If I understand it correctly, one can use $$ -\sum_i^N x_i \ln (x_i) $$ to estimate the entropy a set of values $x_i$. Does the domain matter? Is $x_i \in (0,1)$ the default domain? Anyway, why the above formula can measure the entropy? the wikipedia does not explain.
In physics, the entropy of particles in a box (or a rectangle) is not difficult to conceive. However, how to calcute the entropy of a set of 2d points $\mathbf{x}_i \in (0,1)^2 $ ?