The Search for Symmetric Venn Diagrams

Given a family C = {C 1 , C 2 , ..., C n } of n simple (Jordan) curves which intersect pairwise in finitely many points, we say that it is an independent family if each of the 2 n sets is not empty, where X j denotes one of the two connected components of the complement of C j (that is, each X j is either the interior or the exterior of C j). If, moreover, each of the sets in (*) is connected, we say that the independent family C is a Venn diagram. An independent family or Venn diagram is called simple if no three curves have a common point. Introduced by the logician John Venn in 1880, Venn diagrams with n ≤ 3 curves have been the staple of many finite mathematics and other courses. Over the last decade the interest in Venn diagrams for larger values of n has intensified (see, for example, Ruskey [9] and the many references given there). In particular, considerable attention has been devoted to symmetric Venn diagrams. A Venn diagram with n curves is said to be symmetric if rotations through 360/n degrees map the family of curves onto itself, so that the diagram is not changed by the rotation. This concept was introduced by Henderson [8], who provided two examples of non-simple symmetric Venn diagrams; one consists of pentagons, the other of quadrangles, but both can be modified to consist of triangles. A simple symmetric Venn diagram consisting of five ellipses was given in [6]. As noted by Henderson, symmetric Venn diagrams with n curves cannot exist for values of n that are composite. Hence n = 7 is the next value for which a symmetric Venn diagram might exist. Henderson stated in [8]