## Tuesday, June 21, 2011

### Circle from Square

I got thinking of a thought experiment where you are in a square room and thinking about the four distances to each of the four corners all added together. I wanted to know what that would look like if you graphed it at different points around the room. I was having trouble visualizing it though so I used Maple (a math program) to graph it for me.
The formula up top is adding the four distances to the corners up... I was surprised at the graph, it's like a black hole! Looking at it for a couple seconds starts tricking my eyes like an optical illusion. The darkest part is where the total distance from the four corners is the least,  2*(square root of 2) and it's the most at each corner 2+(square root of 2) . I was suprised at how you can make a circle by saying where this function is equal to say, 3. That's different than the way you usually define a circle as a certain distance from the center.

*EDIT*
I worked some more on the math behind it but if you just want to hear what I think might be practical applications you can skip to the *APPLICATIONS* below....

A professor of mine, Dr. Rose, was asking me how I knew that that where the above graph is the same color creates circles, not some other shape that just appears circular at a glance. Well, the way I defined it above where the four distances to the corners are added up ends up being pretty hard for me to prove. But I did prove that adding the square of each distance together results in a circle. It goes like this:
In polar coordinates, the distances A^2+B^2+C^2+D^2 from a point somewhere within the square to the four corners is:

Which at first doesn't look very simple or nice, but through the magic of my math program Maple an equivalent formula or simplification to that above is:
Now that is much nicer. In polar coordinates the fact that the theta cancels out means the distances squared to the four corners added together only depend on how far you are from the center of the square, not the angle from the center. In other words, you can move around the center of the square in a circle and not change the sum of the squares of the distances to the corners. The graph still looks pretty much the same as the one in the beginning of this post but the dark area is a bit tighter towards the center.

*APPLICATIONS*
So my formula above says that if you have four equal forces pulling towards something from the corners of a square, the place where it should go to get as close as it can to all four is the very center. And it's like a funnel (circular) that will move anything affected by the forces to the center.
So, in fusion experiments, what I've seen is they try to use a bunch of magnets around the middle in a circle to "push" what they are trying to concentrate in the middle, but I've shown that you can concentrate it with just four magnets flipped the other way "pulling" in the four opposite directions. It seems kind of counter intuitive at first, but the math shows that it would work. And with a circle of magnets pushing in, there will usually be dead spots in between the magnets that allow the material to escape, but with just these four magnets pulling you get a perfect funnel type field to concentrate the particles.