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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.

Lou M. Pretorius worked with Noam Elkies and Konrad J. Swanepoel on Sylvester-Gallai theorem analogues over complex numbers and quaternions. The original byline verifies the printed name.^[ref]

They give an elementary complex-case proof and a quaternionic extension. Requiring a third configuration point on each pair's line forces containment in a low-dimensional projective flat, with different bounds over different scalar systems.^[ref]

Noam Elkies coauthored work with Lou M. Pretorius and Konrad J. Swanepoel on finite incidence configurations in projective spaces. Their elementary argument proves the complex Sylvester-Gallai theorem dimension bound and extends the problem to quaternions.^[ref]

The project connects elementary incidence questions with the algebra of the scalar field: complex and quaternionic configurations permit different geometry from real ones.

By Noam Elkies, Lou M. Pretorius and Konrad J. Swanepoel. Discrete & Computational Geometry 35 (2006), 361–373. Author preprint: arxiv.org/abs/math/0403023

Locator: introduction, real Sylvester-Gallai theorem and the extensions to complex and quaternionic projective spaces.