This paper addresses the intrinsic formation control problem of a multi-agent system. The foraging behavior of N agents is modeled as an infinite-horizon non-cooperative differential game under local information, and its Nash equilibrium is studied. The formations are achieved in an intrinsic way in the sense that they are only attributed to the inter-agent interaction and geometric properties of the network, where the desired formations are not designated beforehand. Through the design of individual costs and network topology, patterns of Platonic solids can be achieved as Nash equilibria while inter-agent collisions are avoided. Exponential convergence to the manifold of Platonic patterns is proved. Finally, numerical simulations are provided to demonstrate the effectiveness and feasibility of the proposed methods.