Fractal curves area I was reading about the Sierpinski curve on the wiki page and it says that, considering the sequence of Sierpinski curves $S_n$ such that $\lim\limits_{n\rightarrow \infty}S_n$ completely fills the unit square,  the limit of the area enclosed by those curves is of $\frac{5}{12}$. 
I cannot think how it is possible that the limit of the areas enclosed by curves which are converging to a curve which fills the unit square can be different from $1$

When we say that 
$$\lim_{n\to\infty} S_n = I^2 \text{ (the unit square)},$$
we are referring to a particular notion of distance. In particular, the set $S$ is "close" to the set $T$ if every point of $S$ is close to some point of $T$ and vice-versa. (This is really an intuitive description of the Hausdorff metric.)
The thing is, this notion of distance simply has nothing to do with area. It's quite possible that two sets are close to one another (with respect to this distance), yet have areas that are not close. Once you see a few simple examples, this makes perfect sense. One simple example, is a sequence of finer and finer checkerboard patterns:

Each of these images lies snugly inside the unit square. If we consider the set $S_n$ to be the region shown in black in the $n^{\text{th}}$ checkerboard pattern, then $S_n\to I^2$, yet the area of $S_n$ is 1/2 for all $n$.
For that matter, we could let $S_n$ denote the set of vertices in the $n^{\text{th}}$ checkerboard pattern. Then each $S_n$ is a finite set (so it has area zero), yet $S_n\to I^2$.
