  Borcea and Streinu showed that the upper bound of the number of congruence
classes of a minimally $d$-volume rigid $(d+1)$-uniform hypergraph on $n$
vertices in $\mathbb{R}^d$ increases exponentially in $n$ and $d$. We show that
this result also holds for triangulations of $\mathbb{S}^2$ in $\mathbb{R}^2$,
and then find a geometrically motivated bound linear in $n$ for bipyramids. By
the methods used to deduce this bound, we show that, in general, global
$d$-volume rigidity in $\mathbb{R}^d$ is not a generic property of a
$(d+1)$-uniform hypergraph.
