The Many Names of ( 7 , 3 , 1 ) EZRA BROWN

In the world of discrete mathematics, we encounter a bewildering variety of topics with no apparent connection between them. But appearances are deceptive. For example, combinatorics tells us about difference sets, block designs, and triple systems. Geometry leads us to finite projective planes and Latin squares. Graph theory introduces us to round–robin tournaments and map colorings, linear algebra gives us (0, 1)–matrices, and quadratic residues are among the many pearls of number theory. We meet the torus, that topological curiosity, while visiting the local doughnut shop or tubing down a river. Finally, in these fields we encounter such names as Euler, Fano, Fischer, Hadamard, Heawood, Kirkman, Singer and Steiner. This is a story about a single object that connects all of these. Commonly known as (7, 3, 1), it is all at once a difference set, a block design, a Steiner triple system, a finite projective plane, a complete set of orthogonal Latin squares, a doubly regular round-robin tournament, a skew-Hadamard matrix, and a graph consisting of seven mutually adjacent hexagons drawn on the torus. We are going to investigate these connections. Along the way, we’ll learn about all of these topics and just how they are tied together in one object—namely, (7, 3, 1). We’ll learn about what all of those people have to do with it. We’ll get to know this object quite well! So let’s find out about the many names of (7, 3, 1).