One of our recent problems is a really lovely geometry result concerning circles. When it’s finished, hopefully those who have solved it will feel a nice sense of accomplishment. And if they are anything like me, they’ll reflect a little bit on what just happened, check to see if they learned anything, think about some of the important themes of the problem, stuff like that. They might also ask a very natural question: “am I going to use this anywhere?”
The answer to that question depends on the problem, of course. For this particular problem, the answer is probably a firm “no”. In other words, the theorem being explored in this problem, in all likelihood, is never going to be used by any scientist, mathematician, or engineer to do anything useful for humanity. This problem will likely not benefit anyone in their daily lives, now or in the future. This lovely geometry tidbit, as lovely as it is, will likely never be used to help spawn the next big invention, or to help solve any one of society’s pressing issues of the day. And this knowledge leads us to our next very natural question: “um… why did I just do it, then?”
For lots of really good reasons! First of all, if you’re concerned about applications, then geometry is a lovely place to start. Geometry is incredibly useful. Whether you’re designing a building, launching a satellite, or just building a doghouse out in your backyard, knowledge of geometry is essential. But beyond particular applications that you might be able to think of, geometry and geometric thinking is the foundation for quite a bit of mathematics, which in turn forms the backbone for engineering and physics. So, learning geometry is pretty important, and if you want to learn it, you have to DO it. The more you do, the better you’ll be. And if you take the time to study some of its more arcane and strange corners, like this week’s problem, you’ll be on the road to becoming an expert! Sure, this particular problem is not “applied”, but the subject it lives inside of most assuredly is, so why not devour it all? Like they say, you are what you eat!
But what if it turned out that the subject of geometry wasn’t particularly useful, out there, in the real world? Should you then ignore it? What if you searched high and low, and couldn’t find any applications of geometry anywhere? Is that reason enough to throw it in the dumpster? Certainly not! Again, if you’re really interested in applications, you can study things that aren’t particularly applied, as long as they foster good kinds of applied thinking! In other words, if you want to go out in the world and make a difference, then you should study things that can train your brain to make the difference you want to make. Whatever you’re going to devote your life’s work to, you are going to need to be a logical thinker. You will want to be able to analyze a situation, understand what you know and what you don’t know, and think things through to obtain the conclusion you want. All of these skills need serious training, and one of the training grounds is definitely geometry! Studying geometry involves constructing proofs, analyzing diagrams, finding patterns, and making conjectures, all of which help to turn your brain into a lean, mean, rational thinking machine! This, then, is one of the best applications of geometry: teaching yourself how to think. When you have that under your belt, you’re ready for anything!
There’s another reason to tackle problems that aren’t incredibly applicable to the real world, though. And that’s for the joy of experiencing them. That’s all! Why do you listen to music? Why do you watch movies, look at paintings, do puzzles, read novels? Because you like to, because these things are entertaining, and because many of these things rise to the level of art. For many, many people, mathematics is an art! A beautiful and unexpected theorem, a clever proof that leaves you breathless, or just a pretty diagram, can all qualify as art. Of course, art is appreciated differently by different people; one person can love a painting, and another can hate it. Still, for the person who loves it, it is worthwhile, even if it has no actual use, out there, in the “real world”. We study fascinating things simply because they are fascinating! That’s part of living and enjoying life. And if you don’t think mathematics is artful, that’s cool too. But at least give it a chance, and don’t feel guilty if you enjoy a piece of mathematics that isn’t “useful”!
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