A Wild Obtuse Chase: Discovering the Hard Truth of Mathematical Identity in a Foggy World

Words are dangerous things. They're confusing, blurry, overly-complicated, emotional, and sometimes downright unpleasant. Unfortunately, words possess a rather complete monopoly on communication, and so all we can do is try our hardest to avoid falling into their numerous traps. However, being dangerous things, words always seem to find a way to get the best of you in the end, no matter how carefully you tread; experience will postpone this moment, but it will come. This is a tale of how two young, foolish students of mathematics managed to trip over the very first hurdle, and how they then spent several minutes running in completely the wrong direction, and how they eventually realized their mistake and turned around, and how they then, finally, wound up back at the starting line, having made absolutely no progress in the race, but having become a little bit wiser for it.

Let's start with angles. We are all familiar with angles in two dimensions, and how we represent them as fractions of a 360 degree circle. However, my esteemed colleague Vikram and I decided to try and extend our logic. What about three dimensional angles? That is to say, how can we better visualize the relationship between 3D angle and sphere, in the same we we do for 2D angle and circle? The answer is rather simple, actually.

Above, we have two images, on with a circle, the other with a sphere. In both cases, the point labeled 'O' is the center. As you can see, in the first image, two rays extend from the center of the circle, with the points at which they intersect the perimeter labeled 'A' and 'B'. The segment of circle in between the two rays is a certain number of degrees out of 360, and this is an angle. In the second image, we have a sphere, with three rays rather than two extending from the center and carving a patch out of the surface area. And as you can see, the patch it carves is in the shape of a triangle. And, since we know how to deal with the angles of triangles, this is cause for rejoice. So far, so intuitive. However, a problem quickly arises. Look below.

In this image, the 3D angle is more cone-y, and not so neat. Notice that when we looked at our 2D angle, the patch the rays carved out of our circle was a 1D shape (i.e., a line). And now, looking at our 3D angle, the patches cut out of the sphere's surface by the rays will all be 2D polygons. Because of this, finding a way to talk about and visualize these angles is a daunting task.

Now, Vikram and I faced up against this problem, but immediately wasted a good amount of our name doddling about with words and descriptions, rather than getting to the point of mathematical identity. We kept trying to express the 3D angle in terms of degrees (out of 3600, perhaps), in some attempt to tie it back to familiar 2D territory. However, the clear folly in this is that the number 360 is fully arbitrary. The patch of circle is 1/4 of the whole, the patch of sphere is 2/7 of the whole, and any further description just confuses the matter.

So, we wound up without having made any progress on the language front. And since that is where our research stopped, it seems right to end my report here as well. However, I am fully aware that we haven't even scratched the surface of these ideas; I'm particularly interested in the relationship between areas and angles. This has simply been an overview of the thoughts we had that day, and of how we came to better understand what the word 'angle' means. And so... The End.