Exotic magnetic order in frustrated systems can be engineered by precisely tailoring assemblies of nanomagnets known as artificial spin ices. While artificial spin ices and their temperature-dependent behavior have been extensively explored in two dimensions, their behavior in three dimensions remain largely unexplored. Here, we introduce a family of three-dimensional artificial spin ices based on the five Platonic and thirteen Archimedean solids in which dipolar-coupled Ising spins are constrained to point along the edges of each polyhedron. This construction creates closed, boundary-free spin lattices in which the collective behavior is governed by geometry and dipolar interactions. Because of their small size, interactions between spins on located opposite sides of the lattice play an important role. Using extensive Monte Carlo simulations, we uncover a rich variety of magnetic behavior whereby the systems undergo between one and five distinct thermodynamic crossovers before reaching their ground states. We identify the mechanisms underlying these crossovers, linking them to the emergence of ice rules, charge crystallization, and partial spin freezing, and classify the eighteen lattices into four categories based on their ground states and freezing pathways. Strikingly, we find that five of the lattices exhibit spontaneous phase separation analogous to the Arctic Circle effect, in which a fluctuating spin liquid coexists within an ordered, frozen matrix. Unlike previous realizations of this effect, which rely on externally imposed boundary conditions, phase coexistence here emerges intrinsically from the closed geometry and dipolar interactions between spins in the lattice. These results establish closed artificial spin ices based on polyhedra as a distinct paradigm of frustrated magnetic systems and provide a foundation for future experimental studies of three-dimensional artificial spin ices, where geometry alone can drive multiple crossovers, partial spin freezing, and emergent phase coexistence.