  Graphs triangulating the $2$-sphere are generically rigid in $3$-space, due
to Gluck-Dehn-Alexandrov-Cauchy. We show there is a \emph{finite} subset $A$ in
$3$-space so that the vertices of each graph $G$ as above can be mapped into
$A$ to make the resulted embedding of $G$ infinitesimally rigid. This assertion
extends to the triangulations of any fixed compact connected surface, where the
upper bound obtained on the size of $A$ increases with the genus. The assertion
fails, namely no such finite $A$ exists, for the larger family of all graphs
that are generically rigid in $3$-space and even in the plane.
