n-Venn diagrams

As a member of the American Mathematical Society, I receive its monthly Notices, which arrives about 2 months after its publication. In the December 2006 issue is an interesting article on n-Venn diagrams (pdf file with pretty colour diagrams), the first para of which is as follows:

Many people are surprised to learn that Venn diagrams can be drawn to represent all of the intersections of more than three sets. This surprise is perfectly understandable since small Venn diagrams are often drawn with circles, and it is impossible to draw a Venn diagram with circles that will represent all the possible intersections of four (ormore) sets. This is a simple consequence of the fact that circles can finitely intersect in at most two points and Euler’s relation F − E + V = 2 for the number of faces, edges, and vertices in a plane graph. However, there is no reason to restrict the curves of a Venn diagram to be circles; in modern definitions a Venn diagramis a collection of simple closed Jordancurves. This collection must have the property that the curves intersect in only finitely many points and the property that the intersection of the interiors of any of the 2n subcollections of the curves is a nonempty connected region.

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