I was recently pondering certain graphs related to the recent Polymath project on the CNP Problem and I saw a nice unit-distance graph that was not 3-colourable with just 10 vertices and which could be "moved around", i.e. interpreting it as a bunch of linkages it was not rigid.
A natural question to me then arose: are there graphs which can embedded into the plane in an uncountable number of ways but, interpreting the graph as a linkage, one cannot "move" the graph any embedding to any other embedding?
I am NOT including trivial movements, such as rotations or translations of the entire graph, in such a question.
EDIT: Here is the graph I was talking about on Dusting Mixon's blog. It's quite a nice catch!
I believe the following construction gives such an example:
What I want to consider is a "triangle which grew a triangle which grew a triangle which grew a triangle which ..."
Start with a (say, equilateral with side length $1$) triangle with vertices $P_0, P_1, P_2$. $G_0$ is this graph: it has three vertices and three edges.
Add a new vertex $P_3$ and edges connecting $P_3$ to $P_1$ and $P_2$. This is $G_1$: it has four vertices and five edges.
In general, $G_{n+1}$ is gotten from $G_n$ by adding a new vertex $P_{n+3}$ and new edges connecting $P_{n+3}$ to $P+{n+2}$ and $P_{n+1}$.
Now of course this depends on how I pick my $P_i$s. Here's how I want to do that:
$P_{3+i}$ will be on the bisector of $\overline{P_{1+i}P_{2+i}}$ for $i\ge 0$.
The angle $\theta_i=\angle P_{3+i}P_{2+i}P_{1+i}$ will be really really small relative to all the angles introduced so far. I think $\theta_0={\pi\over 30}$, $\theta_{i+1}={\theta_i\over 10}$ works (and is in fact overkill).
Now leg $\mathcal{G}$ be the graph gotten "at $\infty$" according to this construction. Basically, to embed this graph in the plane we first embed the triangle $P_0P_1P_2$, and then we make a series of "toggles" - does $P_{3+i}$ point "in" or "out" relative to the triangle $P_{0+i}P_{1+i}P_{2+i}$?
Since the $\theta_i$s decrease quickly enough, each choice of toggles does in fact yield an embedding into the plane, so we have uncountably many meaningfully different embeddings of $\mathcal{G}$ into $\mathbb{R}^2$; however, any one of these embeddings should be inflexible (modulo trivial actions).