In his e book The Mathematical Universe, mathematician William Dunham wrote of John Venn’s namesake legacy, the Venn diagram, “No one in the long history of mathematics ever became better known for less.” Whereas Venn diagrams might not have solved any long-standing open issues, certainly these interlocking rings deserve extra credit score. Their compact illustration of group relationships explains their enduring enchantment in lecture rooms, infographics and Web memes.
Not merely visible aids, Venn diagrams might help us resolve on a regular basis logic issues, and so they give rise to stunning geometric questions. Have you ever ever seen a correct Venn diagram with 4 overlapping circles? No, as a result of it’s unimaginable. Venn himself found this and got here up with a intelligent repair, however this solely begot deeper geometric puzzles that mathematicians nonetheless examine as we speak.
Venn debuted his diagrams in 1880 as a technique for visualizing up to date advances in logic. They then discovered utility within the intently associated department of math referred to as set concept, which focuses on collections of objects. Venn diagrams usually encompass overlapping circles, with every representing some set of parts, (e.g., issues which might be cuddly or Broadway reveals). The overlapping area between two circles incorporates parts that belong to each units (e.g., “cats”). Very like in utilizing scatter plots in statistics or drawing shapes in geometry, seeing one’s drawback typically clarifies it.
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Think about you’re planning a cocktail party and navigating your pals’ fickle preferences. If Wilma attends, then so will Fred. If Barney attends, then so will any person else. Barney gained’t come if Wilma comes, however he’ll if she doesn’t. If Fred and Barney each attend, then so will Wilma. Who must you anticipate to indicate up? This poser is difficult to work by after we are solely given the textual content. A Venn diagram gives a scientific technique to visualize and resolve it. Every assertion precludes some attainable outcomes, which we point out by shading the corresponding areas of the Venn diagram.
Most Venn diagrams you encounter depict both two or three overlapping circles, however what in case you have 4 or extra units to contemplate?
Did you see the issue? There is no such thing as a area the place solely A and C overlap that doesn’t additionally embrace one other area, and likewise for B and D. A correct Venn diagram depicts each mixture of intersections. Rejiggering the format gained’t assist. Each four-circle drawing suffers the identical flaw.
To see why, begin with a single circle and observe that it establishes two areas—inside and exterior. After we add a second set of parts (a brand new circle), we double the probabilities, so we have to double the variety of areas (first set, second set, each units and neither set). The one method to do that is to have the second circle intersect the primary at two factors (touching at just one level would lead to solely three areas: first set, second set or neither). This development continues, the place every new circle should double the variety of areas if we wish to characterize all logical potentialities. However the variety of new areas can’t exceed the variety of new intersections, and a brand new circle can intersect the present circles at solely two factors every. This works positive when including a 3rd circle as a result of we have to add 4 areas, and the brand new circle can intersect the 2 present circles at two factors every for 4 whole new intersection factors. Nevertheless it breaks down with a fourth circle, the place we’d like eight new areas however can solely muster six new factors of intersection.
After all, we don’t want to limit ourselves to circles. We might simply hint a wiggly loop by a three-circle diagram in order that it carves out the mandatory variety of areas, however we might lose the class within the diagram. 4 intersecting spheres also can characterize the fitting variety of areas, however three-dimensional visuals are laborious to parse. John Venn knew of the shortcoming with circles, so he proposed ellipses to characterize 4 units.
Amanda Montañez; Supply: “Venn Diagrams and Independent Families of Sets,” by Branko Grünbaum in Arithmetic Journal, Vol. 48, No. 1; January 1975 (reference)
Not like circles, two ellipses can intersect at 4 factors. This overcomes the constraints with circles however solely briefly. Ellipses work for 4 and 5 units earlier than failing in the identical method that circles did. Because the variety of units grows, we’d like an increasing number of unique shapes to painting them.
One might moderately argue that past 4 units of parts, Venn diagrams lose their utility. The four-ellipse picture is already fairly chaotic. Possibly for five-plus units we should always abandon visible representations. However utility doesn’t animate the mathematician a lot as magnificence and curiosity. Though Venn diagrams initially utilized to logic and set concept, the four-circle conundrum raised an attention-grabbing geometry query. That seed has blossomed into an interesting investigation into the geometry of Venn diagrams that continues as we speak.
Venn and his successors believed that ellipses couldn’t painting all 32 areas required for a five-set diagram. Not till 1975 did mathematician Branko Grünbaum show them improper by instance:
Amanda Montañez; Supply: “Venn Diagrams and Independent Families of Sets,” by Branko Grünbaum in Arithmetic Journal, Vol. 48, No. 1; January 1975 (reference)
Discover additionally that Grünbaum’s diagram shows a lovely rotational symmetry. Spinning it one fifth of a full rotation lands it again on itself, leaving the unique form unchanged. Typical two- and three-circle Venn diagrams share this property. Rotate a two-circle Venn diagram by 180 levels (or a three-circle one by 120 levels), and it seems the identical. However the four-ellipse diagram doesn’t have rotational symmetry. Can that be mounted? What do two, three and 5 have in frequent that 4 doesn’t?
In 1960 a then undergraduate scholar at Swarthmore School, David W. Henderson, answered this query with a stunning discovery (Stan Wagon and Peter Webb crammed in some gaps later): Rotationally symmetric Venn diagrams are attainable solely when the variety of units is a prime quantity—a quantity divisible solely by 1 and itself, resembling 2, 3 and 5 however not 4. Henderson solely confirmed {that a} prime variety of units is critical, not that you may at all times design a symmetric Venn diagram for each prime quantity. Thus started a contest to seek out bigger and bigger examples. Right here’s a wild-looking 11-set Venn diagram from Peter Hamburger.
Mathematicians on the College of South Carolina settled the query in 2004 by displaying that rotationally symmetric Venn diagrams exist for each prime variety of units. Should you suppose this brought about mathematicians to pack up their pencils and lay the examine of Venn diagrams to relaxation, then you definately haven’t been following alongside. As a substitute the group has raised their aesthetic requirements, looking for figures with even extra refined properties.
Our opening quote contended that Venn diagrams are overrated. Even those that agree should admit that they’ve a curious attract. Take the units of attention-grabbing subjects in logic, in geometry and in visualization, and also you’ll discover Venn diagrams on the intersection.
