  Rigid frameworks in some Euclidian space are embedded graphs having a unique
local realization (up to Euclidian motions) for the given edge lengths,
although globally they may have several. We study the number of distinct planar
embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo
planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx
4^n$. We also exhibit several families which realize lower bounds of the order
of $2^n$, $2.21^n$ and $2.88^n$.
  For the upper bound we use techniques from complex algebraic geometry, based
on the (projective) Cayley-Menger variety $CM^{2,n}(C)\subset P_{{{n}\choose
{2}}-1}(C)$ over the complex numbers $C$. In this context, point configurations
are represented by coordinates given by squared distances between all pairs of
points. Sectioning the variety with $2n-4$ hyperplanes yields at most
$deg(CM^{2,n})$ zero-dimensional components, and one finds this degree to be
$D^{2,n}={1/2}{{2n-4}\choose {n-2}}$. The lower bounds are related to inductive
constructions of minimally rigid graphs via Henneberg sequences.
  The same approach works in higher dimensions. In particular we show that it
leads to an upper bound of $2 D^{3,n}=
{\frac{2^{n-3}}{n-2}}{{n-6}\choose{n-3}}$ for the number of spatial embeddings
with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to
rigid motions.
