Pappus configuration

In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points ABC and abc (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines Ab, aB, Ac, aC, Bc, and bC, and their three intersection points X = Ab·aB, Y = Ac·aC, and Z = Bc·bC. These three points are the intersection points of the "opposite" sides of the hexagon AbCaBc. According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points X, Y, and Z, called the Pappus line. The Pappus configuration can also be derived from two triangles XcC and YbB that are in perspective with each other (the three lines through corresponding pairs of points meet at a single crossing point) in three different ways, together with their three centers of perspectivity Z, a, and A. The points of the configuration are the points of the triangles and centers of perspectivity, and the lines of the configuration are the lines through corresponding pairs of points. The Levi graph of the Pappus configuration is known as the Pappus graph. It is a bipartite symmetric cubic graph with 18 vertices and 27 edges. The Desargues configuration can also be defined in terms of perspective triangles, and the Reye configuration can be defined analogously from two tetrahedra that are in perspective with each other in four different ways, forming a desmic system of tetrahedra. For any nonsingular cubic plane curve in the Euclidean plane, three real inflection points of the curve, and a fourth point on the curve, there is a unique way of completing these four points to form a Pappus configuration in such a way that all nine points lie on the curve.

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Related concepts (2)

Configuration (geometry)

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 (for instance by Thomas Kirkman in 1849), 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.

Levi graph

In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942. The Levi graph of a system of points and lines usually has girth at least six: Any 4-cycles would correspond to two lines through the same two points.