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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
nauty
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