Venn and the art of diagrams

ANYONE WHO HAS ever spent time in a conference room equipped with an overhead projector is familiar with the basic Venn diagram -- three overlapping circles whose eight regions represent every possible intersection of three given sets, the eighth region being the space around the diagram. Although it resembled the intertwined rings familiar in Christian (and more recently, Led Zeppelin) iconography, when Cambridge University logician John Venn devised the diagram in 1880 it was hailed as an innovative way to represent complex logical problems in two dimensions.

There was just one dilemma, according to British statistician and geneticist A.W.F. Edwards, author of the entertaining new book "Cogwheels of the Mind" (Johns Hopkins): Venn's diagram didn't scale up. When a problem involves four sets, circles don't have enough possible combinations of overlaps. Ovals work for four sets, Venn found, but after that one winds up drawing unreadable messes. What to do?

In Venn's day, a rival lecturer in mathematics at Oxford by the name of Charles Dodgson -- better known to us as Lewis Carroll -- tried to come up with a superior logical diagram using rectangles but failed (though not before producing an 1887 board game based on his design). In fact, it wasn't until 1988 that the problem of drawing visually appealing Venn diagrams for arbitrary numbers of sets was solved -- by A.W.F. Edwards himself, as it happens. Inspired by the seam of a tennis ball, Edwards came up with a five-plus-set diagram (original version, above right) that was later nicknamed the "Edwards-Venn cogwheel."

Bonus: In homage to the source of his eureka moment, the author includes step-by-step instructions for drawing complex Venn diagrams on a tennis -- or "Vennis" -- ball.