How can one find shortest paths between 2 specified points on fractals, or (since I'm pretty sure this is quite complicated) make useful generalizations about them?

Since the above question is broad, how about this one: What is the general formulation (in a direct equation, recursive formulation, or other form) for distance between 2 points on the sierpinski carpet?

Obviously for some fractals all points are infinite. Identifying these is often easy, but are there any edge cases where it's hard to decide whether all paths are infinite length? And if so, how does one decide?

Edit: This question was inspired, by the way, by this thread on a different website (where it became clear that it was beyond the average math knowledge there). http://echochamber.me/viewtopic.php?f=3&t=40348#p1618494 That particular post shows paths(whose presence is recursive) in the carpet.