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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
A variant of the Pappus configuration provides a solution to the orchard-planting problem, 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.
In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, 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.
History and construction
This configuration is named afterPappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
. Pappus's hexagon theorem
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that
*given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and ...
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''.
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
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we fo ...
of the Pappus configuration is known as the Pappus graph
In the mathematical field of graph theory, the Pappus graph is a bipartite 3- regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek ...
. It is a bipartite symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
cubic graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.
A bicubic graph is a cubic bi ...
with 18 vertices and 27 edges.
The Desargues configuration can also be defined in terms of perspective triangles, and the Reye configuration
In geometry, the Reye configuration, introduced by , is a configuration of 12 points and 16 lines.
Each point of the configuration belongs to four lines, and each line contains three points. Therefore, in the notation of configurations, the Reye ...
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 mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation
:
applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an ...
in the Euclidean plane, three real inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s 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, 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