Symmetry, Chirality, Symmetry Measures and Chirality Measures: General Definitions

© Michel Petitjean, 2017
Author's professional address:
MTi, INSERM UMR-S 973, Université Paris 7
35 rue Hélène Brion, 75205 Paris Cedex 13, France.
Formerly: CEA/DSV/iBiTec-S/SB2SM (CNRS URA 2096), Saclay, France
(and formerly: ITODYS, UMR 7086, CNRS, Université Paris 7).

Related topics:

The C3v nut (1)
© 2013 Michel Petitjean

The C3v nut (2)
© 2013 Michel Petitjean

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, and 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 may be found in my paper, downloadable in PDF format (ref. [1]).
This definition 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.

The historical definition of chirality is based on the assumption that the space is orientable. The orientability of the space is by no way required to define chirality: chirality can be defined in any metric space, only with the help of group theory [2,3].

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.
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 [4].
The reader is also welcome to visit the Symmetry website (numerous links), and learn more about symmetry and the symmetry community.

Symmetry 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 [5].

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 [6], 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.


    A Definition of Symmetry.
    Symmetry: Culture and Science 2007,18[2-3],99-119.
    (Zbl 1274.58003)
    Download PDF paper from here or from the HAL repository: hal-01552499
    (both copies were posted with permission from Symmetrion)

    Chirality in Metric Spaces.
    Symmetry: Culture and Science 2010,21[1-3],27-36.
    (Zbl 1274.58004)
    Download PDF paper (posted with permission from Symmetrion)

    Chirality in Metric Spaces. In Memoriam Michel Deza.
    Optim. Letters 2017 (Open Access paper).
    (DOI 10.1007/s11590-017-1189-7)
    This paper is an enhanced version of the one of ref. 2.

    Chirality in Metric Spaces.
    Symmetry Festival 2009, 31 July - 5 August, Budapest, Hungary.
    Download PDF file of the lecture.

    Chirality and Symmetry Measures: A Transdisciplinary Review.
    Entropy 2003,5[3],271-312 (Open Access paper).
    (DOI 10.3390/e5030271; Zbl 1078.00503)

  6. ROSEN J.
    Symmetry in Science. An Introduction to the General Theory.
    Chap. 6.1. Springer-Verlag: New-York, 1995.