Deep dive

 

In the course of reading reference 1 about the world of synchronisation, I came across the trefoil. Which looked ordinary enough, but I found that the trefoil comes in two varieties. Two trefoils which are not quite identical, with one being the mirror image of the other. This rather niggled me: there must be some better way, some more mathematical way of saying that two objects are not quite identical without having to resort to talk of mirrors. Which might, after all, for example in the Garden of Eden, not be known, let alone available. I decided to investigate, an investigation which occupied most of the next couple of days and turned up all manner of interesting stuff.

The first step was to review what we mean here by two objects in three-dimensional space being identical. Is it enough to say that two objects are identical if there is some distance preserving map from one onto the other, with the distance being taken from some metric space in which both objects are embedded? In this context, ordinary, three-dimensional space. Then, given that, in practise, all measurements are approximations and that we can only make some finite number of them, how exact does this distance preserving have to be?

And what about the special case of mapping an object onto itself in some non-trivial way. The sort of mapping derived, for example, by a 90° clockwise rotation of a square about its centre. Such a mapping preserves the shape and position of an object but does not preserve all the features of an object, in particular features like its corners and edges, except in aggregate. A corner will map onto a corner, but not the same corner. Corners are not invariant. In fact, depending on the object, there may be plenty of such mappings but no invariance, no invariant points, no fixed points at all.

The next step was reference 2, informative, but complete with an irritating rotating image. I remember from the world of work the tendency of the IT geeks to do clever stuff on public computers, just because they could, even though every else found this stuff irritating rather than useful.

Where we get the terms chiral, clockwise, left- and right-handed and ambient isotropic. I took the last of these first, starting with reference 3. Not terribly helpful, although reminding me of the homotopy theory which I have failed to get to grips with, despite having first been exposed to it something over half a century ago. So I moved onto the rather more helpful reference 4. From which I learn that, roughly speaking, in two dimensions, a figure is achiral if and only if it possesses an axis of symmetry. Which result generalises, with qualification, to the three dimensions of present interest. Which I gloss by saying that an axis of symmetry allows one to leak from rotations to reflections, to expand the world of rotations to take in reflections. Our trefoil has no such axis and is, therefore, chiral.

And from there to some work by one Michel Petitjean, to be found at references 5 and 6, a little marred by lapses in English, presumably the second or subsequent language rather than the first. That apart, he defines chirality in terms of the properties of the group of isometries of a metric space, a group which inherits properties from that metric space. A group which is closely related to the point group which is known to Wikipedia. 

He manages without mirrors at all: at least, the mirrors have been translated into a special sub-group of his starting group. Glossing again, the mirror images are isomorphisms which do not have square roots. Unlike the regular isomorphism, reflections cannot be expressed as the sum of fractional parts, or, in multiplicative rather than additional jargon, as the product of the square root with itself. You either do a mirror image or you don’t; there are no stations on the way. They are fundamentally different.

And the objects for which chirality is so defined are more or less arbitrary functions defined on the metric space. In the simple case, I think our object is the set of points of that space for which the function takes the value 1 and all other points take the value 0. I dare say Petitjean is interested in other possibilities. He is also interested in defining chirality in all kinds of spaces which are not Euclidean and in which the concept of orientation does not apply. Working out what this orientation is will have to wait for another day.

Note that chirality is like clockwise in the sense that it all depends on the point of view. An object which is chiral in two dimensions may not be in three; an object which is chiral in three dimensions may not be in four.

Another branch of inquiry concerned chirality of more or less complicated chemical molecules. It seems that chiral pairs of chemicals are almost identical, in the sense of having identical properties, of working in the same way – but not quite. A subject of great interest, not least to the sort of chemists who study the chemical goings on in and around the cells making up the human body.

Other matters

I came across one Clifford Dowker at reference 2, a mathematician, then at Birkbeck College, with whom I once had fleeting contact. I had thought of him as an algebraic topologist, while Wikipedia has him down for his work on knots, of which the trefoil is one of the simplest examples. Perhaps the simplest example.

Given that clockwise and anti-clockwise seemed to be mixed up with all this, I wondered where that came from too. Even going so far as to ask Google’s Bard about it, a product which has since been replaced by Gemini. I thought that his reply was quite helpful in pointing up various concepts, basics and problems, even if what he said was not particularly accurate. One of the basics being that something can only be clockwise or anti-clockwise from a point of view. I also learn, for example, that it was not true, as I had thought, that the notion of clockwise rotation (or not) was straightforward enough in the case of a spiral galaxy.

There had been another dive a few days previously. That one was all about Seifert surfaces, manifolds and Hausdorf spaces. Housdorf measures being something else with which I once had fleeting contact.

Digressing, about 18 months ago I started keeping a dream diary, as noticed at reference 7, a diary in the form of an Excel worksheet, now running to just over 700 rows. A second report is due, but to close today, I just offer a curiosity from that dream world. The other morning, I woke early, at around 03:00, to a reasonably complicated dream, a dream which as is often the case with me is strong on setting the scene, on context, but weak on narrative, on action. I did get up, but I did not note the dream down at that time, leaving that until around 08:00 when I woke up for the second time. At that point, some of the dream was still there, but I very much had the sense that what I was remembering this second time around was based on my verbalisation of the dream the first time around, rather than the dream itself. Words rather than images. And I could not say what the verbal content of the dream itself was.

I might add that while the interaction between the phenomenon and the collection of data about it is not as strong as that reported for the Descriptive Experience Sampling at reference 8, it is still there. The business of observing does disturb that which is being observed.

Conclusions

It all serves to help keep retired people out of the pub and otherwise out of mischief!

References

Reference 1: Sync – Steven Strogatz – 2003.

Reference 2: https://en.wikipedia.org/wiki/Trefoil_knot. 

Reference 3: https://en.wikipedia.org/wiki/Ambient_isotopy. 

Reference 4: https://en.wikipedia.org/wiki/Chirality. 

Reference 5: A definition of symmetry – Michel Petitjean – 2017. Searching for the string ‘PMP.SCS_2007’ will turn it up. Petitjean  is a recently retired Parisian mathematician.

Reference 6: Chirality in metric spaces: In memoriam Michel Deza – Michel Petitjean – 2017. Searching for the string ‘s11590-017-1189-7’ will turn it up.

Reference 7: https://psmv5.blogspot.com/2022/10/dream-diary-first-report.html. 

Reference 8: https://psmv3.blogspot.com/2017/01/progress-report-on-descriptive.html.