  As a model to provide a hands-on, elementary understanding of chaotic
dynamics in dimension three, we introduce a $C^2$-open set of diffeomorphisms
of $\mathbb R^3$ having two horseshoes with different dimensions of
instability. We prove that: the unstable set of one horseshoe and the stable
set of the other are of Hausdorff dimension nearly $2$ whose cross sections are
Cantor sets; the intersection of the unstable and stable sets contains a
fractal set of Hausdorff dimension nearly $1$. As a corollary we detect
$C^2$-robust heterodimensional cycles. Our proof employs the theory of normally
hyperbolic invariant manifolds and the thicknesses of Cantor sets.
