It is interesting for me to go back and see what I wrote here several years ago when I was just starting this blog; I notice that I treated it as more of a personal notebook, and less of a platform for what the NSF would characterize as “outreach activity” (incidentally, there was a “part 2″ to this post, which is http://lamington.wordpress.com/2009/06/01/groups-with-free-subgroups-part-2/ ). Curious to see that I am still preoccupied with Gromov’s surface subgroup question in one way or another (eg see http://lamington.wordpress.com/2012/11/04/surface-subgroups-of-sapirs-group/ for a recent example)

The graph you picked above is not optimal in this sense; if you delete edge 8 (set the resistance to infinity) and contract edge 2 (set the resistance to 0), you get another graph that can realise the same set of shapes with an appropriate choice of resistances (i.e., tiling with rectangles of fixed aspect ratio rather than squares). That graph is much simpler, and maybe this explains why the aspect ratio doesn’t seem to be changing (much).

Or maybe you’re thinking about the torus case. Do you have an easy proof in that case?

It doesn’t seem like the aspect ratio of the overall torus changes (very much) as you rotate in the family you picked. Did you do anything to make this happen?

Thurston touches on many different issues in his article. One of them is the question of what a proof in mathematics is supposed to accomplish. Thurston makes the point that one of the functions of a proof is to achieve insight, and different kinds of proofs or ways of thinking about mathematical objects can convey different *kinds* of insights. I am sorry that the W-Z proof does not communicate an insight about mathematics to you different in kind from that communicated by the classic geometric proof; that’s too bad. Perhaps if you find it so unpleasant to contemplate you should just ignore it?

On the other hand, in the classic geometric proof, it’s sufficient to draw only one example triangle, it’s not error prone because it doesn’t require an accurate construction, it uses simple principles available to an unsophisticated mathematical mind, and the existence of the incircle about the incenter is a trivial deduction. Also requiring proof, certainly: but much cheaper in terms of effort than the W-Z horror.

But acceptable? Only if it’s used to illustrate the difference between ugly and elegant. I would only use it comparatively, to show what can be lost or gained; otherwise, not at all.