Visualization to illustrate the crazy fact that a random walk on the integer lattice in dimension d <= 2 will return to its starting point with probability 1 whereas a random walk in dimension d >= 3 has a finite probability of never returning. Or in other words:
"A drunk man will find his way home, but a drunk bird may get lost forever." - Shizuo Kakutani
Click here for a random walk in 3D.
For a cool proof of this theorem using Fourier analysis check e.g. the Fourier Transform and its Applications.