### 5 Ways and a Headache

August 9, 2006 5 Ways and a Headache## "I use variety to see where mathematics connects on itself."

I didn't really want to do anything due to spending most of the day with a headache. Toward the end of the day however, I was concentrating on the point that is opposite to a point given on a circle. These form the endpoints of the diameter. And I figured out 5 ways to do the same thing. Though I haven't evaluated all of them. I use variety to see where mathematics connects on itself. One idea can be used to prove another or explore an old theme in unexpected ways. The path is always a bit exciting. Seldom do I come across an idea like this that keeps me so entertained. I suppose the makings of which began a long long time ago with circles.## "The more ways I can solve the same problem mathematically the more likely I can draw connections between odd ends of the mathematical world"

That's why I paused to play my guitar after writing down 4 ways to arrive at the answer. As I played the usual rhythms on my instrument inspiration dawned on me. I'm not sure how but I realized that the point on the opposite side... was just that. It was the opposite of the point given. It was suddenly so simple. Even proving it was simple. I have yet to pin point the idea directly but the basic notion was that I had found the utterly simplest and most elegant solution to the problem. And than I started to think about the former ways to solve in a new light. Opposite point, eh? Well, then that would mean each other answer has to arrive at the same conclusion. Which meant that I suddenly knew something knew about all the ways I phrased the problem. And the one thing that began to intrigue me the most was this one set up. I thought given a point on the circle all one would have to do was walk half the Circumference to arrive at the point on the other side. This took mathematical form as an integral for length. An integral my calculator could not solve directly. I suspected that I could use the trig sub but haven't done it yet... but that was the thing. Knowing the new information about the opposite cast this integral into a whole new light. I now knew the starting and end points ahead of time which meant I could say something about this function I was suppose to find. The difference of the function between an input x and the opposite input x was half the circumference. .. Now where does that get me? ... I solve these things in this way to train myself to see the mathematics more clearly. There are a lot of people out there that stand only upon research. But to really understand mathematics one can not grasp the ideas firmly simply by looking them up. I firmly believe one has to get in and prove out the laws and rules in order to fully appreciate the complexity of the axiomatic systems playing out upon the world.

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