Visualizing Balls in Ultrametric Spaces I've been reading about ultrametric spaces, and I find that a lot of the results about balls in ultrametric spaces are very counter-intuitive. For example: if two balls share a common point, then one of the balls is contained in the other.
The reason I find these results so counter-intuitive is that I can easily picture "counter-examples," the problem being that these "counter-examples" are balls in Euclidean space. 
My issue is not that I cannot prove these results. My issue is that I don't know how to think about/picture balls in ultrametric spaces, which makes it more difficult for me to actually come up with the proofs. Hence, does anyone have any hints as to how to think about/picture balls in ultrametric spaces?

Think of the Cantor set and its basic closed-and-open intervals, pictured below. 
Note that for any two such intervals, they either do not intersect, or one is contained in the other. 

