The Fabulous (11,5, 2) Biplane

After a workshop for new teaching assistants on innovations in teaching, a new sociology graduate student wandered into my office and asked the question, " Tell me. .. how do you make math exciting for students? " By chance, I just happened to have on my computer screen a picture that exhibits some of the symmetries of one of the most intriguing objects in mathematics: the (11, 5, 2) biplane. Figure 1 A fascinating picture I told him of my chagrin on seeing a picture similar to FIGURE 1 (but much prettier , and in color) on the cover of a book [6] on combinatorial designs. The picture was lovely, and the reason for my strong feelings was purely selfish: I was trying to construct such a picture, and somebody else thought of it first. But it wasn't labeled. It was fun finding a labeling compatible with the symmetries of the biplane. To find generators for the symmetry group of the biplane—which turns out to have a name, PSL(2, 11)—was more fun. The best part, however, was learning about the exact connection between the biplane and six pairs of mathematical objects. We find these six mathematical pairs just outside the boundaries of many traditional courses, where a bit of exploration can lead the curious to all manner of interesting mathematics. A good course in coding theory will mention two pairs of perfect error-correcting codes, namely the Golay codes {G 11 , G 12 } and {G 23 , G 24 }, but sometimes only in passing. Look past the usual topics in combinatorics into the world of combinatorial designs and you will meet two pairs of Steiner systems,