If you were to 'color' these like a four color map problem, are they all trivially colorable with just two colors? It looks that way from the pictures, but I lack the math intuition to prove it.
Yes, the intersection of a number of curves defining regions is color-able with two colors: any point covered by an odd number of regions is black and any point covered by an even number of regions is white (i.e. a checkerboard pattern). Works as long as the boundary intersections are transverse. The proof is simply that if you cross a boundary, you must be decreasing or increasing the region count by one, changing the parity.
Each color is touched by at most one color on top and one color on bottom. Looking at the original 7-Venn we get Red->Orange->Yellow->Green->Blue->Purple. Change the odd colors to white and the even colors to black and you're done.
We have a print-out of an N-set Venn diagram on the wall which we use to illustrate the peril of having too many features with subtle interactions. It resonates because many of us have worked on areas of the code preceded by comments indicating which subset of interacting features you need to keep in mind. There's at least 7 such features, so it's not a small space. We've been desperately struggling to make these features more orthogonal and having a succinct diagram of the problem (read: "enemy") has helped us stay motivated.
megrimlock has a poster of a Venn diagram on the wall. The intersections in the diagram represent interdependencies between different parts of a piece of software (in object-oriented programming terms: places where encapsulation is broken). The Venn diagram is there to motivate the removal of those interdependencies.
Sure, there are both interesting and practical uses of mathematics.
However, venn diagrams are not synonymous with set theory. Venn diagrams are a type of visualization, and once you're dealing with 11 overlapping sets... I don't really see the utility, since these are not human friendly and don't help viewers understand the relationship between sets and their intersections.
There are practical uses for much math but in very many cases it's only practical in extremely specialized fields/cases. Meaning, in most cases very impractical if speaking strictly in terms of practicalities. Yet, it's widely taught, widely misunderstood, and so widely hated. The idea has become humorous to me -- math, a natural pervasive wonder of our universe, hated by many. Very impressive feat.
On this topic this book is great: A Mathematician's Lament by Paul Lockhart. It opened my eyes to the concept of viewing much of math not in terms of something strictly practical, but rather in broader terms, ie. natural art and beauty with very practical applications mixed in here and there. Just as one person enjoys painting, another enjoys dabbling in mathematics taking pleasure from the elegance.
(Venn diagrams can be interpreted in a set theoretic way, but this isn't particularly useful within set theory itself, and the symmetric and simple properties of the Venn diagram are unnecessary for this interpretation. We've had a construction for a simple Venn diagram of n curves since Venn's original paper -- it's the symmetry that's new.)
When I think of practical uses for a venn diagram I think of writing labels in each distinct section... this doesn't seem very practical in that regard.
What would be really cool (and practical) is if all the different segments / sections were of relatively similar sizes.
Anyone know if there's a way to draw that? The examples shown in the article have really large areas and really small areas...
EDIT: It seems venn diagrams lose their "functionality" after 5 sets. After that they are far too complicated.
