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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.

MSC:
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
Software:
nauty
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References:
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