Describing chirality: Tutorial

Chirality is well defined in every dimension

If we were to live in a flat world, like inside a plane, the notion of chirality would still exist. Furthermore, its definition would almost be unchanged. We would simply carry out reflections through straight lines instead of planar mirrors, as in the following example:

Triangle (a)	Reflection line	Triangle (b)

Taking triangle (a) and carrying out the reflection through the line, we get triangle (b). The triangle has been somehow inverted in the process, just like a left hand is converted into a right hand by mirror reflection. As a result, triangle (a) and triangle (b) cannot be superimposed by any displacement in the plane. Just try and check. Remember that moving triangles out of the plane is excluded as we agreed to live, hence to perform any movement, inside the plane.

Trying to superimpose (a) and (b)

Conclusion: triangle (a) is chiral as a two-dimensional (2D) object. Let us say that it is 2D chiral. Note that triangle (b) also is 2D chiral. Next, consider the following equilateral triangle:

Triangle (c)	Reflection line	Triangle (d)

Now, surprise, surprise, we can superimpose (c) and (d)! Rotate (d) clockwise by 90° and move it over (c). This works fine:

(c) and (d) can be superimposed

Conclusion: (c) and (d) are one and the same object. So triangle (c) is 2D achiral.

Chirality can be similarly defined in every conceivable dimension; As far as principles go, chirality is not related to dimension three in any way. Since describing chirality is simpler in the 2D case, most of this tutorial is devoted to the 2D case. Chirality refers to 2D chirality unless otherwise stated.

Our first radial loop

Look at the 2D field schematized below ("A 2D field"). The field is made of 4 gaussians. One is centered at the origin A. The other three are put along 90°, 210° and 330° axes (with respect to Ax), but at the distances 3, 2.5 and 2 from A. The upper gaussian is twice as large as the two others. This field is 2D chiral. Note for chemists: Doesn't it look like the 2D analog of a tetrasubstituted carbon atom?

Now look at the curve shown on the righ-hand side. This is the first-order absolute radial loop, or ARL, of the field.

A 2D field	A radial loop derived from the field

Absolute radial loops are original tools deeply connected with gauge geometries. ARLs are extracted from fields by a simple mathematical operation to be discussed below. While in time you will be able to compute your own ARls, for the time being simply take them for granted. Our first aim is to get familiar with their properties.

The ARL is drawn in the complex plane. Red dotted lines intersect at the complex plane origin, zero (do not confuse zero with A, the origin in the physical plane). The 'loop' in 'radial loop' derived from the fact that ARLs always start at the origin and get back there after some wandering in the plane. ARLs have a cusp at zero when the starting slope is different from the return slope, as examplified above.

Radial loops are run in a definite direction; they are oriented. The arrow alongside the loop indicates the loop's orientation.

Rotating the field

Let us rotate the previous field clockwise by 30°. How does its first-order ARL look now?

If the field is rotated by 30°...	... the radial loop also rotates

Answer: the ARL is also rotated, although anticlockwise. The rotation angle is -30°, that is the opposite of the field rotation; More on this later. The capital point is that shape doesn't change.

Reflecting the field

If radial loops simply rotate when the field is rotated, what if we change the field to its reflection with respect to the Ay axis?

The reflected field...	...has a symmetrical radial loop

Chemists say that the original field and its reflection are enantiomers. So, the moral is that enantiomer fields are represented by symmetrical ARLs. Radial loops provide a graphical representation of enantiomerism.

Radial loops of achiral fields

An achiral field is basically a field that is identical to its reflection, possibly up to rotation. In other words, reflection doesn't generate a new object. So, we expect that the radial loop of an achiral field is somewhat identical to its reflection. In fact, theory shows that radial loops of achiral fields degenerate to line segments containing the origin 0:

If the field is achiral...	...the radial loop turns to a line segment

This is no accident. The following theorem is a basic result of the gauge description of chirality [3]: A field is achiral if and only if all its radial loops degenerate to line segments.

Why 'degenerate'? Because ARLs naturally spread somehow in the plane, so that they span some two-dimensional surface area. On the contrary, achiral fields ARLs shrink to one-dimensional line segments. Did we say 'all' radial loops? Yes, two series of radial loops - absolute (ARL) and relative (RRL) radial loops - can be obtained from every field. All these loops have similar properties, so we can safely postpone a detailed discussion and focus on first-order ARLs.

Continuous evolutions

As soon as we pull one of the two lower gaussians apart from A, the field becomes chiral. What if the lower left gaussian were pulled by a small extent?

Pulling apart one gaussian...	...the ARL spreads to a narrow loop

Answer: the ARL spreads in the complex plane, reflecting the fact that the field now is chiral. However, the ARL changes continuously, so it remains quite narrow, close to the previous line segment.

Diastereomers and enantiomers

Chemical reactions