Press [space] to add a dot, or 't' to add ten dots, or 'c' to clear. (You might need to "click" on the applet first.) Do you see any clumping?
In this applet, each dot is placed uniformly at random in the applet area, independently of all the other dots.
However, if you add LOTS of dots (by holding [space] for a while), then just by chance, various "patterns" will seem to appear. This phenomenon is related to "Poisson clumping", see e.g. Chapter 2 of this book.