  We present our exhaustive exploration of the densest ternary sphere packings
(DTSPs) for 45 radius ratios and 237 kinds of compositions, which is a packing
problem of three kinds of hard spheres with different radii, under periodic
boundary conditions by a random structure searching method. To efficiently
explore DTSPs we further develop the searching method based on the piling-up
and iterative balance methods [Koshoji et al., Phys. Rev. E 103, 023307
(2021)]. The unbiased exploration identifies diverse 38 putative DTSPs
appearing on phase diagrams in which 37 DTSPs of them are discovered in the
study. The structural trend of DTSPs changes depending especially on the radius
of small spheres. In case that the radius of small spheres is relatively small,
structures of many DTSPs can be understood as derivatives of densest binary
sphere packings (DBSPs), while characteristic structures specific to the
ternary system emerge as the radius of small spheres becomes larger. In
addition to DTSPs, we reveal a lot of semi-DTSPs (SDTSPs) which are obtained by
excluding DBSPs in the calculation of phase diagram, and investigate the
correspondence of DTSPs and SDTSPs with real crystals based on the space group,
showing a considerable correspondence of SDTSPs having high symmetries with
real crystals including $\mathrm{Cu}_2 \mathrm{GaSr}$ and $\mathrm{ThCr}_2
\mathrm{Si}_2$ structures. Our study suggests that the diverse structures of
DBSPs, DTSPs, and SDTSPs can be effectively used as structural prototypes for
searching complex crystal structures.
