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One should notice several properties of this sketch. First, there are 8 numbered points. The small unmarked point in the center represents the origin in R2. The three thin lines colored blue, green, and red and passing through the origin represent 1-dimensional subspaces. There are an additional 5 unmarked points. These have a special significance as explained below This configuration of 8 points has ghost symmetry: If one projects the set of points orthogonally to one of these three colored subspaces, then one obtains a configuration having two-fold symmetry. Thus, while the 2-dimensional configuration of 8 points has no apparent symmetry, there are three 1-dimensional projections which each have two-fold symmetry. The purpose of the thick lines is to assist in showing that the configuration has these three ghost symmetries. Notice that there are 8 thick lines of each color, each line has precisely one point of the configuration on that line, and each point lies on a line from each of these three families. One can think of these lines as the "lines of projection", as they are parallel to the kernels of the projection maps. The configuration of 8 points has ghost symmetries because this configuration of 24 lines has two-fold rotational symmetry. Associated to this configuration is a graph:
The 8 vertices of this graph correspond to the points of the configuration. One draws an edge colored by C between vertices V and W if projection onto the line colored by C yields a ghost symmetry which interchanges the images of V and W. For example, there is a blue edge between 2 and 6 because the blue lines passing through the points marked 2 and 6 lie at the same distance to the origin. Thus, projecting points 2 and 6 onto the blue subspace yields images which are the same distance to the origin. This graph is trivalent and 3-edge colored. Thus, every vertex has precisely 3 edges connecting to it, one of each color. There are 12 edges and each color yields a 1-factor (also known as a "perfect matching") of the graph. This type of graph is often called "Tait-colored". The Theorem. Given a planar configuration of points which has three ghost symmetries, one may always associate such a Tait-colored graph. However, the converse is not automatic. That is, given a Tait-colored graph representing a triple of permutations corresponding to three desired ghost symmetries, one cannot always construct a corresponding Euclidean configuration. The theorem given here gives a characterization of these graphs. Before stating the main theorem, it is necessary to give some terminology. A "hyperinvolution" on a set S is a permutation of order 2 on S which has at most one fixed point. Thus, if S has even cardinality, then a hyperinvolution has no fixed points. The blue edges in the graph above correspond to the hyperinvolution (1,3)(2,6)(4,7)(5,8), and so on. In graph-theoretic terminology, a hyperinvolution coincides with a 1-factor. Hyperinvolutions arise for the simple reason that a two-fold symmetry of a line is necessarily a hyperinvolution of the points on the line. One may now state: Theorem. Sketch of Proof. If one knows some things about convex polytopes, then one may see a similarity to Steinitz's theorem. This theorem states that a graph is the 1-skeleton of a convex 3-dimensional polytope if and only if it is simple, planar, and 3-connected. In fact, the proof of the theorem stated above is similar to several well-known proofs of Steinitz's theorem. One of the basic techniques in the proof is to use the fact that one may perturb such a configuration in an interesting way. Consider perturbing those five unmarked points lying in the the three subspaces colored blue, green and red. If one perturbs these slightly, then one will notice that some of the points and lines in the sketch move. For example, if one perturbs the point on the blue subspace corresponding to point 4, then the result is a perturbation of points 4, 5, 6, and 7 and the corresponding lines. Notice that points 4, 5, 6, and 7 are the vertices of the cycle in the graph. As it turns out, there is a similar type of perturbation for every cycle in the graph, and this allows one enough wiggle room in order to exploit 3-connectivity and wrestle out a proof by induction. The details are given below in a preprint. (I welcome corrections!)
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