  Let $\Omega \subseteq {\bf R}^d$ be an open set of measure 1. An open set $D
\subseteq {\bf R}^d$ is called a ``tight orthogonal packing region'' for
$\Omega$ if $D-D$ does not intersect the zeros of the Fourier Transform of the
indicator function of $\Omega$ and $D$ has measure 1. Suppose that $\Lambda$ is
a discrete subset of ${\bf R}^d$. The main contribution of this paper is a new
way of proving the following result (proved by different methods by Lagarias,
Reeds and Wang and, in the case of $\Omega$ being the cube, by Iosevich and
Pedersen: $D$ tiles ${\bf R}^d$ when translated at the locations $\Lambda$ if
and only if the set of exponentials $E_\Lambda = \{\exp 2\pi i \lambda\cdot x:
\lambda\in\Lambda\}$ is an orthonormal basis for $L^2(\Omega)$. (When $\Omega$
is the unit cube in ${\bf R}^d$ then it is a tight orthogonal packing region of
itself.) In our approach orthogonality of $E_\Lambda$ is viewed as a statement
about ``packing'' ${\bf R}^d$ with translates of a certain nonnegative function
and, additionally, we have completeness of $E_\Lambda$ in $L^2(\Omega)$ if and
only if the above-mentioned packing is in fact a tiling. We then formulate the
tiling condition in Fourier Analytic language and use this to prove our result.
