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The Pappus configuration can also be derived from two triangles and 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
The Levi graph of the Pappus configuration is known as 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 a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
, with three points per line and three lines through each point.
History and construction
This configuration is named afterPappus of Alexandria
Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...
. Pappus's hexagon theorem states that every two triples of collinear points and (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines , , , , , and , and their three intersection points , , and . These three points are the intersection points of the "opposite" sides of the hexagon . According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points , , and , called the ''Pappus line''.
The Pappus configuration can also be derived from two triangles and 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
In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point.
Graphics
The science of graphical perspective uses perspectivities to make realistic images in p ...
, , and . 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
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 ...
. It is a bipartite symmetric
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
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 bip ...
with 18 vertices and 27 edges.
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.
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 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 (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph ...
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 Greek mathematics