  The exploration of the densest sphere packings is a fundamental problem in
mathematics and a wide variety of sciences including materials science. We
present our exhaustive computational exploration of the densest ternary sphere
packings (DTSPs) for 451 radius ratios and 436 compositions on top of our
previous study [Koshoji and Ozaki, Phys. Rev. E 104, 024101 (2021)]. The
unbiased exploration discovers diverse 22 putative DTSPs, and thereby 60
putative DTSPs are identified in total including the 38 DTSPs discussed by the
previous study. Some of the discovered DTSPs are well-ordered, for example, the
medium spheres in the (9-7-3) structure are placed in a straight line with
comprising the unit cell, and the DTSP has the $Pm \bar{3}m$ symmetry if the
structural distortion is corrected. At a considerable number of radius ratios,
the highest packing fractions are achieved by the phase separations consisting
of only the FCC and/or the putative densest binary sphere packings (DBSPs) for
all compositions, and the tendency is getting evident as the small and medium
spheres are getting larger. The result seems to indicate directly that the
local structures in the DBSPs may be denser than those consisting of three
kinds of spheres. However, the unit cell of undiscovered DTSPs might only be
much larger than in this study due to the complexity of the ternary local
structures. Finally, we discuss the correspondence of the DTSPs with real
crystals based on the space group. Our study suggests that the diverse
structures of DTSPs can be effectively used as structural prototypes for
searching ternary, quaternary, and quinary crystal structures.
