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Configuration (geometry)
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a>, a configuration in the plane consists of a finite set of <a href="/facts/Point_(geometry)/6YPAvX2a">points</a>, and a finite <a href="/facts/Arrangement_of_lines/15uaSKRu">arrangement of lines</a>, such that each point is <a href="/facts/Incidence_(geometry)/dlzk9bUl">incident</a> to the same number of lines and each line is incident to the same number of points. </p><p>Although certain specific configurations had been studied earlier (for instance by <a href="/facts/Thomas_Kirkman/R9TtyI3j">Thomas Kirkman</a> in 1849), the formal study of configurations was first introduced by <a href="/facts/Theodor_Reye/D8t4ssDq">Theodor Reye</a> in 1876, in the second edition of his book <i>Geometrie der Lage</i>, in the context of a discussion of <a href="/facts/Desargues%2527_theorem/vJzx0uPh">Desargues' theorem</a>. <a href="/facts/Ernst_Steinitz/jX6HsXLf">Ernst Steinitz</a> wrote his dissertation on the subject in 1894, and they were popularized by <a href="/facts/David_Hilbert/1vhlHn4b">Hilbert</a> and <a href="/facts/Cohn-Vossen/mUfH8SzR">Cohn-Vossen</a>'s 1932 book <i>Anschauliche Geometrie</i>, reprinted in English as Hilbert & Cohn-Vossen (1952). </p><p>Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the <a href="/facts/Euclidean_plane/tAUtRFcn">Euclidean</a> or <a href="/facts/Projective_plane/p6ot8oNv">projective planes</a> (these are said to be <i>realizable</i> in that geometry), or as a type of abstract <a href="/facts/Incidence_geometry/F1yPfCGu">incidence geometry</a>. In the latter case they are closely related to <a href="/facts/Regular_graph/j9dSoLE6">regular</a> <a href="/facts/Hypergraph/qIR2o2I3">hypergraphs</a> and <a href="/facts/Biregular_graph/g5ToSgcr">biregular</a> <a href="/facts/Bipartite_graph/xWcXV9MB">bipartite graphs</a>, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point. That is, the <a href="/facts/Girth_(graph_theory)/rPwRGgxj">girth</a> of the corresponding bipartite graph (the <a href="/facts/Levi_graph/pMnp7zkv">Levi graph</a> of the configuration) must be at least six. </p>