Theorems of points and planes in three-dimensional projective space. (English) Zbl 1187.51001

By an \(n_k\) configuration of a \(3\)-dimensional projective space we mean a set of \(n\) points and a set of \(n\) planes with \(k\) points on each plane and \(k\) planes through each point. An \(n_k\) configuration is called ‘theorem’, if the last of the \(nk\) incidences is determined by the first \(nk-1\).
In order to find \(n_k\) configurations the author uses the embedding algorithm of D. G. Glynn [Bull. Aust. Math. Soc. 76, No. 1, 15–31 (2007; Zbl 1122.51002)], implemented in the general mathematical programm MAGNA and the graph-theoretic program NAUTY. Starting point are all non-isomorphic connected regular bipartite graphs of \(2n\) vertices and of valency \(k\). Among others, the algorithm yields two non-isomorphic \(8_4\) configurations, one being the well-known Möbius \(8_4\) configuration, and one \(9_4\) configuration.
The non-Möbius \(8_4\) configuration is a ‘theorem’ in projective \(3\)-space over a field; this is proved by the author synthetically as well as analytically. Furthermore, the author shows that the \(9_4\) configuration is a ‘theorem’ in projective \(3\)-space over a field or a skew field; the \(9_4\) ‘theorem’ is related to Desargues’ ten-point configuration.
The author describes how to construct nice models of the \(8_4\) and \(9_4\) configurations in Euclidean \(3\)-space. Moreover, the connection between the \(8_4\) and \(9_4\) ‘theorems’ with minimum-energy configurations of electrons and with forbidden minors in graph and matroid theory is mentioned.
The text is accompanied by two very aesthetical figures.


51A20 Configuration theorems in linear incidence geometry
51A45 Incidence structures embeddable into projective geometries
05B25 Combinatorial aspects of finite geometries
05B35 Combinatorial aspects of matroids and geometric lattices


Zbl 1122.51002


Full Text: DOI


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