Konrad J. Swanepoel coauthored an extension of Sylvester-Gallai theorem geometry with Noam Elkies and Lou M. Pretorius. Their paper gives an elementary proof that a finite Sylvester-Gallai configuration over the complex numbers lies in a projective plane, and extends the conclusion over the quaternions to a three-dimensional flat.[ref]
The result shows that incidence constraints depend on the underlying scalars: the corresponding real configuration must already be collinear.
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