There is a gif of this at http://en.wikipedia.org/wiki/Wiener_process#Self-similarity http://en.wikipedia.org/wiki/Wiener_process#Self-similarity which is rat

There is a gif of this at http://en.wikipedia.org/wiki/Wiener_process#Self-similarity http://en.wikipedia.org/wiki/Wiener_process#Self-similarity which is rather nice. If you get some infrastructure for making Brownian motion sample paths, another good one is that it crosses zero infinitely often in the neighborhood of the origin. Keep zooming in, and see more and more crossings. This is another instance of the scaling rule you mention, but it is neat to see.