We have experimentally studied small impact parameter heavy-ion collisions in the (nearly) symmetry entrance channels $^{12}\mathrm{C}$${+}^{12}$C, $^{20}\mathrm{Ne}$${+}^{27}$Al, $^{40}\mathrm{Ar}$${+}^{45}$Sc, $^{84}\mathrm{Kr}$${+}^{93}$Nb, and $^{129}\mathrm{Xe}$${+}^{139}$La, each at many intermediate beam energies. The results from a number of analyses based on a projection of the ``shapes'' of the experimental events called the sphericity are presented. Comparisons of the relative efficiencies of various experimental methods for the selection of central events are made. The importance of autocorrelations between the sphericity and the various impact-parameter--dependent variables is evaluated. Searches for beam energy-dependent transitions from sequential binary disassembly to multifragmentation in the central events are described. Comparisons to dynamic and hybrid model code calculations will be discussed. The average sphericities of the intermediate mass fragments (IMF's, for which 3\ensuremath{\le}Z\ensuremath{\lesssim}20), are presented. The possibility that the IMF emission occurs following the formation of transient toroidal or disk-like geometries in the central events is explored. Increases in the average sphericities of the central events for increasing beam energies are observed which is attributed to transitions from sequential binary disassembly to multifragmentation. The transitional beam energies for the central $^{40}\mathrm{Ar}$${+}^{45}$Sc, $^{84}\mathrm{Kr}$${+}^{93}$Nb, and $^{129}\mathrm{Xe}$${+}^{139}$La reactions are near \ensuremath{\sim}50, \ensuremath{\sim}40, and \ensuremath{\sim}40 MeV/nucleon, respectively.