TITLE: Why can a Venn diagram for $4+$ sets not be constructed using circles?
QUESTION [210 upvotes]: This page gives a few examples of Venn diagrams for $4$ sets. Some examples:


Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-set Venn diagram using only circles as we could do for $<4$ sets. Yet it is doable with ellipses or rectangles, so we don't require non-convex shapes as Edwards uses.   
So what properties of a shape determine its suitability for $n$-set Venn diagrams? Specifically, why are circles not good enough for the case $n=4$?

REPLY [2 votes]: I'm quite late to the party, but I'd like to add a very simple but much less rigorous answer for variety.
Each region of a Venn diagram must represent some combination of true or false for each category. For instance, the two circle Venn diagram has 4 regions: the intersection, the outside, and the last two circle parts. The intersection contains everything true to both categories, the outside neither category, and the last two only one of the two categories. (You could list them like TT, TF, FT, FF)
In order to fully cover all cases of true/false combinations, we need $2^n$ sections. Each new category we add doubles the number of regions as it must split each existing region into 2 smaller regions. (One for the true case of the new category, the other for false, so the TT region is now split into TTT and TTF)
We can see how the third circle intersects all 4 regions, but there's no way to draw a 4th circle that intersects all 8 of the new regions, making it impossible to draw a 4 circle Venn diagram.