Gale transformations

In order to obtain a Gale diagram for a given polyhedron, the polyhedron is first subjected to a Gale transformation. Consider a polyhedron with v vertices as a set of v points Xy in three-dimensional space 31 These points... 

The final result of this construction is the assignment of a point xj in (v-4)-dimensional space (91 ) to each vertex xj of the polyhedron. The collection of V points x, ..., Xv in 91v-4 is called a Gale transform of the set of vertices xi,..., Xv of the polyhedron in question. The following features of a Gale transform of a polyhedron should be noted ... 

Gale transforms xj and Xk of two or more vertices of a polyhedron may lead to the same point (i.e., the same v-4 coordinates) in (v-4)-dimensional space (91 ). In other words some points of a Gale transform may have a multiplicity greater than one so that the Gale transform of a polyhedron in such cases contains fewer distinct points than the polyhedron has vertices. 

V fl,x, = 0 for 1 < fc < 3 (17a) In practice, it is easier to work with Gale diagrams corresponding to Gale transforms of interest. Consider a Gale trans-... 

Hofstadler, S. A., Swanek, E D., Gale, D. C., Ewing, A. G., and Smith, R. D. (1995). Capillary electrophoresis-electrospray ionization Eourier transform ion cyclotron resonance mass spectrometry for direct analysis of cellular proteins. Anal. Chem. 67, 1477—1480. 

R. Gillard and his co-workers at Kent University have extended these studies, and shown that organic complexes of rhodium produce similar effects. G. Gale and his associates at the Medical University of South Carolina developed a parallel story of the photochemical transformations and fi-lamentation of bacteria by the cA-isomer of the neutral complex with iridium instead of platinum. Thus, the experience now accumulated suggests a... 

Salmons S, Gale DR, Sreter FA. Ultrastructural aspects of the transformation of skeletal muscle fibre type by long-term stimulation changes in Z discs and mitochondria. J Anat 1978 127 17-31. 

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