*Introduction*

### Kelvin’s definition

“I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself.”

Lord Kelvin stated this celebrated definition in 1904, in his Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light. Presently, this statement is universally accepted as

thedefinition of chirality. Other statements simply transpose it in other contexts.Here are some features of chirality

in Kelvin’s framework:## Chirality is dichotomous

Take an arbitrary “geometrical figure.” Create its mirror image. Then try to match the two figures. There are only two possible outcomes: “they are identical” (the figure is achiral) or “they are not identical”(the figure is chiral). It works just like a toggle switch: the switch is either on or off. Accordingly, a chiral aminoacid like L-phenylalanine is just as chiral as the parent compound L-alanine, because Kelvin’s definition simply tells us that L-alanine and L-phenylalanine are

not achiral. The chiral objects are the objects that fail to be achiral.## Chirality belongs to the realm of geometry

Indeed, Kelvin himself spoke of “geometrical figures” and “groups of points.” Constructing mirror images is applying a plane symmetry. Comparing an object with its mirror image proceeds by rotating or translating them. Now, plane symmetry, rotations and translations are geometrical operations known as

isometries. The moral is that chirality is a purely geometrical notion. Moreover, in Kelvin’s framework chirality depends on a very special class of geometrical operations: isometries.Accordingly, whilst chirality indeed plays a tremendous part in organic chemistry or biochemistry, it is not related in any particular way with chemistry or biology. Kelvin’s definition applies to molecules just as it applies to knots and to any stuff that is “sufficiently geometrical” to be handled through isometries.

### Can we go beyond Kelvin’s definition?

Now, there is a long-standing tendency among chemists to consider some molecules “more” chiral than others. This intuition is based on a wealth of experimental evidence. Occasionally, but regularly, this idea is explicitly stated in research articles.

However, Kelvin’s definition definitely prohibits such statements. There lies a clear-cut and seemingly hopeless contradiction, and this contradiction originates in Kelvin’s definition itself. With no way out of this contradiction, it seems that the chemist’s intuition that molecules could be assigned different chiralities on a firm basis is to remain a dream.

### But yes, we can *describe* chirality

A way out of this contradiction was found in the years 95-98, and two (now four) research articles presenting this approach were published in 97 and 98. It was mathematically proved that chirality is amenable to a rich description for a large class of objects that includes molecules. This approach was called the

gaugedescription of chirality, because it is associated with a new class of geometries called gauge geometries. It may not be clear how a geometry could allow us to go further than Kelvin’s definition. Besides, what mathematicians callageometry may not be clear to everybody. Last but not least, what we mean by describing chirality is not evident at all at this point.Let us keep these questions open for the moment. Let us first look at the way this description works practically, in the following tutorial. Afterwards, explanations will come, first as a non-technical discussion, then as a more mathematically oriented presentation.

### Overview of the following pages

- TutorialA practical tutorial on the gauge description of chirality, in two dimensions for the sake of simplicity.
- DiscussionA formal, albeit non-technical discussion can be found here.
- Mathematical presentation (not written yet!)For afficionados of Lebesgue integration and Group theory only.