By examining the basic characteristics of clathrin lattices, we discover that simple topological rules impose strict constraints on clathrin lattice transformations. These constraints require that internal bond rearrangements take place in conjunction with the addition or removal of pairs of clathrin triskelions within the interior of existing clathrin lattice patches. Similar constraints also are relevant to coated-vesicle shape changes and their budding-off from pit lattices. Via specific illustrations, successive vesicles with hexagonal-barrel and other coats are shown to grow out from the interior of a initially flat clathrin-coated pit so long as free triskelions are available from cytoplasm. Concomitantly, we present mathematical derivations of several simple and useful topological equations. These equations govern the numbers of nonhexagonal clathrin lattice facets and their variations during internal shape transformations and justify the proposed mechanisms of triskelion pair insertion and removal.