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We first present a general overview of Venn diagrams and their properties and uses, focusing on diagrams with rotational symmetry, which can be quite beautiful. We then discuss some recent results about Venn diagrams. Killian, Griggs, and Savage proved that symmetric Venn diagrams exist if and only if the number of curves, $n$, is prime. However, the resulting diagrams are highly non-simple, where a simple diagram is one with at most two curves passing through any point. In fact, the KGS diagrams are minimal with respect to number of intersection points within the class of diagrams drawable with convex curves, since they have ${n \choose n/2}$ points of intersection. We show how to modify their construction so that the resulting diagrams are ``half-simple" in the sense of having asymptotically $2^{n-1}$ points of intersection, whereas a simple diagram has $2^n-2$ points of intersection. Time permitting, we also present some recent constructions of pseudo-symmetric and area-proportional Venn diagrams.

Adrian Vetta 2004-03-05