A question of H.G. Wells' mathematics
Wells’s explanation seems perfectly correct to me. Think of $\mathbb R^3$ as embedded into $\mathbb R^4$ by $(x,y,z)\mapsto(x,y,z,0)$. Then apply, in four-space, the rigid rotation $$ R_\theta\colon\quad \pmatrix{1&0&0&0\\0&\cos\theta&0&-\sin\theta\\0&0&1&0\\0&\sin\theta&0&\cos\theta}\,. $$ Here $\theta=0$ gives the identity, and $\theta=\pi$ gives a $180^\circ$-rotation in four-space, which sends $(x,y,z,0)$ to $(x,-y,z,0)$, thus exchanging left and right.
This reminds me of this answer of mine, which explores chirality in higher (and lower) dimensions.
Firstly, let's use the definition of dimension1 of an object as the last value of $n$ such that the object or a rotated copy of it can be said to cover an open subset of $\mathbb R^n$.
Now, the concept of "chirality" is where a reflection cannot be composed of rotations.
One can show that if an $n$ dimensional object is taken and place/projected in $m$ dimensional space for $n\neq m$ and viewed from the larger of the two, the object is achiral.
Yes, there are chiral objects in 2D and 1D. Take the following sequence of dots:
... .. .
If it is living in a 1D space, it cannot be "turned around". This means that these two mirror images:
... .. .|. .. ...
are not the same object.
In two dimensions, take a look at these two mirror images:
X | X
H-+ | +-H
T | T
Note that one cannot rotate one into the other if one is confined to two dimensions (i.e., we cannot make the object "jump" out of the page).
Now, to get a bit more mathematical. This is not going to be rigorous, but I'll try to be convincing.
I'm only going to consider $180^\mathrm{o}$ rotations along a coordinate axes ($R_{x_i}$) and reflections along coordinate planes ($F_{x_i}$), but this can be generalized. Let us note these things of the two operations:
- A rotation $R_{x_i,x_j}$ or $R_{ij}$ has the following properties (check these out for yourself):
- It takes the object and flips the signs of $x_i$ and $x_j$ of all of the points on the object.
- It is commutative (note: I'm only talking about 180 degree rotations here)
- Two rotations of the same kind lead to an identity operation $I$
- $R_{ij}R_{jk} = R_{jk}$
- A flip $F_{x_i}$ or $F_i has the following properties: - It flips the sign of $x_i$ only - It is commutative - Two flips of the same kind lead to an identity operation - $F_iF_j = R_{ij}$
Now, for an object to be chiral, we should not be able to chalk up any flip as a composition of rotations. Let us take an $n$ dimensional object in $m$ dimensional space.
If $n=m$, for an asymmetric object, the above is not necessary. Note that any composition of rotations can be reduced down to something where there is at most one of each kind. In three dimensions, this gives us the options $I, R_{xy},R_{yz},R_{zx}, R_{xy}R_{yz},R_{yz},R_{zx}R_{yz},R_{xy}R_{zx}, R_{xy}R_{yz}R_{zx}$, or $8$ combinations. However, using $R_{ij}R_{jk} = R_{jk}$ we can reduce the rotations to the $4$ compositions $I, R_{xy},R_{yz},R_{zx}$
However, the total number of sign flip combinations (i.e. all the ordered tuples obtained by selectively flipping the signs of coordinates) is $2^3=8$. But we can obtain all $8$ by composing flips and rotations. Given a sign flip combination: If there are an even number of coordinates with flipped signs, we just compose the $R_{ij}$s corresponding to pairs of coordinates which have their sign flipped. If there are an odd number, we ignore one of the sign flips, do the same thing as in the even sign flip case, and append a $F_{j}$ around the originally ignored coordinates.
This means that there are objects belonging to the set formed by composing flips and rotations that do not belong to the set formed by just composing rotations. Chirality exists.
This can be easily generalized, we can show that in any number of dimensions, we can form all sign flip combinations ($2^n$) by composing flips and rotations, but only half by composing only rotations as, for any composition of $\left\lfloor\frac{n}{2}\right\rfloor +1$ or more rotation objects, there will be at least a pair of objects sharing an index 2, so we can always use $R_{ij}R_{jk} = R_{jk}$ to reduce the composition to one of less than or equal to $\frac{n}{2}$ objects.
However, if the object is lesser-dimensional, this no longer works.
In this case, we can always align one of the "extra" axes of the object with $x_1$. Now, $x_1=0$ for all points on the object. This makes the operation $F_{1}$ impotent (and equivalent to the identity operation $I$). Now, we had shown that we can make the $2^N$ unique sign flip combinations by composing a number of rotations and possibly a single flip. Now, we can show that the composition of a number of rotations and a flip is simply a composition of rotations, as we can multiply the former by $F_{1}=I$ and then reduce the composition of two flips to the composition of multiple rotations.
Thus, a lower dimensional object in a higher dimensional space cannot be chiral.
1. There are many ways to define dimension which don't always give the same result. My favorite is the Lebesgue dimension which is a pretty beautiful concept, but it's not what I need here.
2. By pigeon hole principle — there are $n$ possible indices and $n+1$ (if $n$ is odd) or $n+2$ total indices from the composition of $\left\lfloor\frac{n}{2}\right\rfloor +1$ rotation objects each with a nonequal pair of indices
I think the key here is that a flat object living purely in the $xy$-plane is not affected by reflection in $z$. Thus, an apparent reflection across the $x$-axis can be achieved by reflecting across both the $x$ and $z$ axes — this is a net rotation and hence can be achieved by continuous transformation without breaking orientability.
The same principle would apply to a 3D object embedded in $\mathbb R^4$. Having no width along the fourth dimension makes one invariant under reflections across that dimension.
Martin Gardner's The Ambidextrous Universe discusses this and related issues (mathematical, physical, chemical, anatomical...) in detail, with Gardner's inimitably readable style.
Jeffrey Weeks's The Shape of Space investigates the geometry and topology of $3$-manifolds in more mathematical detail, yet in a friendly, engaging way.
As Thomas Andrews notes, it's possible for a traveller to get "reflected" by traversing a closed loop in a non-orientable universe. The impossibility of doing so in $\mathbf{R}^3$ (or in any orientable $3$-manifold) amounts to disconnectedness of the set of linear frames and the definition of orientability.