  This work analyzes the distribution and size of interparticle gaps arising in
an ensemble of hexagonal unit structures in the xy plane when packing disks
with a Gaussian distribution of radii with mean (r) and standard deviation
$\Delta r$. During the course of this investigation an equivalency is
established between gaps arising in hexagonal unit structure packs and
nine-ball billiard rack patterns. An analytic expression is derived for the
probability distribution and location of interparticle gaps of magnitude
$\Gamma$. Due to the number of variables and large number of possible
arrangements, a Monte Carlo simulation has been conducted to complement and
probe the analytic form for three very different systems: i) billiard balls
with Billiard Congress of America (BCA) specifications, ii) US pennies with
specifications of the US Mint, and iii) a hypothetical system with $r = 1.0 m$
and $\Delta r = 1x10^{-10}$ m corresponding to the scale of one atomic radius.
In each case, probability density distributions of gap sizes have been
calculated for those $\Delta r$ above, and also for 2$\Delta r$ and 0.5$\Delta
r$, respectively. A general result is presented for the probability of a
nonzero normalized ($\frac{\Gamma}{\Delta r}$) gap size arising, $P(\Gamma \geq
\alpha \Delta r)= 1-0.124\alpha$, where $\alpha$ is a constant $\leq 5.0$. This
curious result reflects the phenomenon of geometric frustration; the inability
of the system to simultaneously satisfy all geometric constraints required by a
perfect-rack sans interparticle gaps.
