  Zorich maps are higher dimensional analogues of the complex exponential map.
For the exponential family $\lambda e^z$, $\lambda>0$, it is known that for
small values of $\lambda$ the Julia set is an uncountable collection of
disjoint curves. The same was shown to hold for Zorich maps by Bergweiler and
Nicks.
  In this paper we introduce a topological model for the Julia sets of certain
Zorich maps, similar to the so called \textit{straight brush} of Aarts and
Oversteegen. As a corollary we show that $\infty$ is an \textit{explosion
point} for the set of endpoints of the Julia sets. Moreover we introduce an
object called a \textit{hairy surface} which is a compactified version of the
Julia set of Zorich maps and we show that those objects are not uniquely
embedded in $\mathbb{R}^3$, unlike the corresponding two dimensional objects
which are all ambiently homeomorphic.
