Configuration (geometry)

Configuration (geometry)

In <a href="/en/Mathematics" title="Mathematics" class="wl">mathematics</a>, specifically <a href="/en/Projective_geometry" title="Projective geometry" class="wl">projective geometry</a>, a configuration in the plane consists of a finite set of <a href="/en/Point_(geometry)" title="Point (geometry)" class="wl">points</a>, and a finite <a href="/en/Arrangement_of_lines" title="Arrangement of lines" class="wl">arrangement of lines</a>, such that each point is <a href="/en/Incidence_(geometry)" title="Incidence (geometry)" class="wl">incident</a> to the same number of lines and each line is incident to the same number of points.<a href="#cite_note-1" style="counter-reset: mw-Ref 1;">[1]</a>

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Points and lines with equal incidences</div>
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<figure class="mw-default-size wiki-thumb"><a href="/en/File:Complete-quads.svg" class="mw-file-description image media-thumb" data-hash="File:Complete-quads.svg"><img loading="lazy" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Complete-quads.svg/300px-Complete-quads.svg.png" decoding="async" data-file-width="630" data-file-height="266" data-file-type="drawing" height="127" width="300" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Complete-quads.svg/450px-Complete-quads.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Complete-quads.svg/600px-Complete-quads.svg.png 2x" alt="Complete-quads.svg"></a><figcaption>Configurations (4362) (a <a href="/en/Complete_quadrangle" title="Complete quadrangle" class="wl">complete quadrangle</a>, at left) and (6243) (a complete quadrilateral, at right).</figcaption></figure>
Although certain specific configurations had been studied earlier (for instance by <a href="/en/Thomas_Kirkman" title="Thomas Kirkman" class="wl">Thomas Kirkman</a> in 1849), the formal study of configurations was first introduced by <a href="/en/Theodor_Reye" title="Theodor Reye" class="wl">Theodor Reye</a> in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of <a href="/en/Desargues'_theorem" title="Desargues' theorem" class="mw-redirect wl">Desargues' theorem</a>. <a href="/en/Ernst_Steinitz" title="Ernst Steinitz" class="wl">Ernst Steinitz</a> wrote his dissertation on the subject in 1894, and they were popularized by <a href="/en/David_Hilbert" title="David Hilbert" class="wl">Hilbert</a> and <a href="/en/Cohn-Vossen" title="Cohn-Vossen" class="mw-redirect wl">Cohn-Vossen</a>'s 1932 book Anschauliche Geometrie, reprinted in English as <a href="#CITEREFHilbertCohn-Vossen1952" target="_blank">Hilbert &amp; Cohn-Vossen (1952)</a>.

Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the <a href="/en/Euclidean_plane" title="Euclidean plane" class="wl">Euclidean</a> or <a href="/en/Projective_plane" title="Projective plane" class="wl">projective planes</a> (these are said to be realizable in that geometry), or as a type of abstract <a href="/en/Incidence_geometry" title="Incidence geometry" class="wl">incidence geometry</a>. In the latter case they are closely related to <a href="/en/Regular_graph" title="Regular graph" class="wl">regular</a> <a href="/en/Hypergraph" title="Hypergraph" class="wl">hypergraphs</a> and <a href="/en/Biregular_graph" title="Biregular graph" class="wl">biregular</a> <a href="/en/Bipartite_graph" title="Bipartite graph" class="wl">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="/en/Girth_(graph_theory)" title="Girth (graph theory)" class="wl">girth</a> of the corresponding bipartite graph (the <a href="/en/Levi_graph" title="Levi graph" class="wl">Levi graph</a> of the configuration) must be at least six.

In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points. Although certain specific configurations had been studied earlier , the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book Anschauliche Geometrie, reprinted in English as Hilbert &amp; Cohn-Vossen . Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes , or as a type of abstract incidence geometry. In the latter case they are closely related to regular hypergraphs and biregular bipartite graphs, 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 girth of the corresponding bipartite graph must be at least six. 