The Curious History of Venn Diagrams

A look at how Venn diagrams blend logic with geometry

- JACK MURTAGH

IN HIS 1994 BOOK The Mathematical Universe, William Dunham, now an emeritus professor of mathematics at Muhlenberg College, wrote of English mathematician John Venn, "No one in the long history of mathematics ever became better known for less." Although Venn's namesake legacy, the Venn diagram, might not have solved any long-standing problems, surely these interlocking rings deserve more credit. Their compact representation of group relations explains their enduring appeal in classrooms and infographics.

Venn diagrams are more than mere visual aids: they can help us solve everyday logic problems, and they give rise to surprising geometric questions. Have you ever seen a proper Venn diagram with four overlapping circles? No, because it's impossible to make one. Venn himself discovered this predicament and came up with a clever fix, but it only begot deeper geometric puzzles that mathematicians still study today.

Venn debuted his diagrams in 1880 as a means of visualizing contemporary advances in logic. People then applied them in the related branch of math called set theory, which focuses on collections of objects. Venn diagrams typically consist of circles, with each circle representing some set of elements (for instance, things that are cuddly or Broadway shows). The region where two circles overlap contains elements that belong to both sets (cats, perhaps, in our case). Much like one finds when using scatter plots in statistics, actually seeing a problem often clarifies it.

Imagine you're planning a dinner party and navigating your friends' fickle preferences. If Wilma attends, then so will Fred. 1. If Barney attends, then so will somebody else 2. Barney won't come if Wilma comes, but he will if she doesn't 3. If Fred and Barney both attend, then so will Wilma (which says nothing about what she will do if only one of the guys attends) 4. Which people should you expect to show up?