Gale diagrams

Consider a polytope V in -dimensional space where the term polytope refers to the generalization of the concept of polyhedron to any number of dimensions [36]. The minimum number of vertices of such a polytope is -h 1 and there is only one such polytope, namely the /-simplex in which each possible pair of the - -1 vertices are connected by an edge corresponding to the so-called complete graph [51] Kd+ - The combinatorially distinct possibilities for i-dimensional polytopes having only d + 2 and d + 3 vertices (polyhedra with few vertices) are also rather limited. They can be represented faithfully in a space of less than d dimensions through a Gale 

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. 

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 

Equation (l.lla) may also be viewed as three orthogonality relationships between the v-dimensional vector A = (ai. v) and the three v-dimensional vectors (xi,, x, k.Xy,k), I k 3. Now consider the locations of the ver- 

Consider the columns of Do as vectors in DP. Since Do has rank 4, the four columns of Do are linearly independent. Hence the subspace A4(X) of CP represented by these four linearly independent columns has dimension 4. Its orthogonal complement A4(A) = A e CP A X = 0 for all X e A4(X) coincides with V(A) defined above by equations (l.lla) and (1.11b). Therefore  

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The following properties of Gale diagrams corresponding to three-dimensional polyhedra are of interest since they impose important restrictions on configurations of points which can be Gale diagrams ... 

Any (v-5)-dimensional plane passing through the central point of the Gale diagram bisects the space of the Gale diagram into two halfspaces. Each... 

Figure 1.9 (a) Gale diagrams for the two five-vertex polyhedra. (b) Standard Gale diagrams for the seven six-vertex polyhedra. 

The central point is a vertex of a Gale diagram if and only if the corresponding polyhedron is a pyramid. The central vertex of such a Gale diagram corresponds to the apex of a pyramid which is the coface corresponding to the base of the pyramid. 

Figure 1.10 Gale diagrams for the triple dsd degenerate isomerization of an octahedron through a trigonal prismatic intermediate corresponding to the Bailar or Ray/Dutt twists.

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 Xi,...,X in three-dimensional space 91. These points may be regarded as three-dimensional vectors X = ( / ,I. n,2. 1 5 < V, from the origin to the vertices of... 

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-... 

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