Heh wrote about this (on my blog) a long, long ago. Was a fun numerical experiment to break the boringness of my own numerical experiments with fractals
I found your explanation to be much clearer than this one, as I'm still having trouble seeing why the uniform density is that of a poisson RV.
Thanks, but I just redacted for clarity a solution I found elsewhere. At the time I was teaching at a university, and redacting for clarity was ingrained :)