Oh boy. What can I say? I've been away a couple of weeks, so sorry not to respond.
I've put some thought into this and will post more fully next week. I came up with a series of numbers of triangles produced at each iteration up to about 10th. I can work out an algorithm for the number of triangles at the k-th iteration given the number at the k-1 th iteration; that is, I can't tell you the number at iteration N.
The only obvious side to this problem is that the perimeter's infinite. Heh.
Guys, I have to confess my status here. I'm a high school level (though not age..) mathematician. For all I know I could have posted a problem that's a classic with an easy answer. I don't know what an eigenvalue is. I don't know whether I'll have any useful insights.
Lastly, I hope I haven't contravened etiquette - is it OK to post problems when you don't have much of a solution yourself?
I hope the puzzle gave you some pleasure, anyhow...!