Questions about the Nature of Chirality (with some focus on dimensionality)

Question

Are all chiralities the same? (Not in the sense of "is the right hand the same as the left hand?" but in the sense of "is the way in which X is chiral the same (or negative of the same) as the way in which Y is chiral?") Can all chiralities be described (by the right hand rule) as right handed or left handed? What are the pertinent features of chirality? (Please answer this question in the case of three dimensions specifically and then dimension-generally.)

Is the (fact of the) non-superposability of the reflection of X with itself (being true) equivalent to the fact that X is chiral? And can a chiral object be relieved of its chirality by embedding it in higher dimensions? For example, could my three-dimensional hands be superposed via reflection (or whatever) in four (or more) dimensions (resulting in their not being chiral any longer)? In what dimensions can chirality occur (my guess: just the ones in which the cross product is defined)?

Answer 1

Chirality can be defined in any dimension (the condition is non-superposability of the reflection about a hyperplane):

http://www.chirality.org/research.htm

http://www.chirality.org/twodim.htm

Adding another dimension, you can reverse a chiral figure.

Essential reference: The Ambidextrous Universe.

Answer 2

Chirality is not a well-defined concept, its use depends on the context. Here are just three definitions, one in geometric and the other two in topological context. In what follows, $r: R^n\to R^n$ denotes the reflection in the hyperplane $x_n=0$. (I could have taken any other hyperplane, definitions will not change.) I use the notation $G$ for the full group of rigid motions (Euclidean isometries) of $R^n$ and $G_+<G$ its orientation-preserving subgroup.

Note also that one can only talk about chirality or achirality of a subset of $R^n$, not the "different" chiralities. This is due to the fact that the group $G$ has exactly two connected components (for every $n\ge 1$); the reflection $r$ (and reflection in any hyperplane) belongs to the cnnected component of $G$ which is different from $G_+$.

Definition 1. (Geometric) A subset $X\subset R^n$ is achiral if there exists an element $g\in G_+$ such that $g\circ r(X)=X$. A subset $X$ is called chiral if it is not achiral.

The existence of chiral subsets of $R^n$ (for each $n\ge 1$) can be proven as follows. Observe that if we take the standard basis in $R^n$ (regarded as a vector space) $e_1,...,e_n$, then there is no orientation-preserving isometry $g$ of $R^n$ which fixes the origin, and such that $g(e_i)=r(e_i)$, $i=1,...,n$. Now, define $X$ to be the union of closed balls centered at the points $0, e_1,...,e_n$ and having radii $0, 2^{-1}, 2^{-2},...,2^{-n}$ respectively. Then $X$ is chiral (follows from the above elementary observation, since every isometry $h=g\circ r: X\to X$ would have to preserve centers of the above balls).

Chirality can be also defined in the topological context:

Definition 2. A subset $X\subset R^n$ is called topologically achiral if there exists a homeomorphism $g: R^n\to R^n$ isotopic to the identity such that $g\circ r(X)=X$. A subset of topologically chiral if it is not topologically achiral.

Again, there is only one notion of chirality in this context, due to the (much more difficult) fact that the group of self-homeomorphisms $R^n\to R^n$ has exactly two connected components (one containing the identity and the other containing $r$. The example given above is chiral but topologically achiral. There are examples of topologically chiral 9say, compact) subsets of $R^n$, but proofs are no longer elementary.

One can also impose the smoothness assumption in Definition 2 and require $g$ to be a diffeomorphism of $R^n$. You still have just one notion of chirality (proving this is even more difficult). To make things more interesting, you can consider subsets of the unit $n$-sphere $S^n\subset R^{n+1}$ and diffeomorphisms $S^n\to S^n$.

Before defining the relevant chirality, note that the group $Diff_+(S^n)$ of orientation-preserving diffeomorphisms $S^n\to S^n$ has finitely many cnnected components, their number $N_n$ can be $\ge 1$ (for instace, if $N_6=28$. Pick elements $h_1,...,h_N$ from these components and define smooth reflections $r_i=h_i\circ r$, where $r: S^n\to S^n$ is the isometric reflection in the equatorial subsphere in $S^n$ given by the equation $x_{n+1}=0$.

Definition 3. For $i=1,...,N$, a subset $X\subset S^n$ is $i$-achiral if there exists a diffeomorphism $g: S^n\to S^n$ isotopic to the identity such that $g\circ r_i(X)=X$. A subset which is not $i$-achiral is $i$-chiral.

Thus, we now have $N$ different notions of chirality!