Preface: I'm almost embarrassed to post up a math discussion of such a simple problem, but the answer is nonintuitive enough to be interesting. Please share your thoughts!
Today is something a little different: math. I'm sure some of you have already lost your appetite - but hang on for a second. This is sort of interesting.
TL;DR I math'd a graph for Anteaters who don't know whether to take inner or outer ring mall.
As I'm sure many people do, I always wonder if I'm walking to class via the most direct path. It's a matter of efficiency and elegance. It's also a matter of trying to get to class on time.
So in the midst of mentally minimizing my mileage at UC Irvine, I considered the two main paths that make up the campus.
If the above image is not clear, the campus is pretty much a donut (geographically, demographically). (I live right next to the star-trek library on the bottom left). In fact, it's two concentric circles, one slightly smaller than the other. One can more or less travel between the two circles. The idea can pretty much be distilled as in the following image:
Lets say you're at your computer science class on the south west side of campus, and you want to get some Peets coffee on the north of campus. Both points are on the outer circle, but you have full access to the inner circle. You want to get caffeinated as soon as possible, and you ask yourself: should I cut to the inner path (red), or travel on the circumference (orange)?
We can find the conditions for the paths to be equal by setting the last equation to zero. By inspection, we find that at two radians exactly, the paths are equal.
I arrived at this answer the first time I formulated this problem, but it seemed so strange for two reasons. Firstly, the functional is independent of radius. Secondly, my intuition thought that it was strange to find an integer value of radians.
I asked my friends to find my error, and it wasn't until Mitchell Hsing pointed out that math doesn't lie that I reconsidered the validity of this conclusion. Assume for a second that it's correct:
Absurd!! Regardless of the relative or absolute sizes of the circles, the two paths are always equal at an angular distance of two radians. If you know any reason why this should be intuitive, or why it isn't, add a comment!
Anyways, I since made a handy cheat sheet for UCI to help you decide whether you should take the inner or outer path. It's a little simplified, but I hope you enjoy it! You can download the full version here.
3 comments:
My boyfriend is a genius, theoretically speaking. In practice, the guy takes the longest route possible by the longest means possible. So if you see him walking along the side of the road, please give him a ride and remind him of his genius idea of taking the shorter way home.
Theoretically speaking, wouldn't it be faster to just to cut across the park?
It makes sense for me that 2 radians is the critical angle thinking of the limiting case of the smaller circle being the point.
At that point, it makes perfect sense that the 2 paths are equal at 2 radians, the outside path you travel 2*r, the inner path you literally just traverse 2 times the radius.
So then, at any point you are travelling to the inner circle, you could turn, and take a *different circle*, but THAT path will be the same as going to the origin and back again, for the same reasoning as before. So it doesn't matter what the size of the inner circle is, 2 radians is the magic number.
As to WHY 2 radians is the magic number, it's because you go in, and then out, and this can again be seen in the limiting case of the inner circle being a point.
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