  For any $d\geq 1$, we obtain counting and equidistribution results for tori
with small volume for a class of $d$-dimensional torus packings, invariant
under a self-joining $\Gamma_\rho<\prod_{i=1}^d\mathrm{PSL}_2(\mathbb{C})$ of a
Kleinian group $\Gamma$ formed by a $d$-tuple of convex cocompact
representations $\rho=(\rho_1, \cdots, \rho_d)$. More precisely, if $\mathcal
P$ is a $\Gamma_\rho$-admissible $d$-dimensional torus packing, then for any
bounded subset $E\subset \mathbb{C}^d$ with $\partial E$ contained in a proper
real algebraic subvariety, we have $$\lim_{s\to 0} { s^{\delta_{L^1}({\rho}) }}
\cdot \#\{T\in \mathcal{P}: \mathrm{Vol} (T)> s,\, T\cap E\neq \emptyset \}=
c_{\mathcal P}\cdot \omega_{\rho} (E\cap \Lambda_\rho).$$ Here
$0<\delta_{L^1}(\rho)\le 2/\sqrt d$ is the critical exponent of $\Gamma_\rho$
with respect to the $L^1$-metric on the product $\prod_{i=1}^d \mathbb{H}^3$,
$\Lambda_\rho\subset (\mathbb{C}\cup\{\infty\})^d$ is the limit set of
$\Gamma_\rho$, and $\omega_{\rho}$ is a locally finite Borel measure on
$\mathbb{C}^d\cap \Lambda_\rho$ which can be explicitly described. The class of
admissible torus packings we consider arises naturally from the Teichm\"{u}ller
theory of Kleinian groups. Our work extends previous results of Oh-Shah on
circle packings (i.e. one-dimensional torus packings) to $d$-torus packings.
