In geometry, the Pappus configuration is a

configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice board ...

of nine points and nine lines in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

, with three points per line and three lines through each point.

History and construction

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 , , and . 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''. Pappus hexagon

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.

Related constructions

The Levi graph of the Pappus configuration is known as the Pappus graph. It is a

bipartite Bipartite may refer to: * 2 (number) * Bipartite (theology), a philosophical term describing the human duality of body and soul * Bipartite graph, in mathematics, a graph in which the vertices are partitioned into two sets and every edge has an en ...

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 Two desmic tetrahedra. The third tetrahedron of this system is not shown, but has one vertex at the center and the other three on the plane at infinity. In projective geometry, a desmic system () is a set of three tetrahedra in 3-dimensional p ...

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.

Applications

A variant of the Pappus configuration provides a solution to the

orchard-planting problem In discrete geometry, the original orchard-planting problem asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. It is also called the tree-planting problem or simply the orchard ...

, the problem of finding sets of points that have the largest possible number of lines through three points. The nine points of the Pappus configuration form only nine three-point lines. However, they can be arranged so that there is another three-point line, making a total of ten. This is the maximum possible number of three-point lines through nine points.

References

External links

* {{Incidence structures Configurations (geometry) Dot patterns