Venn Diagrams’ History and Popularity Outside of Math Explained

In his book mathematical universe, Mathematician William Dunham wrote of John Venn’s legacy, the Venn diagram: “No one in the long history of mathematics was ever better known for less.” Venn diagrams may not solve anything long-standing open problemssurely these interlocking rings deserve more credit. Dense representations of group relationships explain their enduring appeal in classrooms, infographics, and classrooms internet memes.

In addition to visual aids, Venn diagrams can help us solve everyday logic problems, and lead to puzzling geometric questions. Have you ever seen a suitable one Friend chart four overlapping circles? No, because it’s impossible. Venn discovered it himself and came up with an ingenious fix, but this led to deeper geometric puzzles that mathematicians still study today.

Ven In 1880 he released his charts as a method of seeing the progress of the age of logic. Then they found a narrow application the branch of mathematics called set theorywhich focuses on collections of objects. Venn diagrams typically consist of overlapping circles, each representing a set of elements (such as jewelry or Broadway shows). The overlapping region between the two circles contains items belonging to both sets (eg, “cats”). As in use scatter plots in statistics or drawing shapes in geometry, seeing It often solves one’s problems.

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Imagine you are planning a dinner party and navigating the changing preferences of your friends. If Wilma goes, so does Fred. If Barney goes, so does someone else. Barney won’t come if Wilma comes, but if she won’t come. If Fred and Barney both go, so does Wilma. Who should you expect to show up? It’s hard to practice this pose when we’re only given text. A Venn diagram provides a systematic way to visualize and solve problems. Each statement excludes a number of possible outcomes, which we indicate by shading the corresponding regions of the Venn diagram.

Most Venn diagrams you come across depict two or three overlapping circles, but what if you have four or more sets to consider?

Did you spot the problem? There is no region that overlaps only A and C that does not also include another region, and also for B and D. A proper Venn diagram represents all combinations of intersections. Changing the layout won’t help. Each drawing of four circles suffers from the same error.

To see why, start with a single circle and note that it establishes two regions: the inside and the outside. When we add a second set of elements (a new circle), we double the choices, so we have to double the number of regions (first set, second set, both sets, and neither). The only way to do this is to make the second circle intersect the first at two points (touching it at a single point would only affect three regions: the first set, the second set, or neither). This trend continues, where each new circle must double the number of regions if we want to represent all logical possibilities. But the number of new regions cannot exceed the number of new ones intersectionsand a new circle can intersect existing circles at only two points each time. This works well when adding a third circle because we need to add four regions, and the new circle can intersect the two existing circles at two points for a total of four new intersection points. But it breaks with a fourth circle, where we need eight new regions but can only collect six new intersection points.

Of course, we don’t have to limit ourselves to circles. We could easily trace a squiggly loop through a three-circle diagram to cut the required number of regions, but we would lose the elegance of the diagram. Four crosses the spheres it can also represent a decent number of regions, but three-dimensional visuals are difficult to parse. John Venn knew the emptiness of circles, so he proposed ellipses to represent four sets.

Unlike circles, two ellipses can intersect four points This breaks the boundaries with circles but temporarily. Ellipses work in sets of four to five in the same way that circles did before they fail. As the number of sets grows, we need more and more exotic shapes to portray them.

Beyond sets of four elements, it could reasonably be argued that Venn diagrams lose their usefulness. The four-ellipse image is already quite chaotic. Perhaps we should abandon visual representations of more than five sets. But utility does not inspire the mathematician as much as beauty and curiosity. Although Venn diagrams were initially applied logic and set theoryThe four-circle riddle posed an interesting geometry question. That seed has blossomed into fascinating research geometry Venn diagrams that continue today.

Venn and his successors believed that ellipses could not portray all 32 regions required for a five-set diagram. Not until 1975 did the mathematician Branko Grünbaum prove them wrong with the example:

Note also that Grünbaum’s diagram exhibits a pleasing rotational symmetry. If rotated a fifth of a full rotation, it returns to itself, leaving its original shape unchanged. Two- and three-circle Venn diagrams share this property. Rotate the two circle Venn diagram 180 degrees (or a three circle 120 degrees), and it looks the same. But the four ellipse diagram has no rotational symmetry. Can that be fixed? What do two, three and five have in common that do not have four?

In 1960, then an undergraduate student at Swarthmore College, David W. Henderson, answered this question with a surprising discovery (Stan Wagon and Peter Webb filled in some gaps later): Venn diagrams of symmetric rotations are possible only when the number of sets is a number one—1 and a number that can only be divided by itself, such as 2, 3, and 5 but not 4. Henderson only showed that a prime number set is necessary, not that you can always design a symmetric Venn diagram for each prime number. Thus began a competition to find ever larger examples. Here’s a wild look 11 set Venn diagram Peter Hamburger.

Mathematicians from the University of South Carolina solved the question 2004, that there are rotationally symmetric Venn diagrams for each prime set. If you think that made mathematicians pack up their pencils and put the study of Venn diagrams to rest, then you haven’t followed. Instead, the community has raised its aesthetic standards, seeking equal figures more refined properties.

Our initial estimate was that Venn diagrams are overrated. Even those who agree have to admit it has a strange appeal. Take clusters of interesting topics in logic, geometry, and visualization, and you’ll find Venn diagrams at the intersection.