Pop Math

One of my favorite movies is Mr. Holland’s Opus, starring Richard Dreyfus. It’s about a man (Mr. Holland) who, contrary to his expectations finds himself teaching high school music classes rather than being a professional performance musician. At first Mr. Holland has a hard time getting through to his students. He lectures them about the technical details of music. Finally, he realizes that he can get through to his students by playing them the pop music that they already listen to and teaching them about the musical theory that happens there and how it relates to other forms of music, including classical. (I’m not a musician, so forgive me if I am getting something technically wrong here).

The point is that Mr. Holland was able to get his student interested in a subject that initially held little interest for them by approaching it through the channel of pop culture, in particular pop music.

So what is this doing on a math blog? Well, if you’ve read Lockhart’s Lament you realize that my little music analogy here is not original. What I was thinking of this morning is: why don’t we use a similar thing in math: come at math from a pop culture angle. The answer is obvious: THERE IS NO SUCH THING AS POP MATH! There is pop music, pop art, there are pop icons, there is pop fiction (and pulp fiction), Popular Mechanics, even Popular Science, but no pop math. It makes sense. After all, it is natural and healthy to like music, people are born liking music and listen to it all the time, so why shouldn’t there be pop music? But pop math? Nobody enjoys doing math. It’s not a natural thing that people enjoy from birth, or that people use to pass the time! So naturally there is no pop math.

OK, could you tell that I wrote the last paragraph with my tongue in my cheek? What I want to claim in this post is that there is pop math. We just don’t call it pop math (unfortunately). Math is enjoyed by children and it is for everybody (anyone can cook, right Gusteau?). Not all of us have the same mathematical ability. Then again, we aren’t all Mozart. Heck, I’m tone deaf! I still like music. In future posts (sorry, no time tonight) I hope to provide examples of pop math that will convince the reader that it is a real phenomenon, that pop math does exist. (Meanwhile, send me your own examples of pop math).

Stay tuned…

More Experimental Topology

I was thinking that there might be a way to get a hexagon along the lines of the methods of the post Math with scissors. I told my kids about the idea and they were excited to do some math experiments.

I started with three strips of paper.

Taped them together like this:

Then I wanted to tape the three ends up like this,

but I knew there had to be some twisting of the strands involved. In the end we are going to cut each of the strands down the middle.

So we experimented. I knew that at least two of the strands had to have some odd number of half twists in them to have any chance of getting a hexagon (can you see why?). It took us several tries, but my son and I both came up with our own solutions for how to get a hexagon:

Try it and see if you can get it. Here are our solutions:
My son’s solution: Put a single half twist in each of the three strands, twisting two in one direction and the other one in the opposite direction.
My solution: Put a half twist in each of two of the strands, twisting them in opposite directions. Leave the third strand untwisted.

Of course, the experts will want to conjecture and prove necessary and sufficient conditions to get a nice flat hexagon. I also have an idea for making a five-(or more)-pointed-star along similar lines. I’ll let you know what I come up with.