Let $E_\lambda(z)=\lambda e^z$ be the exponential family for $\lambda>0$. It is well known and proven by Misiurewicz that for $\lambda >1/e$ the Julia set is the entire complex plane while for $0<\lambda \le1/e$ the Julia set is a 'Cantor bouquet'. In this talk we will discuss the dynamics of a higher dimensional generalization of the exponential map called the Zorich map, denoted $Z$, in $R^3$. Bergweiler has shown that for small values of $\lambda >0$ the Julia set of $\lambda Z$ is a disjoint collection of curves. We will show that for large values of $\lambda$ the Julia set is the entire $R^3$. We will also discuss how other well known theorems about the exponential hold for Zorich maps as well.