  The special geometries of two recently discovered Calabi-Yau threefolds with
$h^{11}=1$ are analyzed in detail. These correspond to the 'minimal
three-generation' manifolds with $h^{21}=4$ and the `24-cell' threefolds with
$h^{21}=1$. It turns out that the one-dimensional complex structure moduli
spaces for these manifolds are both very similar and surprisingly complicated.
Both have 6 hyperconifold points and, in addition, there are singularities of
the Picard-Fuchs equation where the threefold is smooth but the Yukawa coupling
vanishes. Their fundamental periods are the generating functions of lattice
walks, and we use this fact to explain why the singularities are all at real
values of the complex structure.
