Friday, December 30, 2011

Merry Christmas

Did Santa disappoint this year? Just how good do you have to be in order to get what you've always wanted? Is a generalization of the Law of Cosines applicable to pentagons really too much to ask for? Well, whether you wanted it or not, here it is. I doodled it today during my morning coffee.

As you'll recall, the Law of Cosines gives one side of a triangle in terms of the other two sides and the opposite angle. (See Wikipedia here, and why not give them five bucks while you're at it?) So as I sat down with my coffee, I decided there ought to be a Law of Cosines that gives the unknown side of a pentagon in terms of the other four sides and the "opposite angles" (i.e., the three angles of which the unknown side is not a part).
First I treated u, v, w and x as vectors with appropriate orientations; then I computed t-squared as the inner product of u + v + w + x with itself. This led easily to the desired relation, which has some nice rhythms in it pointing to the general case (click to enlarge):
Here is an example problem:
I find t = 1.3826 or so. See how useful the formula is? I don't know how I ever got by in theoretical physics without it.

 As an extra stocking stuffer, the simpler Law for quadrilaterals:
Note, these formulas also work for non-simple polygons (closed polygonal chains with self-intersections; see Wikipedia here and you'll have another chance to give them five bucks).

By the way, I donated $10 to Wikipedia myself today. Merry Christmas, Jimmy Wales!