Because of its ad hoc development, the present paradigm for describing stereoisomerism lacks a unifying framework. We conclude that it is only through the lens of a systematic mathematically complete framework, that the relationships between the different facets of stereoisomerism can be determined. Further, such a framework provides the means of determining the completeness of stereoisomerism. We are developing such a unifying systematic framework and here we detail its implementation to one aspect of stereoisomerism: the geometries of coordination centres ABn. In the 1960s and 1970s, Muetterties et al. and Mislow developed a general approach called the polytopal rearrangement model for analysing this type of stereoisomerism. In this context, a "polytope" is the coordination polyhedron and a "polytopal rearrangement" describes any shape changing "ligand permutating" process that interconverts between different coordination geometries. Familiar examples of polytopal rearrangements are the "inversion" (more accurately called a geometric reflection) of trigonal bipyramidal tricoordinate (TPY-3) NH3 and simple amines , and the Berry pseudorotation mechanism as occurs in trigonal bipyramidal pentacoordinate (TBPY-5) PF5 (Fig. ).