Even in the absence of external stimuli, neural activity is both highly dynamic and organized across multiple spatiotemporal scales. The continuous evolution of brain activity patterns during rest is believed to help maintain a rich repertoire of possible functional configurations that relate to typical and atypical cognitive phenomena. Whether these transitions or "explorations" follow some underlying arrangement or instead lack a predictable ordered plan remains to be determined. Here, using a precision individual connectomics dataset, we aimed at revealing the rules that govern transitions in brain activity at rest. The dataset includes individually defined parcellations and ~5 hours of resting state functional Magnetic Resonance Imaging (fMRI) data for each participant – both of which allowed us to examine the topology and dynamics of at-rest whole-brain configurations in an unprecedented detail. We hypothesized that by revealing and characterizing the overall landscape of whole-brain configurations we could interpret the rules (if any) that govern transitions in brain activity at rest. To generate the landscape of whole- brain configurations we used Topological Data Analysis (TDA)-based Mapper approach that could reveal shape of the underlying dataset as a graph (a.k.a. shape graph). We observed a rich topographic landscape in which the transition of activity from one canonical brain network to the next involved a large, shared attractor-like basin, or "transition state", where all networks were represented equally prior to entering distinct network configurations. The intermediate transition state and traversal through it seemed to provide the underlying structure for the continuous evolution of brain activity patterns at rest. In addition, differences in the manifold architecture were more consistent within than between subjects, providing evidence that this approach contains potential utility for precision medicine approaches.