  Motivated by breakthroughs in the synthesis of faceted nano- and colloidal
particles, as well as theoretical and computational studies of their packings,
we investigate a family of truncated triangular bipyramids. We report dense
periodic packings with small unit cells that were obtained via numerical and
analytical optimization. The maximal packing fraction $\phi_{\max}$ changes
continuously with the truncation parameter $t$. Eight distinct packings are
identified based on discontinuities in the first and second derivatives of
$\phi_{\max}(t)$. These packings differ in the number of particles in the
fundamental domain (unit cell) and the type of contacts between the particles.
In particular, we report two packings with four particles in the unit cell for
which both $\phi_{\max}(t)$ and $\phi'_{\max}(t)$ are continuous and the
discontinuity occurs in the second derivative only. In the self-assembly
simulations that we perform for larger boxes with 2048 particles, only one out
of eight packings is found to assemble. In addition, the degenerate
quasicrystal reported previously for triangular bipyramids without truncation
[{http://prl.aps.org/abstract/PRL/v107/i21/e215702}{Haji-Akbari \emph{et al.},
\emph{Phys. Rev. Lett.} \textbf{107}: 215702 (2011)}] assembles for truncations
as high as 0.45. The self-assembly propensities for the structures formed in
the thermodynamic limit are explained using the isoperimetric quotient of the
particles and the coordination number in the disordered fluid and in the
assembled structure.
