#generalized quadrangle

Take a set of six elements, {a, b, c, d, e, f}. Every pair of elements is called a duad. A triple ot three duads that partition the original set, such as {a, d}, {b, c}, {e, f}, is called a syntheme. A quick check shows that there are 15 possible duads in total, and also 15 synthemes.

We can adopt a more geometric viewpoint by declaring the duads to be “points” and the synthemes to be “lines” through three such “points”. Then—if we allow to curve the “lines” to make our life easier—we obtain the figure above. In this form, the configuration is known as the doily.

One defining property is that no three “points” make a “triangle” of three “lines” in this structure. But, if we look at any “point” P not on a “line” L, then there is a unique “point” Q on L such that P and Q lie on a “line”. Hence this configuration does not contain combinatorial “triangles”, but it does contain lots of “quadrangles”. It makes it a prime example of a generalised quadrangle.

With a slightly different viewpoint we can realise this configuration using only familiar straight lines; this version is known as the Cremona–Richmond configuration.