If you press the button above, you will get a window and see a pond for water lilies. The mathematical model consists of a number of randomly distributed points in the plane, which begin to grow circles at equal rate. The growth of each circle stops, when The circle touches another circle. The above applet simulates this growth. If you press the animation button, you can view the growth.

It is unknown, which percentage of the plane is expected to be covered this way. If the circles stop growing half way to their nearest neighbors, the expected coverage would be 1/4 (1/2^d in d dimensions). But the circles can grow larger. So this is lower limit for the expected coverage.