## See things differently than others? This science is for you

Have you ever stared at something until it starts to look like something else? I mean, when you haven’t been drinking. I haven’t, but I am told some people can. It must be true because there are numerous people who can look at something and see it completely differently from the way I do.

If you’re really good at seeing things differently you might consider a career in topology. This is a field of mathematics that studies geometric problems that depend not on the exact shape of the object but on the way the object is put together. In other words, topologists study the properties that are preserved in an object when it is deformed, twisted or stretched. Tearing, however, is not allowed.

In topology a circle is considered equivalent to an ellipse because it can be deformed into that by stretching. A sphere is topologically equivalent to an ellipsoid. I don’t know about you, but I just flat disagree with that. I don’t know who makes up these rules. Well, actually I do know. Topologists make up the rules. As usual with mathematicians, I don’t know what they are talking about.

Topologists do have some useful ideas though. For example, a topologist might have a hard time determining a doughnut from a coffee cup. See, if one were to twist and poke a pliable doughnut into the right shape, the hole could become the handle and one could poke a cavity to become the cup portion. It’s the same structure, just twisted differently.

Now before you get all excited, I have already patented the idea. This is going to be huge — a doughnut with the coffee poured into it! No more messy dipping.  We’ll start with a simple glazed doughnut, but there should be no reason we can’t expand quickly into cinnamon and cherry jelly.

Anyway, that gives you a feel for how important topology can be in solving scientific problems. Another major contribution of topology was the famous “hairy ball theorem.” This theorem was first stated by Henri Poincare, a French mathematician in the 19th century. He stated that “there is no non-vanishing continuous tangent vector field on even dimensional n-spheres.”

That’s what I’m told, anyway. I think it’s French. Translated, I think it means that if you try and comb a hairy ball flat, there will always be a cowlick. It took years of experimentation with comb-overs to prove the validity of this now-accepted theorem.

I had a topological experience once, and now I have a greater appreciation for this field of endeavor. It turns out that when a retina becomes detached (the retina is the field of nerve endings in your eye) someone either has to solder it back in place with lasers or put a rubber band around your eyeball to push it back to the retina. It’s sort of a “if the mountain won’t go to Muhammad, Muhammad will go to the mountain” kind of situation. Wait, can I say that kind of thing?

Anyway, I ended up with a cool rubber band that distorts my left eyeball. Now, when I close my right eye and look at a round clock on the wall, it looks oblong. It used to, anyway. Apparently one’s brain can somehow correct for these kinds of things because now the clock looks round again. I’m back to normal, but it’s not as entertaining as when I could switch back and forth looking at the clock with each eye.

This raises the interesting question about whether topology is even real or not. I mean, if we stare at something and deform it in our minds, will our minds just un-deform it after a while? This is a hopeful thought. Perhaps someday the brains of all those people who see things differently from the way I see them will eventually un-deform and they will finally begin to agree with me.

Gary McCallister, .(JavaScript must be enabled to view this email address), is a professor of biology at Colorado Mesa University.

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