How to generate a point cloud with known symmetry?

Question

So I would like to know if there are any published algorithms to generate point clouds with known symmetry groups, such as $D_{3h}$ or $O_h$ and stuff like that. I know lots and lots of point clouds (infinitely many) would fit in each group, but I don't mind, as long as the algorithm can guarantee that whatever collection of points it generates will have the given symmetry (probably to a certain error margin, since computations will suffer from machine precision limits).

Answer 1

Having looked up the definition of "point cloud", my comment of taking a Cayley graph and removing the edges may (theoretically) work, but you need a couple of fudges and in reality it all depends on where you are working. There is lots of literature about algorithms and Cayley graphs (they have applications in computer networking), so I thought it would be worthwhile to expand on my comment here.

I should say, however, that it is not clear to me that every group can be realised as the symmetry group of a point-cloud! I expand on this a bit, below.

So, a Cayley graph $\Gamma(G, S)$ of a group $G$ with generating set $S$ is a specific graph whose symmetry group is precisely $G$. Embedding this into some space where all edges have length $1$, say, and removing the edges gives you a point cloud whose symmetry group contains $G$ (most probably strictly contains $G$). To make sure that the symmetry group is precisely $G$, you need to differentiate between the (now removed) directed edges somehow. For example, the usual Cayley graph of $\mathbb{Z}\times\mathbb{Z}$ embeds into $\mathbb{R}^2$ but when we remove the edges you have a copy of $D_4$ in there, corresponding to rotation and flipping. Similarly, the Cayley graph of $D_4$ embeds as a cube into $\mathbb{R}^2$ so the resulting point-cloud has a larger symmetry group (each face yields a copy of $D_4$!).

There are two issues here.

1. It is not obvious that you can embed the Cayley graph of a given group into a nice space. See this MO question for a discussion on this. (For example, I am pretty sure that the Cayley graph of the group $B=\langle a, b; ab^2a^{-1}=b^3\rangle$ does not embed like this into $\mathbb{R}^n$ for any $n$, because the embedding would (I think) imply that the group was linear, and the group $B$ is not linear.)

2. I do not know how to get around that differentiating-between-edges issue. One fudge might be to change the length of the edges when switching between generators. However, this does not quite work (as the Cayley graph of $\mathbb{Z}=\langle a\rangle$ can always be flipped).

A more fundamental issue is that it is not clear to me that every group can be realised as the group of symmetries of a point-cloud. Indeed, I do not believe that the infinite cyclic group can be, because you can always "flip" it. So:

Questions you should answer before proceeding too far:

1. Does there exist a point cloud whose symmetry group is infinite cyclic?

2. Does there exist a point cloud whose symmetry group is cyclic of prime order?

As I said above, I believe that the answer to (1) is no. I am unsure about (2), but I suspect "no".