Monday, 8 September 2014

Back to rotationally symmetric Venn diagrams

Over the past few posts we have been considering Venn diagrams, their properties and their uses. Although we are usually more concerned with what goes into the Venn diagram sets than their actual shape mathematicians like to abstract everything they can get their hands upon. Thus, they very quickly stopped thinking about the things inside the sets and simply began to consider the properties of the diagrams themselves.

An algorithm for creating a Venn diagram with any numbers of sets was quickly found. For example, Anthony Edwards showed that a diagram containing any number of sets can be constructed using symmetric wavy curves as shown in the animation below.
Anthony Edwards' construction of a diagram which will contain the intersection of all sets. However, it is not a true Venn diagram as, although all possible intersections do appear, some of the intersections appear more than once.
Having solved the basic problem of showing existence further constraints where added to the problem. In particular, mathematicians asked whether it was possible to create rotationally symmetric Venn diagrams. To be honest, I have no idea why rotational symmetry is so highly prized, other than it quite aesthetically pleasing.

A rotationally symmetric Venn diagram of $N$ sets is simply a Venn diagram that can be rotated around its centre, such that after $360/N$ degrees (or $2\pi/N$ radians) the graph looks the same as it did initially. With small numbers of sets rotationally symmetric Venn diagrams are fairly easy to produce. For example below are $N=$2, 3 and 5 set diagrams.
Unfortunately, we are unable to create a rotationally symmetric Venn diagram with 4 sets. As we have seen previously, the 4 set diagram cannot be created using circles. Instead, ovals can be used, as seen below.
Of course, just presenting one 4 set diagram that is not rotationally symmetric is not a proof that such a representation does not exist. Perhaps there is a 4 set Venn diagram with non-regular shaped sets that is rotationally symmetric? Fortunately, there is a simple proof that shows that only prime number set diagrams could possibly be rotationally symmetric. Next time I will reproduce the proof that for any composite number $N$ the accompanying $N$ set Venn diagram cannot be rotationally symmetric.

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