# Continuous Symmetry Measure: an old work

I decided to finally post an old work I did in science. The actual working paper is CSM_paper. It did never got peer reviewed for different reasons but it still deserves some notice. It is a bit off topic from the usual Unix and/or programming subjects you find in this blog. Nevertheless you can find some patterns, techniques or the like.

It is indeed interesting for the more IT oriented guys, for subjects like image recognition and fuzzy comparison. The approach here is pure scientific with the exception that there is no scientific connection or background beside a pure statistical approach. For the IT guys I suggest to follow D.Avnir work who has more dedicated papers.

The main subject is: Continuous Symmetry Measure (CSM). The term, I think, was defined in 1992 by D.Avnir in *J.Am.Chem.Soc.* 7843 (114) 1992 (Site PDF). Please follow D.Avnir pubblications on the subject. A second important publication I deal with in this blog and paper comes from 1998 from S.Grimme in Chem.Phys.Lett. 15 (297) 1998 where he includes some quantum properties in the measure.

Here below I will present a very reduced comment on the paper. An important question I was trying to answer is if there exist a reference object when we introduce a/the measure or it is an intrinsic property of the object. From this work it comes out that we have ideed an intrinsic reference object which depends on our mathematical construction and definition of symmetry and more in general of measure. If you are interested just read further.

The paper extracts.

## Abstract

In this contribution we address the problem of near symmetry and present a Continuous Symmetry Measure (CSM) based on the electron density function. In particular we propose an algorithm which generalizes the formalism proposed by Avnir (Encyclopedia of computational Chemistry, pp 2890 ,*J.Am.Chem.Soc.* 7843 (114) 1992)toward an exploitation of point symmetry groups properties. Correlations with existing definitions of CSM are discussed and an implementation which uses a simplified electron density of the proposed approach is presented. Advantages and disadvantages of the different approaches are reported.

## Introduction

You will need to read the PDF version for a proper introduction. The main points are the definition of “near” symmetry and/or a distance measure from symmetry. A side problem but very important in life science is chirality, and it is discussed within the method.

## Description of the CSM model

It is introduced the “symmetriser”. It works perfectly with any continuous 3d quantity.

The idea is simply to apply all the symmetry operators of a given symmetry group to a random 3d continuously defined density. If the system is continuos the result object is simply “symmetrised”.

This object is then defined to set up a measure or symmetry. There is indeed some background information on the PDF paper.

## Relation with previous CSM definitions

There are 2 other approaches. One is discrete the other uses a wave function.

The discrete approach does not differ from this one with the exception that using discrete values need a classification of the “points” (the discrete part). The classification can be minimised and defined via algorithms in advance but gives also space to specific patterns (in this case it is disjoint from the described approach).

The wave function approach has to be the same thing, as the density is defined by the wave function. The problem is that the wave function itself has no 3d properties if not expressed in some measure. The symmetry (the 3d version) till now is not a measure. Better, this is what we try to find it out. The Grimme approach lacks of scientific approach to a 3d geometrical problem. As we all are far away from any hamiltonian form of measure.

## Implementation

Beside a solid scientific background there are some statistical data which drive us to try an implementation. As we try a fast approach I propose this method: Gaussians. We define the object to be made of the best gaussian fit. 3D continuous, but also single points represented as gaussian. The Algorithm works in the continuous space. The integration of gaussians is a simple numerical problem.

The code is given within the freemol framework on github.

## Applications

Read D.Avnir work. My current applications are trends. You can always read the paper.