Why do chiral objects only come in pairs?

Question

My question arise and is connected to the "strange" fact that many things seem to come in pair or in number of two similar "objects".

Why are there chiral "pairs" and not groups of 3,4, or more? What happens in higher dimensions.

Answer 1

Related answer: https://math.stackexchange.com/a/532746/24293

Looking at the comments, you seem to be asking why there are chiral 'pairs' and not chiral multiplets. Looking at the tag, it looks like you want an analysis of higher dimensions as well.

TL;DR

Short answer: In any number of dimensions, chiral objects come in pairs. This is because numbers come in pairs; positive and negative. You could say that numbers come in reciprocal pairs ($x$ and $\frac{1}{x}$) as well, but the fact is, in the real world, only addition of coordinates has a meaning.

Chiral group?

This is my interpretation of your question:"In multiple dimensions, will we have a 'chiral group' instead of a chiral pair by doing multiple reflections over different planes?"

Let's see:

In the next few paragraphs, I'll explain it considering reflections along the coordinate planes only, and 180 degree rotations only. Intuitive generalisation is easy, but I don't want to try mathematical generalisation. Actually, all of this ought to be easily shown by group theory. Maybe someone else will post an answer using group theory later.

In the real world, there are two operations which we can do: rotation and translation. These two do not affect the object in question. Reflection does not affect the relative structure, but it may not keep the object the same (chirality). Affine transformations don't preserve the object in general; stretching/skewing changes the object. I'm neglecting translation and stretching here, and focusing on rotation/reflection. Since both of these (atleast for 180 degree rotations) only deal with sign-flips of coordinates, we can analyse these effectively:

Alright. A simple reflection can be defined as flipping the sign of one coordinate. Similarly, a simple rotation can be defined as flipping the sign of two coordinates. We know for sure that:

• A rotation does not change the object
• A single reflection can change an object.

Since we're dealing with chiral stuff only, we'll say that a single reflection always changes the object.

Now, any even number of sign flips can be represented as a sum of pure rotations. Thus, an even number of sign flips is rotation-like (does not change the object). An odd number of flips can be represented as a bunch of rotations plus a reflection. The rotations don't change the object, but the reflection does. So, an odd number of flips is reflection-like, i.e. the structure of the object remains the same, but the object is not the same (non-superimposible). Also, all reflection-like states will be identical; as they can be written as one reflection followed by n rotations (which keep it identical).

As you can see, I have split up the relavant transformations into TWO groups, and every transformation in a group gives the same result. So, we have TWO structurally identical non-superimposible objects.

I've not really assumed any dimension specific things. I've taken a few intuitive axioms like the coordinate-flip definitions of reflection/rotation and the fact that rotation/reflection don't/do change the object--these could be wrong in higher dimensions, but I doubt it--they're pretty basic and work in the lower dimensions. So, in any dimension, chiral objects come in pairs. Reflection-like transformations switch them.

Examples in lower dimensions

Believe it or not, we have chirality even in lower dimensions. Here's a 1D example:

... .. .|. .. ...

The | represents a mirror; and . is a 0D dot. In 1D, we can only have dots and horizontal lines (In fact, | doesn't exist, but I've drawn it to avoid confusion--the mirror itself is another dot). Here we have two mirrored dot-sequences. Here, within one dimension, we cannot rotate/translate the first to get to the other. In fact, rotation isn't defined for one dimension. We can only reflect it. On your 2D screen, you can see that this mirroring can be done via a 2D rotation.

In two dimensions, this can be chiral:

X   |   X
H-+ | +-H
T   |   T

The |s are mirrors again, the rest are just random symbols (ASCII art is not my forte). If you wish, draw the two 2D objects on small squares of paper. Within the plane of the table, you can't rotate the squares to show equivalent objects. But, you can flip one over (let's assume the paper is transparent), and get something equivalent to the other. This flipping requires taking it through a third dimension.

One can see the same thing happening here. It's a reflection in 2D, but a rotation in 3D.

In fact, a chiral object in 2D will be achiral in 3D, and the same holds true for a 2D projection of a 3D object, when viewed from the third dimension. More generally, an chiral object in $N$ dimensions projected into $M$ dimensions and viewed from the larger of the two dimensions will be achiral if $N\neq M$. So, each dimension can be said to have one unique type of chirality. We'll half-intuitively prove this in the next section.

Inductive generalization

Now, here's one argument for having more chiralities as the dimensions increase: with every dimension, we should get one more 'mirror' or reflection operation orthogonal to the new axis, so the number of elements in a chiral group should be $2^{N}$ in $N$ dimensions. This logic is correct except for one thing: when we increase the number of dimensions, we also turn a reflection into a rotation, as shown in the examples. So, we simultaneouslty gain and lose a degree of chirality. The net result is that there is no change in the number of chiralities. Now, in 1D, we have only one degree of chirality (there's only one reflection, and no rotation), so, by induction, there is only one degree of chirality for any number of dimensions. One degree of chirality corresponds to one reflection, which corresponds to a PAIR of objects.

Diastereomers

Something worth mentioning is that, in chemistry, we have something akin to a "chiral group". Such a group consists of (optical) diastereomers. Diastereomers are compounds which can be broken up into structurally identical chiral bits (as in, for each bit in one compound, there is a corresponding structurally identical bit in one of its diastereomer). These chiral bits are arranged in a structurally identical manner in the diastereomers. Now the question is, are all the diastereomers structurally identical? After all, they consist of a structurally identical arrangement of structurally identical units. The answer is NO. Let's take a non-chemistry example, since I don't want to start drawing Fischer projections(a way of turning 3D chirality into 2D chirality).

You have two cars which are identical except one is American, the other is British (a chiral pair due to the position of the wheel). You also have two small, identical models of another make of car (again, it can be of either type). Your mission, should you choose to accept it, is to glue a model car to the driver's seat of a large car, facing forwards. Note that you have four possible combinations. Also note that the constituent bits are structurally equivalent (both model cars are equivalent, the same goes for the large cars). They are arranged in a structurally identical way (one is stuck to the driver seat of the other, and faces a particular direction). So they're basically diastereomers,

But, these guys are not all structurally equivalent. In one pair of them, the drivers side window of the model is close to the drivers side window of the large car. In another pair, the drivers side window is closer to the stickshift of the main car. So they're not structurally equivalent. So you've just seen that this apparent chiral group is just a pair of chiral pairs, which are not structurally equivalent to each other.

Bottom line

The two comes from:

• Reflection can give a pair of two unidentical objects
• Numbers come in pairs with their negatives
• Multiple reflections can be reduced to one reflection.

Man, this question was fun to answer =D. I wonder why it was neglected...