  If $P\subset \R^d$ is a rational polytope, then $i_P(n):=#(nP\cap \Z^d)$ is a
quasi-polynomial in $n$, called the Ehrhart quasi-polynomial of $P$. The period
of $i_P(n)$ must divide $\LL(P)= \min \{n \in \Z_{> 0} \colon nP \text{is an
integral polytope}\}$. Few examples are known where the period is not exactly
$\LL(P)$. We show that for any $\LL$, there is a 2-dimensional triangle $P$
such that $\LL(P)=\LL$ but such that the period of $i_P(n)$ is 1, that is,
$i_P(n)$ is a polynomial in $n$. We also characterize all polygons $P$ such
that $i_P(n)$ is a polynomial. In addition, we provide a counterexample to a
conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart
quasi-polynomial.
