Describing chirality: 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."

This is the celebrated definition stated by Lord Kelvin in 1904, in his Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light. This statement is universally accepted as the definition of chirality. Equivalent statements simply transpose Kelvin's definition in other contexts, such as symmetry groups. Kelvin's definition illustrates the issue to be discussed here. Let's review some features of chirality in Kelvin's framework:

Chirality is a dichotomous property

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-alanine is just as chiral as the parent compound L-phenylalanine. Kelvin's framework simply tells us that L-alanine and L-phenylalanine are not achiral. Kelvin's definition is basically the definition of achirality. Chirality is only defined negatively; the chiral objects are the objects that fail to be achiral.

Chirality is a geometrical property

Kelvin himself spoke of "geometrical figures" and "groups of points" only. Thus, take a geometrical figure. Constructing its mirror image is applying the geometrical operation known as a plane symmetry. Comparing the initial object and its image proceeds by rotating or translating any one of them. Now, plane symmetry, rotations and translations are geometrical operations known as isometries. Thus, chirality is a purely geometrical property. Moreover, this geometrical property depends on a definite way of handling the objects considered: in Kelvin's framework, objects are handled through isometries. A first consequence is that chirality is not related in any particular way with chemistry or biology. Kelvin's definition applies to molecules just as it applies to knots and other mathematical constructs because they are "sufficiently geometrical" to be handled through isometries. Whereas chirality indeed plays a tremendous role in organic chemistry or biochemistry, chirality does not depend on carbon atoms properties.

Can we go beyond Kelvin's definition?

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, it must be emphasized that 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 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 gauge description 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 call a geometry 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. The best method probably is to first have a look at the way this description works practically. This is the object of 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

• Tutorial

  A practical tutorial on the gauge description of chirality, in two dimensions for the sake of simplicity.

• Discussion

  A formal, albeit non-technical discussion can be found here.

• Mathematical presentation (not written yet!)

  For afficionados of Lebesgue integration and Group theory only.