The binding polynomial formalism is used to calculate binding isotherms for nearest neighbor interacting ligands on the platonic solids, i.e., tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The activity of ligand (concentration multiplied by intrinsic binding constant of ligand to single sites) at half occupancy is precisely determined by the product of the coordination number or valence of the vertices and the free energy of interaction, regardless of other geometrical and topological features. More importantly, the sharpness of binding curves increases with interaction strength, with Hill exponents approaching the number of binding sites per solid particle, i.e., the number of vertices or the number of faces. Potential applications of the model to viral capture nanotherapy are suggested.