  Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic
complement. We describe $L$ explicitly and study its properties, and discover
that $L$ is in many aspects similar to the Borromean rings in $S^3$. In
particular the following hold: (1) Any two tori in $L$ are unlinked, but three
are not; (2) The complement $M=S^4\setminus L$ is integral arithmetic
hyperbolic; (3) The symmetry group of $L$ acts $k$-transitively on its
components for all $k$; (4) The double branched covering over $L$ has geometry
$\mathbb H^2 \times \mathbb H^2$; (5) The fundamental group of $M$ has a nice
presentation via commutators; (6) The Alexander ideal is explicit; (7) Every
class $x\in H^1(M, \mathbb Z)=\mathbb Z^5$ with $x_i\neq 0$ is represented by a
perfect circle-valued Morse function; (8) By longitudinal Dehn surgery along
$L$ we get a closed 4-manifold with fundamental group $\mathbb Z^5$; (9) The
link $L$ can be put in perfect position.
  This leads also to the first descriptions of a cusped hyperbolic 4-manifold
as a complement of tori in $\mathbb R \mathbb P^4$ and of some explicit
Lagrangian tori in the product of two surfaces of genus two.
