  In this paper, we give a complete description of the language of geodesic
words for the discrete Heisenberg group $H(\mathbb{Z})$ with respect to the
standard two-element generating set. More precisely, we prove that the only
dead end elements in $H(\mathbb{Z})$ are nontrivial elements of the commutator
subgroup. We give a description of their geodesic representatives, which are
called dead end words. The description is based on a minimal perimeter
polyomino concept. Finally, we prove that any geodesic word in $H(\mathbb{Z})$
is a prefix of a dead end word.
