Fun with Moebius Noodles

About 2 weeks ago, I stumbled across the Moebius Noodles group referenced on the Let's Play Math blog. Inspired by the idea of sharing cool math with my kids, we spent last Wednesday playing with Moebius noodles. For those of you who don't know, a Moebius noodle (also known as a Moebius band or strip) is a loop of a flat material (usually paper) that is twisted once before the ends are joined. Here, B demonstrates one.



Both P and E's fine motor skills are at a level where they can make their own, so they pasted together both regular loops and Moebius noodles. I had them each draw a line down the middle of "only one side" of each loop. One fascinating thing about Moebius noodles is that they only have one side. We then experimented with cutting our loops in half lengthwise. With a regular loop, of course, you get 2 loops. With a Moebius noodle, well... try it yourself if you've never done it. It's really fun, and surprising the first time.



Since we were still enjoying ourselves and it wasn't lunch time, I thought I'd try out another idea I stumbled upon. The gist is that you take two loops, glue them together, and then cut them both in half lengthwise. First, I took two regular loops and glued them at right angles. It looked like the beginnings of a chain you might put on a Christmas tree, only the connection was on the outside. I cut it in half down the middle of each loop, through the intersection point, and ended up with this:



P thought this was interesting, and tried it herself (by this time, E was busy attaching yellow loops to a piece of paper to make a "rocket", which seemed valuable in its own way). Meanwhile, I tried what happened when one of the loops is a Moebius noodle and one is a regular loop, which turns out to be the same as when both are regular loops. Finally, I tried both of the possible ways of joining two Moebius noodles: one when the loops are twisted the same direction, and one when they're twisted in opposite directions. I found it completely mind-bending! When the loops are twisted in the same direction, you get this:



But when they're twisted in opposite directions, this is what you get. (P tried this one as well, though I had to help her make the Moebius noodles so that they'd be even enough to cut in half neatly).



An instant "I LOVE MATH" logo!