Topological models for classical configurations
Let \(S_k\) be an orientable surface (a sphere with \(k\) ``handles'' attached), \(k\geq 0\). The Fano, Pappus and Desargues configurations are considered. The Menger graph have the points of a configuration as the vertices of the graph, and two vertices be adjacent if the points belong to a common line. The Menger graph of the Fano configuration is the complete graph \(K_7\). Any imbedding of \(K_7\) on the torus \(S_1\) models the Fano plane. As there is no imbedding of the Fano plane into \(S_0\), we say that this geometry has genus one. The Menger graph of the Pappus configuration is the complete multipartite graph \(K_{3\times 3}\). The Coxeter imbedding of \(K_{3\times 3}\) on the torus models the Pappus configuration. The Menger graph of the Desargues configuration is the triangular graph \(T(5)\) (the complement of the Petersen graph). An imbedding of \(T(5)\) into \(S_2\) is established (there is no imbedding of \(T(5)\) into \(S_1\)).
scientific article; zbMATH DE number 1471124