The most beautiful present

By Jeffrey Ayers

Lenore Edman/Flickr

Ever since I became interested in math, I’ve always wanted a Klein bottle. It therefore comes as no surprise that ever since receiving one, I’ve been completely enthralled with it. I stare at it every day and marvel at the glasswork and incredible mathematical concepts behind it. To me, the Klein bottle represents everything I love about math: Beautiful shapes, confusing results and a connection with other subjects and seemingly opposite areas of math.

I realize that what appears to be a glorified water glass may seem like a weird thing to get so excited about, but trust me, it’s much more than that. The Klein bottle is a three-dimensional representation of a four-dimensional object with no edges. Technically, we would need to live in four spatial dimensions to observe a true Klein bottle, but since we don’t, we’re relegated to the not as precise, but still mathematically and aesthetically pleasing three-dimensional object.

The Klein bottle, named after its inventor, German mathematician Felix Klein, was created in the 1800s. Klein thought of his bottle when asking what two Möbius loops would look like if you sewed their edges together. Möbius loops themselves are fascinating things that are created when you take a strip of paper, give it a half twist and tape the edges together, creating an object with one surface and two edges. Klein knew that when you take a piece of paper and make a Möbius loop, you turn something with four edges into two. Logically, if you took two Möbius loops and sewed their edges together, you’d get something with no edges, right? Well, that’s exactly what a Klein bottle is! If you’re still unable to picture what a Möbius loop looks like, find any bottle and look at the recycling symbol of three twisted arrows in a triangle. That’s a Möbius loop in action.

This incredibly nifty object has no edges and only one side. When I say no edges, I mean that with a traditional bottle, you have a lip where there is a distinct separation between the inside and outside of the bottle. If you squished the bottle to make the lip very sharp, a little ladybug would cut itself before crossing to the inside of the bottle. But that ladybug doesn’t have to worry about that with the Klein bottle. It can travel on the surface and then move in the hole of the bottle, through the tube and into the inside of the bottle without crossing an edge. And another little fun fact is that a true Klein bottle has no volume.

The Klein bottle is just one of so many incredible creations from an area of math called topology, defined as “the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects.” Did you know that a coffee cup and a donut are the same shape topologically? They both have only one hole. Crazy, right? And that’s just the beginning of the wonderful world of topology.

This Klein bottle reminds me why I fell in love with math in the first place. Every time I look up at it in my room I can’t help but smile and think about the first time I saw one and how amazing I thought, and still think, it is.

Jeffrey Ayers is a Collegian columnist and can be reached at [email protected]