Question about symmetry groups in 2D, and how symmetric transformations generalize to higher dimensions

I am currently taking an introductory abstract algebra course, and learning about group theory, etc. One thing we have covered is symmetry groups of simple geometric shapes in 2D, and to a cursory extent in 3D.

Our definition for the symmetries of a 2D group (i.e. a square) includes rotations ($r, r^2, r^3, \ldots$), a horizontal flip ($h, h^2$), and compositions of the two ($rh, r^2h, \ldots$)

In looking briefly at a 3D cube, we are asked to consider two opposite sides of the cube (along an axis; i.e. swap all vertices that are directly above or below each other), and whether there is any way to swap the locations of the vertices of the two sides with each other as a rigid transformation. My text then goes on to say that this is not possible without distorting the cube, and that such a transformation is not considered to be part of the symmetry group of a cube.

My question is, why is this? As the definitions for a 2D shape vs. a 3D shape seem to be inconsistent in dimensionality with each other.

That is; we cannot perform a horizontal flip in 2D as a rigid transformation; this can only be done in 3D. So a purist 2D symmetry set for a 2D shape would seem to include only the rotations. However, we lump in what is basically a 3D transformation into the set of symmetries for a 2D square, without really commenting on it.

Then, in the case of the 3D cube, I suspect (but admittedly have not proved) that such a swapping of two opposing sides' vertices might be possible in 4D (so I may just be blindly overgeneralizing). But we only include in the symmetry group those rotational symmetries that occur purely in 3D for this 3D shape. Any horizontal-flip-equivalent (as seen in the 2D square case), which might take place in 4D for a cube, is seemingly ignored. I would tentatively argue the same for a 4D shape having a 5D "horizontal flip", and so on.

Is there a reason for this seeming discrepancy in format beyond simple human convenience? Am I overlooking something obvious/non-arbitrary in how these symmetries are restricted? For a subject so fixated on clear generalizations, I found this to be a minor but glaring wrinkle in the ointment.

I hope I have been clear in my framing of my question; thanks for your consideration.

You're absolutely right that there's an odd asymmetry here, that feels unnatural to some extent. Assuming you're interpreting the author correctly -- that transformations of the type $(x, y, z) \mapsto (x, y, -z)$ are under consideration -- these reflections are, in many contexts, considered to be symmetries of the cube.

I can only imagine that the author wants you to think of both polygons and polyhedra as sitting in $\Bbb R^3$, and that the only "symmetries" of these objects are those rigid motions (distance-preserving transformations; "isometries") which also preserve orientation. In general, there is no need to do so; I would consider reflections to be rigid motions.

However, there is also some topological stuff happening, which justifies our different stances, to some extent: we can start from the identity and continuously perform a rotation; this is not so for a reflection; the fancy word here is "homotopy".

So, it's true that isometries do fall into two categories: Those that are said to "preserve orientation", and those that don't. I still have a hard time thinking in terms of orientation, but thankfully, there's a helpful shorthand: those linear maps $\varphi : \Bbb R^3 \to \Bbb R^3$ which preserve orientation have positive determinant; those which do not have negative determinant. For our $(x, y, z) \mapsto (x, y, -z)$ example, we can check that the associated matrix indeed has determinant $-1$.

Perhaps stranger still, this "handedness" (or lack thereof) of a transformation really depends upon more than the abstract group structure of the elements. As you rightly observe, things which are achieved by reflections in $\Bbb R^2$ can be achieved by rotations in $\Bbb R^3$ (a more elementary example: The characters S and A both have symmetry groups of order $2$, but these groups are realized in very different ways).

And, your hunch that this pattern continues (jump up a dimension to achieve a reflection by a rotation) is also right: Given a reflection $M: \Bbb R^n \to \Bbb R^n$ (so $\det M = -1$), we can find a transformation $\hat{M} : \Bbb{R}^{n + 1} \to \Bbb{R}^{n + 1}$ whose restriction to $\{(a_1, \ldots, a_n, 0) : a_i \in \Bbb R\} \cong \Bbb R^n$ is $M$, and such that $\det \hat{M} = 1$ (it does get tricky to think about how rotations generalize in higher dimensions, but we can still speak of determinants of linear transformations without trouble).

As a closing comment, I'll mention that this idea of jumping up a dimension to achieve reflections as rotations is relatively recent (as is higher-dimensional geometry in general) -- according to Coxeter in Regular Polytopes:

Practically all the ideas in this chapter [...] are due to Schläfi, who discovered them before 1853---a time when Cayley, Grassmann, and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.$^*$

$^*$ Möbius realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. [...] This idea was neatly employed by H.G. Wells in The Plattner Story.

The Plattner Story was published in 1896; really not long after the first mathematicians began to consider higher dimensions!

• Thank you, I think this answers my question pretty handily. We are told explicitly to consider only rotations of the 3D cube as symmetries, at least within the scope of this course, so I am pretty sure I am interpreting the author of my text correctly. I will probably give "The Plattner Story" a read, thanks for the clear and detailed response! I am not sure I fully follow what you describe about orientation, but I will think on it and try to understand how it ties into what I am learning. – C R Mar 30 '18 at 3:44
• Also, thank you for the notation formatting improvements, I appreciate it. – C R Mar 30 '18 at 3:48
• Yeah, like I said, I think "orientation" is sort of a strange thing to think about, even in 2D; basically, whether things going clockwise stay going clockwise after you perform the operation. In 3D, it takes the shape of the "right hand rule" you probably learned in vector calculus (which is why the determinant characterization is so nice; it's a lot easier to digest, I think). Anyway, you're very welcome, and it's a nice question you asked! – pjs36 Mar 31 '18 at 1:30