Reblog: Extreme Venn Diagrams

Richer Ramblings here presents a rather attractive Venn diagram showing the possible combinations of eleven distinct sets. It’s a neat picture and one of the things that people who insist mathematics can be artistic are thinking of when they say it.

Venn diagrams are fairly good ways to visualize data, particularly the ways in which things can be parts of one or more sets simultaneously (or maybe part of no set). I find them most useful, in teaching, in doing probability questions, because so many questions about how probable something is amount to “how many ways can a described outcome happen”, and a nice, clean diagram can show just which outcomes fit which description. (“Coin comes up heads and the first child is a girl; coin comes up heads and the second child is a girl; coin comes up tails and the die roll is a prime number”, etc).

For that, though, I find their use kind of limited: if there are too many things happening (coin, child’s gender, die being rolled, goat behind door number two) the problem becomes one students’ eyes glaze over rather than try solving and I lose the thread of the question too. Worse, if there are too many possibilities, the number of lumpy circles I need to draw becomes smaller than the number of lumpy circles I can draw.

This picture does pretty completely away with the lumpy circles and goes in for much more involved curves. Some of the details are kind of small, but, this covers — at least if it was done correctly and I admit not testing — all the different ways that something can belong or not belong to eleven distinct sets simultaneously.

Thinking about the number of different subsets and shades that are needed — go on, how many are needed to give every distinct combination its own color (which isn’t what’s done here)? — makes me appreciate how choroplethy isn’t my thing.

Richer Ramblings

Venn diagrams are cool, and extremely varied.

“If you think Venn diagrams are just a bunch of interlocking circles, think again. Pushing this iconic branch of mathematics to its limits reveals just how varied – and beautiful – these diagrams can be. This gallery showcases some of the wilder possibilities, including the most recent breakthrough in Venn geometry – the first simple, symmetric diagram to encompass a whopping 11 sets.” (New Scientist)

The picture above is the said first simple, symmetric diagram to encompass 11 sets, and yes, it is beautiful. “One of the sets is outlined in white, and the colours correspond to the number of overlapping sets. The team called their creation Newroz, Kurdish for “the new day”. The name also sounds like “new rose” in English, reflecting the diagram’s flowery appearance.“

Amazing stuff. Onwards!

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What Lewis Carroll Says Exists That I Don’t

I borrowed from the library Symbolic Logic, a collection of an elementary textbook — intended for children, and more fun than usual because of that — on logic by Lewis Carroll, combined with notes and manuscript pages which William Warren Bartley III found toward the second volume in the series. The first part is particularly nice since it’s text that not only was finished in Carroll’s life but went through several editions so he could improve the unclear parts. In case I do get to teaching a new logic course I’ll have to plunder it for examples as well as for this rather nice visual representation Carroll used for sorting out what was implied by a set of propositions regard “All (something) are (something else)” and “Some (something) are (this)” and “No (something) are (whatnot)”. It’s not quite Venn diagrams, although you can see them from there. Oddly, Carroll apparently couldn’t; there’s a rather amusing bit in the second volume where Carroll makes Venn diagrams out to be silly because you can make them terribly complicated.