The Desargues Configuration – A Quick Tour

Configurations are a wonderful way to confront the Euclideanly prejudiced with the abstraction of incidence geometries. Take for instance the Desargues configuration. Its ten points are given by the 10 different subsets with two elements of the set {1,2,3,4,5}. Here they are:

{1,2},{1,3},{1,4},{1,5},{2,3},{2,4},{2,5},{3,4},{3,5},{4,5}

Its ten lines are given similarly by the 10 different subsets with three elements of the set {1,2,3,4,5}. Here they are, likewise:

{1,2,3},{1,2,4},{1,2,5},{1,3,4},{1,3,5},{1,4,5},{2,3,4},{2,3,5},{2,4,5},{3,4,5}

Then we say that a point is incident with a line if is a subset of the line. For instance, the point {1,2} is incident with the three lines {1,2,3}, {1,2,4}, and {1,2,5}. Clearly, every point us incident with exactly three lines. Likewise, every line is incident with precisely 3 point: For instance, {1,2,3} is incident with {1,2},{1,3}, and {2,3}. This makes the Desargue configuration a configuration of type (10,3).

Desargues5a

Above is a nice geometric interpretation of this. Take five planes in space so that no two of them are parallel and no three intersect in a single line. Then these planes will intersect in the 10 lines and 10 planes of the Desargues configuration. Unfortunately, there is no way I can think of to place five planes into space so that this will look pretty, the reason being that the “faces” of this configuration are nonconvex quadrilaterals. But one can do differently. Still using space, consider a tetrahedron. Take as points the four vertices of the tetrahedron as well as the 6 midpoints of the edges. Lines are the six edges of the tetrahedron and the incircles of the triangles. That requires bending our understanding of lines a little, but now this configuration becomes easy to visualize.

Desargue tetra

Where have the five planes gone? You could say that four of them have become the faces of the tetrahedron, and the fifth a wisely chosen sphere. The right image shows in a top view how to label the points and lines. If this is still too concrete, you can also use the following 2-dimensional drawing.

Desargues

This is illustrating Desargue’s Theorem: Take two triangles p1, p2, p3 and q1, q2, q3 which are in perspective, i.e. have the three lines lines p(i)q(i) pass through a common point p. Now consider the intersection r12 of the two lines p1q1 and p2q2, and likewise the intersections r13 and r23. Desargue tells us that these three points lie on a common line. This sounds complicated, but given that we are only using the most elementary notion from geometry, namely incidence, it is surprising enough that there are non-trivial theorems at all. Why is this true? A miracle? Let’s move back into space and pretend for a second that the two triangles lie in different planes.

Desargues space

Then these planes meet in a line, which is the one through r12, r13, r23, because these points are intersections of lines that belong to one of the two planes each. So, in space Desargue was evidently right. Can things go wrong in the plane? Yes and no. No, because a Euclidean plane can always be though of as a plane in space, and we can understand any planar Desarguesian construction of the projection of a similar configuration from space. No, because there are in fact “planes” that do not lie in any space and where Desargues’ theorem is false.

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