The study of Topology begins with such a simple formula that you can explain it to a 3rd grader ... make that a Kindergartner and maybe an astute pre-schooler. It's this:
V - E + F = 2
V = The number vertices, that is to say: corners, of a polyhedron (and not just a regular one).
E = The number of edges
F = The number of faces
That was discovered by the incredible Leonhard Euler of Switzerland (and Russia and Prussia and Russia again) in 1750, the proof of which he published one year later.
It is considered the 2nd most beautiful equation in the entire history of Mathematics.
But wait, it gets better, and points out the beauty of a single equation being a special case of a more general equation, an example being in Physics the Special Theory of Relativity (acceleration being zero) being a special case of the General Theory of Relativity (acceleration included).
And remember folks, if there's one thing Mathematicians love, it's backing up, abstracting if you will, and finding a general case where "a cool thing" is but a special case. We don't have our cell phones or this computer you're using if they didn't.
Here's the general case, and before I state it look at the picture above of the American Pentagon building, which is in the shape of a polyhedron with a hole in it.
V - E + F = 2 - 2R
V, E, and F are as previously stated
R = The number of Holes.
So for one hole as in the case of the U.S. Pentagon building we have:
V - E + F = 2 - 2R = 2 - 2(1) = 0
V - E + F = 0
This is always true for a one hole-y polyhedron, and what amazes is that this simple formula was in the grasp of the Ancients, but nobody "got it" until Euler in 1750.
Thus begins our study of Topology.
This and more is clearly stated and explained in the 4-page "Topology" section of Tony Crilly's "50 Mathematical Ideas You Really Need To Know", and for deeper study of this important subject, read "Euler's Gem" by Dave Richeson. After that read the relevant sections of "Mathematics 1001" then "The Princeton Companion to Mathematics" and the world should be your oyster from there on out.
I'm currently exploring all of those, and will get back when I know more.
Oh, and if you wish to know the MOST beautiful equation, it is also by Euler, the one we've previously talked about that unites the numbers 0, 1, pi, i, and e, which is:
|Mathematician's wedding ring, if you're poor and can only afford pewter. What's the difference between a Mathematician and a large pizza? A large pizza can feed a family of four. Euler was an exception (there's always them)|
UPDATE: So where is all this leading? Well, it will lead many places. To the early 20th century for starters, when Topology took off, and to the present time with Grigori Perelman's proof of The Poincare Conjecture, and Calabi-Yau manifold theory which is SO important in String Theory, and the world's most excellent rap video, which is this, by Mathematics Professor Steve Sawin, moonlighting as rapper "Slim Dorky", here:
Thanks to Dave Richeson of Division by Zero weblog for turning me on to that. Honestly, how can ANYone be depressed when just THINKing of that video is a drug-free instantaneous cure.
People are funny. :-)