Using three-point function techniques, correlations between the vacuum action density and the positions of quarks are used to identify the formation of gluon flux-tubes within baryons. A highstatistics approach based on the translational and rotational symmetry of the four-dimensional lattice volume is adopted to avoid the need for gauge-dependent smoothing techniques. Vacuum field fluctuations are found to be suppressed in the presence of static quarks such that flux tubes represent the expulsion or suppression of gluon-field fluctuations. By considering numerous different link paths in the creation of the static quark sources, we are able to explore the dependence of the observed flux tubes on the source shape. In particular, “T,” “L” and “Y” shapes are considered to access a variety of flux-tube topologies including the ground state. T-shape paths are observed to relax towards a Y-shape topology as opposed to a ∆ shape. L-shape topologies give rise to a large potential. Upon identifying the precise geometry of the flux tube formation, we are able to perform a quantitative comparison between the flux tube length and the associated static-quark potential. For every source considered we find the flux-tube length and associated potential to provide a universal string tension. With this new knowledge, one can conclude that the flux tube configuration of the ground state potential for large quark separations is that which minimizes the flux tube length. The characteristic flux tube radius of the baryonic ground state potential is found to be 0.38(3) fm with vacuum-field fluctuations suppressed by 7.2(6)%. The node connecting the flux tubes is 25% larger at 0.47(2) fm with a 15(3)% larger suppression of the vacuum action at 8.1(7)%.