  We study the time constant $\mu(e_{1})$ in first passage percolation on
$\mathbb Z^{d}$ as a function of the dimension. We prove that if the passage
times have finite mean, $$\lim_{d \to \infty} \frac{\mu(e_{1}) d}{\log d} =
\frac{1}{2a},$$ where $a \in [0,\infty]$ is a constant that depends only on the
behavior of the distribution of the passage times at $0$. For the same class of
distributions, we also prove that the limit shape is not an Euclidean ball, nor
a $d$-dimensional cube or diamond, provided that $d$ is large enough.
