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{{Short description|Geometric configuration of 9 points and 9 lines}}
[[File:Pappusconfig.svg|thumb|Pappus configuration]]

In [[geometry]], the '''Pappus configuration''' is a [[Configuration (geometry)|configuration]] of nine points and nine lines in the [[Euclidean plane]], with three points per line and three lines through each point.<ref>{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | isbn = 978-0-8218-4308-6
 | location = Providence, RI
 | mr = 2510707
 | publisher = American Mathematical Society
 | series = [[Graduate Studies in Mathematics]]
 | title = Configurations of points and lines
 | volume = 103
 | year = 2009}}.</ref>

==History and construction==

This configuration is named after [[Pappus of Alexandria]]. [[Pappus's hexagon theorem]] states that every two triples of collinear points {{mvar|ABC}} and {{mvar|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 {{mvar|{{overline|Ab}}}}, {{mvar|{{overline|aB}}}}, {{mvar|{{overline|Ac}}}}, {{mvar|{{overline|aC}}}}, {{mvar|{{overline|Bc}}}}, and {{mvar|{{overline|bC}}}}, and their three intersection points {{math|1=''X'' = ''{{overline|Ab}} · {{overline|aB}}''}}, {{math|1=''Y'' = ''{{overline|Ac}} · {{overline|aC}}''}}, and {{math|1=''Z'' = ''{{overline|Bc}} · {{overline|bC}}''}}. These three points are the intersection points of the "opposite" sides of the hexagon {{mvar|AbCaBc}}. According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points {{mvar|X}}, {{mvar|Y}}, and {{mvar|Z}}, called the ''Pappus line''.<ref>{{harvtxt|Grünbaum|2009}}, p. 9.</ref>

[[File:Pappus hexagon.svg|thumb|240px|The Pappus configuration from [[Perspective (geometry)|perspective triangle]]s {{math|△''XcC''}} and {{math|△''YbB''}}
]]

The Pappus configuration can also be derived from two triangles {{math|△''XcC''}} and {{math|△''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]] {{mvar|Z}}, {{mvar|a}}, and {{mvar|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==
[[File:Pappus graph 2COL.svg|thumb|The Pappus graph with bipartite coloring]]
The [[Levi graph]] of the Pappus configuration is known as the [[Pappus graph]]. It is a [[bipartite graph|bipartite]] [[symmetric graph|symmetric]] [[cubic graph]] with 18 vertices and 27 edges.<ref>{{harvtxt|Grünbaum|2009}}, p. 28.</ref>

Adding three more parallel lines to the Pappus configuration, through each triple of points that are not already connected by lines of the configuration, produces the [[Hesse configuration]].<ref>{{citation
 | author-link = Harold Scott MacDonald Coxeter
 | first = H. S. M. | last = Coxeter
 | title = Self-dual configurations and regular graphs
 | journal = [[Bulletin of the American Mathematical Society]]
 | volume = 56 | year = 1950 | pages = 413–455
 | doi = 10.1090/S0002-9904-1950-09407-5
 | issue = 5| doi-access = free
 }}</ref>

Like the Pappus configuration, the [[Desargues configuration]] can 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 point]]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.<ref>{{citation
 | last1 = Mendelsohn | first1 = N. S.
 | last2 = Padmanabhan | first2 = R.
 | last3 = Wolk | first3 = Barry
 | editor1-last = Colbourn | editor1-first = Charles J.
 | editor2-last = Mathon | editor2-first = R. A.
 | contribution = Some remarks on ”n”-clusters on cubic curves
 | doi = 10.1016/S0304-0208(08)72903-7
 | isbn = 9780444703286
 | mr = 920661
 | pages = 371–378
 | publisher = Elsevier
 | series = Annals of Discrete Mathematics
 | title = Combinatorial Design Theory
 | volume = 34
 | year = 1987}}.</ref>

==Applications==
[[File:Orchard-planting problem.svg|thumb|The Pappus configuration, augmented with an additional line (the vertical one in the center of the figure), solves the [[orchard-planting problem]].]]
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.<ref>{{Cite OEIS|A003035|mode=cs2}}</ref>

==References==
{{reflist}}

==External links==
*{{mathworld|title=Pappus Configuration|urlname=PappusConfiguration|mode=cs2}}

{{Incidence structures}}

[[Category:Configurations (geometry)]]
[[Category:Dot patterns]]
[[Category:Greek mathematics]]