The Desargues configuration

[revised and updated, 10-5-05]

is one of those wonderful fundamental mathematical structures that is both geometric and combinatoric. In the plane a closed configuration of 10 points and 10 lines where three points lie on each line and three lines concur at each point is a Desargues configuration.

Hilbert and Cohn-Vossen, page 127, 28 relate that there are a number of such configureations most of which do not close except under special circumstances.

Here is a way to notate the points of a Desargues configuration (these words are taken from The Triangle Book

The points and lines of the Desargues configuration can each be labeled (P_ij, L_ij) with the 10 unordered pairs ij = ji from 5 letters a, b, c, d, e in such a way that P_ij is on L_kl just if i, j, k, l are distinct. Then for any two of these letters, x, y, the two triangles with vertices P_xi P_xj P_xk and P_yi P_yj P_yk are in perspective, with perspector P_xy and perspectrix L_xy (the edges of these triangles are L_yi L_yj L_yk and L_xi L_xj L_xk respectively).

Here is a picture with one pair of triangles emphasized to show how the notation works. Please not that each triangle in the configuration has another triangle and a point perspector and a line perspectrix that is equivalent to the one shown.

Since this comes from the permutation of 5 letters, it is invariant under S₅, the group of permutions of 5 objects. My first reaction to hearing this was "permutations of what?" As always Coxeter came to the rescue. It turns out that Cayley discovered that 5 points in space in general determine 10 lines and 10 planes which intersect an arbitrary plane in 10 points and 10 lines. The figure is invariant under permutations of these points.

We can see that each point has 3 lines from it. Call points on those lines nearest neighbors of which it has 6. The 3 others are colinear, forming a line uniquely related to the point. Hence the largest path between any pair of points is 2.

For each point there are 2 triangles that can be formed from its nearest neighbors. This point is the perspector of both triangles. Here is a slideshow that shows all possibilities of perspective triangles in a Desargues configuration (Note: iPhoto renders all transparency into black on thumbnail pictures. Don't worry the actual picture pages are fine.)

Looking at this figures from the level of lines is equivalent. Each line has 6 nearest neighbors and 3 second nearest, which concur.

Looking at this figure from the level of triangles, there is one point that is distance 1 from each vertex and one line that is whose 3 points are distance 1 from some, but not all vertices. There are 3 points, forming a triangle, that are distance two from each vertex.

The 10 points and 10 lines can be divided into two pentagons with each side of one passing though a vertex of the other.