Heilbronner modes in polyhedra

Symmetry also adds useful detail to the picture for polyhedral 3D tt systems. A trivalent polyhedron obeys 

The edge-vertex difference function of equations (9) and (10) then becomes 

The extended Euler theorem can also be used to give a version of equation (15) in which vertex and face, but not edge, terms appear. 

Consideration of the description of leapfrogging in terms of edge crossings suggests formulas for the permutational representations of the leapfrog polyhedron Lf in terms of structural representations of the parent, P. From 

In general, F, (e) and F (c, P) are not equal, being related by a global twist 

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