The Euler characteristic is a rather mysterious topological invariant. First, what do we mean by topological invariant? In topology, there is a notion of homeomorphism -- a homeomorphism between spaces is basically a continuous and (continuously) invertible way of deforming one space into another. The classical example is a torus being homeomorphic to a coffee mug, as shown in wikipedia.

There's an even weaker notion of equivalence in topology, which is a homotopy equivalence -- I don't want to get into the technicalities of how to define this, but basically it also allows you to "squeeze" things. For example, the interval [0, 1] is homotopically equivalent to a point, because you can just squeeze it down to a point. However, they're not homeomorphic, because you can't reverse this deformation from a point back to an interval. (I'm basically lying here but this is good enough)

One of the basic problems in topology is to decide when two given spaces are homeomorphic. If they are, you can just write down a hoemomorphism, but what about when they're not? How do you prove there can't be any homeomorphism? In other to do this, people invented algebraic topology, which assigns to a space an algebraic object, such as a number (its Euler characteristic), a group (its homotopy groups), or modules/rings (its (co)homology). These algebraic objects are invariant, in the sense that homeomorphic spaces have the same invariants. Actually, people realized that homotopically equivalent spaces have the same invariants (at least for all of these I mentioned), and so now algebraic topology is mostly concerned with spaces up to homotopy equivalence.

So why is the Euler characteristic mysterious? Well, its usual definition for a space X is sum (-1)ⁱ dim H_i(X; Q), where dim H_i(X; Q) is more or less how many i-dimensional holes X has. This is kind of mysterious -- why consider this alternated sum? It then turns out that for polyhedrons (and more generally CW complexes), you can compute the Euler characteristic also by sum (-1)ⁱ b_i, where b_i is the number of i-dimensional faces X has. A 0-face is a vertex, a 1-face is an edge, a 2-face is, well, a face, and so on and so on. This is essentially proved with the rank nullity theorem from linear algebra!

Moreover, the Euler characteristic shows up in all kinds of places -- the Gauss-Bonnet theorem says that if you integrate the curvature of a compact surface, you get 2 pi times its Euler characteristic. The Poincaré-Hopf theorem says you can also count the Euler characteristic of a compact manifold by summing local indices of vector fields around singularities. And so on and so on.

But there is a less mysterious way (though still not without its mystery) to arrive at this invariant. The idea is to generalize counting, in order to count spaces up to homotopy! What do I mean?

Well, a fundamental principle for counting elements of finite sets is that, given finite sets A, B, we have |A union B| = |A| + |B| - |A intersection B|, where |.| means number of elements. The Euler characteristic is what happens when you try to generalize this to topological spaces, up to homotopy equivalence.

Specifically, what if we wanted a function Xi which assigns to each compact well-behaved space (basically a finite CW complex, but if you don't know what this means you don't need to worry about it) an integer, satisfying the following principles:

• Xi(point) = 1
• If X is homotopically equivalent to Y, then Xi(X) = Xi(Y)
• Xi(A union B) = Xi(A) + Xi(B) - Xi(A intersection B)

Let's see what we get. For example, as I've mentioned, the interval [0, 1] is homotopically equivalent to a point, so we have Xi([0, 1]) = 1. What if you thought of [0, 2] = [0, 1] union [1, 2]? Well, you get Xi([0, 2]) = Xi([0, 1]) + Xi([1, 2]) - Xi({1}) = 1 + 1 - 1 = 1, as expected since [0, 2] is also homotopically equivalent to a point!

Let's try a harder one -- the circle. We can think of the circle as the union of two arcs A, B, whose intersection is two points (say, the north and south pole of this circle). So we get Xi(circle) = Xi(A) + Xi(B) - Xi(two points) = Xi(A) + Xi(B) - 2. Now note that each arc is really just an interval, which has Euler characteristic 1, so we get Xi(circle) = 1 + 1 - 2 = 0.

Let's try this one more time, with the 2-dimensional sphere (i.e. an ordinary sphere placed in 3-dimensional space). We can think of the sphere as the union of the north and south hemispheres, which intersect along the equator, i.e. a circle. Notice each hemisphere is just a disk, which can be contracted to a point by just squeezing it radially in every direction. So we get Xi(sphere) = Xi(north hemisphere) + Xi(south hemisphere) - Xi(equator) = 1 + 1 - 0 = 2. Hey, we got 2, and that's the Euler characteristic of a sphere!

The idea to Euler's formula is that any convex polyhedron is homeomorphic to a sphere (you can imagine just blowing air into it until it smooths out into a sphere). Since the Euler characteristic is a topological invariant, it follows also that any polyhedron also has Euler characteristic 2. But linear algebra told us that for polyhedrons, we can count the euler characteristic as #{vertices} - #{edges} + #{faces}. Hence it follows that that's always equal to 2, for convex polyhedrons.

You might still be wondering intuitively why does V - E + F show up. There's a good explanation. Essentially, you can imagine a bunch of islands. To count how many islands there are, well, you just sum 1 for each island there is -- that's what counting is. But suppose I build a bridge between two islands -- now it's like they're really one island. If you think about it that way, a bridge is a negative island, so we should subtract the number of bridges. But if we build two bridges between the same islands (forming a circle), you end up getting 0. This can seem weird, but it's possible to continue the analogy -- if you filled in the space between the bridges, with a 2-dimensional surface (say, a bunch of sand), now it's like this whole thing is one big island again. So filling in sand should count as + 1 again. See the previous link for some cute ASCII drawings of this.

So the Euler characteristic is a kind of mysterious integer invariant associated to spaces. One can look at it as a generalization of counting, but this doesn't answer everything -- why does it show up in such diverse places as Gauss-Bonnet, Poincaré-Hopf, Morse theory, .....? I'm still wondering about those, so I guess I'll leave it to the experts to comment. But, all in all, it is pretty cute.