A generalized quadrangle: the doily
Take a set of six elements, say S = {a, b, c, d, e, f}. Consider all two-element subsets or unordered pairs of elements of S, for instance {b, e}, and call these duads. Next, consider all partitions of S into three such unordered pairs, for instance {{a, f}, {b, e}, {c, d}}, and call these synthemes. One can count that there are precisely 15 duads and 15 synthemes.
Out of these combinatorial concepts, we can now define an interesting geometrical configuration. Consider the 15 duads as “points” and the 15 synthemes as “lines” with the obvious incidence relation (so that for instance, {b, e} is one of the three points of the line {{a, f}, {b, e}, {c, d}}). If we allow lines to be curved, we can obtain the following beautiful illustration, where the blue circles are the “points” of the geometry (depicting a duad) and the black lines each connect three such points. Note that there exists precisely three lines through each point.
A little confusing may be that the lines appear to intersect in other points, such as the center of the pentagon, but these do not count: we only refer to the duads in the blue circles as being points of the geometry.
(Remark: in this situation, it is not really necessary to visualize the geometry’s lines as being curved, as the Cremona–Richmond configuration demonstrates. It just looks more balanced this way.)
This incidence structure is commonly known as the doily. It is the smallest non-trivial example of an interesting class of geometries known as generalized quadrangles (yes, even though the illustration clearly shows a pentagon). In general, a generalized quadrangle contains points and lines obeying the following axioms:
1) Two distinct lines intersect in at most one point; 2) There is at most one line through two distinct points; 3) For any point P not on a line L, there is a unique line passing through P which intersects L.
The third axiom implies that there cannot be any “triangle” in a generalized quadrangle, but that there exist lots of “quadrangles”, hence the name.
