposium, which can be downloaded: see [7]). This latter definition works in the classical spacetime [8], and it has been extended to the Minkowski spacetime [9]. In this spacetime, it was shown that the commutative composition of parity reversal with time reversal is an indirect isometry [9]: it is the opposite of what could be expected in Euclidean space. The same conclusion applies to the Dirac spinor field [10]).
How works our modern definition in quadratic spaces is explained in [11] (in particular, see Theorem 13).
The case of translations is considered in [12].

A summary of group theory concepts useful in symmetry and chirality definitions is available in the PDF of my lecture at the Symmetry Festival 2009 in Budapest: see [13].

It is crucial to understand that mathematical constructs are models of real physical situations, and that a mathematical model of symmetry is a simplified image in our mind of some physical situation in which we would like to see symmetry [14]. Following this idea, a real situation in which we would like to recognize some symmetry, can be modelized by several mathematical models of symmetry. This is exemplified in ref [14], where the relations of antisymmetry with symmetry and chirality are discussed.

Symmetry (and chirality) can be measured as a continuously varying quantity (e.g., the skewness of a distribution).
Thus, we need a general way to define symmetry measures.
The following text results from the conversion in html of a part of the latex source of my open access review on symmetry measures [15].

First step: we consider a set E of objects, provided that the equality between any two objects is rigorously defined. Note that the definition of "equality" may be simple or not, depending of what objects are considered: distributions, labelled graphs, lattices,...

Second step: we consider a set of operators {T}, and the product between two operators is defined, mapping {T}*{T} on {T}. The set {T} has a group structure for this product, and the neutral element is I. The group operates on E, offering the associativity property: T1(T2X)=(T1T2)X, the neutral element I being such that IX=X. Note that this model is not adequate for indirect symmetry, since improper isometries have no group structure and no neutral element.

Definition: an object X in E is symmetric if and only if there exists T not equal to I such that TX=X. In other words, symmetry arises when an object is identical to one of its non-trivial transforms.

Some immediate properties follow. Denoting by T-1 the symmetric element of T, a symmetric object X is obviously such that T-1X=X, and it is proved by recurrence that TnX=X for any signed integer n. When X is symmetric for both operators T1 and T2, then T1T2X=T2T1X=X, meaning that T1 and T2 operate commutatively on X.

Third step: the space E is metrized with a distance d. Thus, an object X in E is symmetric if and only if there exists T not equal to I such that d(TX,X)=0.

Fourth step: consider the quantity Inf{T not equal to I} d(TX,X), and normalize it, if possible. Intuitively, quasi-symmetry is related to a small distance between the object and one of its non-trivial transforms [16], the smallness being evaluated by the normalizing factor, such as an upper bound of the distance, if any. This fourth step fails when it happens that Inf{T not equal to I} d(TX,X)=0 despite the fact there is no T not equal to I such that d(TX,X)=0. This failure cannot occur when the set {T} is of finite cardinality, but many direct symmetry models involve an infinite number of isometries.

A similar problem arises with skew indirect symmetry. An object having skew symmetry is the image of a symmetric object through a full rank linear transform A-1. Consider an imperfectly skew symmetric object for which A is specified: its degree of symmetry could be defined as being that of its transformed image through A. Now, evaluating indirect skew symmetry without knowledge of A leads one to consider the lower bound of some chiral index, taken either over the set of all full rank transforms, or more specifically, over a subset of this latter such that there are at least two vectors of A allowed to have arbitrarily close directions. Since A may be arbitrarily close to some non-full rank transform, the lower bound of the chiral index is null, provided that this chiral index has the adequate continuity property (it is built to have it). It should be noted that considering a finite set of transforms avoids this problem, and that a skew symmetry index defined without considering any linear transform is just an ordinary symmetry index.

REFERENCES

1. PETITJEAN M.
A Definition of Symmetry.
Symmetry: Culture and Science 2007, 18[2-3], 99-119 (Zbl 1274.58003)
Download PDF paper from the HAL repository: hal-01552499 (copy deposited with permission from Symmetrion).

2. PETITJEAN M.
Global Symmetries, Local Symmetries and Groupoids.
Symmetry 2021, 13[10], 1905 (open access paper: DOI 10.3390/sym13101905)

3. PETITJEAN M.
Angew. Chem. Int. Ed. 2020, 59[20], 7650-7651 (DOI 10.1002/anie.201904314)
Angew. Chem. 2020, 132[20], 7722-7723 (DOI 10.1002/ange.201904314)

4. PETITJEAN M.
Chirality in Metric Spaces.
Symmetry: Culture and Science 2010, 21[1-3], 27-36 (Zbl 1274.58004)

5. PETITJEAN M.
Chirality in Metric Spaces. In Memoriam Michel Deza.
Optim. Letters 2020, 14[2], 329-338 (DOI 10.1007/s11590-017-1189-7)
This paper is an enhanced version of the one of ref. 5.

6. PETITJEAN M.
Chirality defined: The end of the even/odd paradigm.
Talk presented at the Symmetry Festival, Sofia, Bulgaria, July 9-12, 2021.

7. PETITJEAN M.
Defining and measuring asymmetry.
Poster presented at the First European Asymmetry Symposium,
15-16 March 2018, Université Nice Sophia Antipolis, Nice, France.
(abstract)

8. PETITJEAN M.
Molecular Chirality in Classical Spacetime: Solving the Controversy about the Spinning Cone Model of Rotating Molecules.
Chem. Eur. J. 2020, 26[47], 10648-10652 (DOI 10.1002/chem.201904247)

9. PETITJEAN M.
Symmetry 2019, 11[10], 1320 (open access paper: DOI 10.3390/sym11101320)

10. PETITJEAN M.
Chirality of Dirac Spinors Revisited.
Symmetry 2020, 12[4], 616 (open access paper: DOI 10.3390/sym12040616)

11. PETITJEAN M.
Chirality in Geometric Algebra.
Mathematics 2021, 9[13], 1521 (open access paper: DOI 10.3390/math9131521)

12. PETITJEAN M.
Chirality in Affine Spaces and in Spacetime.
arXiv:2203.04066 [math-ph], 2022.

13. PETITJEAN M.
Chirality in Metric Spaces.
Symmetry Festival 2009, 31 July - 5 August, Budapest, Hungary.