Polyhedral isomerizations planar

Consider a polyhedral isomerization sequence P, rp2 S3 in which Pi and are combinatorially equivalent. Such a polyhedral isomerization sequence may be called a degenerate polyhedral isomerization with Pj as the intermediate polyhedron. Structures undergoing such degenerate isomerization processes are often called fluxional. A degenerate polyhedral isomerization with a planar intermediate polyhedron (actually a polygon) may be called a planar polyhedral isomerization. The simplest... 

Pi and V3) through a square planar intermediate V2. Except for this simplest example, planar polyhedral isomerizations are unfavorable owing to excessive intervertex repulsion. 

Now consider polyhedra in the ordinary three-dimensional space of interest in chemical structures (i.e., d = 3). Gale diagrams of five- and six-vertex polyhedra can be embedded into one- or two-dimensional space, respectively, thereby simplifying analysis of their possible vertex motions leading to non-planar polyhedral isomerizations of these polyhedra of possible interest in a chemical context. 

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