TITLE: geometric argument for van-kampen?
QUESTION [10 upvotes]: I've seen Van-Kampen's theorem presented algebraically many times; and although it provides a useful method of calculation; I don't have a very clear picture for "why" it should be true.  Does anyone know of a more visual argument; or even an example that makes the it easier to see?

REPLY [5 votes]: I will say it less acurately than Arturo and sure enough too much roughly. Consider it as a first approxiation to the theorem.
The way I see Seifert-Van Kampen theorem is like a sophisticated version of the dimension of the sum of vector subspaces:
$$
\mathrm{dim} (F + G) = \mathrm{dim} F + \mathrm{dim} G - \mathrm{dim} (F\cap G) \ .
$$
Because, forgetting torsion, the fundamental group counts the number of "holes" (non contractible loops, generators of $\pi_1$) in your space $X$. Right? 
So, if you have $X = U \cup V$, then the number of holes in $X$ must be equal to the number of holes in $U$ plus the number of holes in $V$..., minus the number of holes in $U\cap V$, because these ones you have already counted them twice.
Hence, in order to count all the holes in $U$ plus those in $V$, you take the free product $\pi_1 (U) * \pi_1(V)$, but you must "identify" the holes "shared" by $U$ and $V$. This is the reason for the amalgamated product $\pi_1(U) *_{\pi_1(U\cap V)} \pi_1(V)$.