Euclidean geometry – Which solid deals with the truncation of an icosahedron?

I was looking at fun math stuff on youtube, and I came across this video: https://www.youtube.com/watch?v=cwWBpjeyRS0 where the guy mentions how a balloon is obtained by truncating an icosahedron in a certain, so I wondered: what happens if you repeat the process? I mean, if you truncate the form of "football" he's talking about, you'll get an object with a lot of triangular faces, but what if you keep doing it? Does it approach a sphere? Does it become "smooth" at some point?