For the past week my kitchen has had an ant problem. Any normal person would get some ant traps and be done with the problem. Not being a normal person I've spent the past week learning about ant behavior. I learned that they only focus on one bowl of cat food, even though I have two in my kitchen. They rarely form lines of ants unless there is something really good at the other end like a used lollypop stick. If they can't find the cat food, they love the cat's water dish but I can't figure out why. Leaving a path of ants killed by Windex doesn't stop them, just slows them down for a day. Killing ant "scouts" is of no use. My goal has been to deter the ants from coming in the house and I've had little luck. After much office discussion and memories of my thesis, I've decided that if I can't beat them, I'll use them. I will use them to learn about the topology of various surfaces. When describing curvature of surfaces one often says "if I were an ant on a surface, it would look like this..." Well, I have ants, and I can make surfaces!
In my ant experiments I'm going to look specifically at one type of curvature, geodesic curvature. This type of curvature is extrinsic meaning that to figure it out, you can't be actually on the surface. Only an observer could figure it out, not an intelligent ant with a ruler. To find geodesic curvature, an ant on a surface would walk the shortest distance between two points as an observer watched (you can see where this is going). Then take this path, jam the wonky surface back into flat space and look at how the path is curved. Think of a sphere. If you tired to get from point A to point B on a sphere but had to stay on the sphere, the shortest path isn't a straight line because you aren't allowed to go through the middle of the sphere. Then if the sphere were flattened, the geodesic curvature would be the curvature of the path projected in a flat space.
I plan on finding the geodesic curvature of wonky shapes with my new pet ants. Ants generally get from point A to point B in the shortest possible way. My plan is to make a funky looking shape with paper and let the ants walk over it to a tasty piece of hard candy at the other end. Then, i can trace their path and flatten on the shape. At this point you are probably saying to yourself "why don't you just use math to do this!?!" Yes, I could technically do that, but I would have to know what equations make that surface exist. And those are hard to calculate for random surfaces. Also, this is a heck of a lot more fun.
I also plan on trying to make an ant trap by smearing ant scent on a Mobius strip and seeing if they will follow it and get stuck. How many times have you said "if an ant were walking on a Mobius strip it would only be walking on one side and never escape." We shall see...