Tuesday, June 2, 2015

Euler, Not Venn

There are Venn Diagrams and there are Euler Diagrams.

Here's a Venn Diagram:
Fruits, Red Things, and Hats.  You can see where Red Things and Hats cross, there are Cardinals caps and Santa hats. Where Red Things and Fruits cross, there are plums (red ones), apples (red ones) and berries (I had to write small; assume strawberries and other red berries). Lastly, where Fruit and Hats cross, there's obviously Carmen Miranda.

Note, however, that there are no Red Things that are also Fruits AND Hats. But the diagram shows the possible location of red fruit hats. It is possible. But, as far as this Venn Diagram goes, no red fruit hats are known. I didn't do a lot of research, frankly.

But they could exist. It could happen. The Venn Diagram says so.

Here's an Euler Diagram.
Note the differences. There are no Red Things that are Fruits and Hats, and so there is no place where they overlap. There are subsets of Red Things, like things with wheels (firetrucks and my 1974 Triumph Spitfire), and things you can eat (beets and red, red wine). Also, there is a set of Maple Trees that are sometimes Red and sometimes not, but are never Hats or Fruits. And of course, the important set of Artists from Bulgaria, who are not Maple Trees, Fruits, Hats, or Red things (except possibly Christo...).

A Venn Diagram of this mess would be, well, a mess. Lots of empty sets where, for instance, Hats and Maple Trees cross. No subsets. Nothing that stands alone and apart. Even though there are no Bulgarian Artists that are also Maple Trees, a Venn Diagram would contain the possibility that they could exist in some ridiculously strained ovoid container.

It would be nearly impossible for me to draw using MS Paint. I'm not even going to try.

Euler Diagrams (named for Leonhard Euler, who happens to be my favorite mathematician) contain only the relationships that actually exist. The way things are is the way things are. All the what-ifs and impossibilities are burned away and what you have is reality. Reality with rounded crayon edges and amusing anecdotes and plenty of strange people.

Hence, this is blog is Euler, not Venn.


  1. I love you Bridgett. I don't think anyone else I know could or would utter/write the phrase, "... who happens to be my favourite mathematician."

    I also learned something - I don't think I knew what a Euler diagram was before reading this.