Rock Hill, South Carolina|
Alright guys, so something really awesome I learned in my Mathematical Reasoning Class. Pi can equal to 4. Yes, this is mathematically accurate. Don't believe me? That's okay, I didn't believe my professor either at first. But it's true. I'll need to inform y'all on a little background information before I go into the details, but it basically boils down to the way we calculate distance.
The numbers that people are generally used to working with are calculated in something that's called the Euclidean Norm. The name sounds intimidating, but it's really not. To begin with what the Euclidean Norm, which for shortness we'll call "2 Norm," think of two points on the xy-plane. One at (1,0) and the other at (0,1). Actually, write this down, it's easier to understand if you can look at it. Now, the shortest distance between these two points is obviously a straight line that connects them. If you draw this line, you'll see that a right triangle forms from the straight line and the x and y axes. You are able to find the length of the straight line that connects the two points using Pythagorean's Theorem, which is the square root of 2. So, that's all fine and good, but not a lot of people bother to memorize what the square root of two is, so it's not always that helpful.
Furthermore, you can't walk a perfectly straight line from one point to another. Like, if you needed to get from one corner of a city block to the opposite corner, you can't walk a diagonal straight line across the block to this other point because there's probably a building between you and this other point. So, if you've still got the two points I mentioned earlier in your mind, think about walking from the point (0,1) to (1,0). You'll need to imagine walking down the y-axis until you reach the origin, and then walk along the x-axis in the positive direction. So, instead of trying to walk through a building, you walk down the first street, and along the second intersecting street. Now, think about how far you've just walked. By doing this, you are calculating the distance between two points by this formula: d((x1, y1),(x2, y2)) =
| x2 - x1 | + | y2 - y1 |. This is mathematically mapping the distance between two opposing corners of a city block by walking down the streets. By calculating distance this way, you are no longer working with numbers in 2 Norm, you are in what's called the Taxicab/Manhattan Norm (1 Norm). It makes sense why. You drive up the street, turn a 90 degree corner and drive down a different street. You've just gotten from one corner of a block to another.
Now with this idea in mind, start thinking about the unit circle in 2 Norm, where √(x2+y2) = 1 and Pi = C⁄2r; where C is the circumference of the circle, and r is the radius. Now in 1 Norm, the unit circle looks like this. Yup, it's a square, but it's still called the unit circle. You're supposed to be confused by this, but just go with it anyway. Remember that the unit circle isn't just a line on a graph. It's the infinitely many points on the graph that form the shape of a circle. In 1 Norm, each of these coordinate points will add up to the number 1; (x,y) -> x + y = 1. Remember the distance formula above?
Let's use this to find the points between (1,0) and (0,1).
| 0 - 1 | + | 1 - 0 | = 2. A square has 4 sides (in case you were unaware of this) so C = 8 and the radius is 1 (because it's a unit circle). Since Pi = C⁄2r, in the Taxicab/Manhattan Norm, Pi = 8⁄2(1) = 4!
I realize this might be difficult to follow, so if you're still unsure that all this means, take a look at this PowerPoint. It was make by my awesome professor and it goes more into detail about how Pi is derived and how you can calculate it in different Norms. Thank for reading. I hope you enjoyed it.