I'm new here so sorry if I use improper format and whatnot.

So basically I was just surfing the web and saw a post about calculating the average distance between any 2 points in a triangle, and was wondering whether it would be possible to do something like this but for various iterations of the Sierpinski Triangle (unit sides). For example, what is the average distance between 2 points in the second iteration of the Sierpinski triangle? Problem is I have no idea how to start.


  • $\begingroup$ This is an interesting question. I guess that to make this formally correct you'd have to take some care defining a probability measure which does not depend on a concept of area which fails for such a fractal. But I wouldn't be surprised if most “sane” definitions of such a measure would actually be equivalent. When you speak about iterations, what iterative procesture do you have in mind? Taking triangle and cutting out smaller triangles? $\endgroup$ – MvG Sep 19 '17 at 11:10
  • $\begingroup$ Numeric experiments suggest a mean distance of approximately $0.422679101$, based on $2^{35}$ pairs of points generated using the chaos game method. $\endgroup$ – MvG Sep 19 '17 at 14:32
  • $\begingroup$ How are you measuring distance? $\endgroup$ – Xander Henderson Sep 21 '17 at 2:04

The average distance between two independent uniformly distributed points (defined in the natural way) on the Sierpinski triangle (with unit side) is 466/885. This is for the internal distance, so the path between the points must be entirely within the Sierpinski triangle. The result is in the paper The Tower of Hanoi by Andreas Hinz. It applies only to the actual Sierpinski triangle, not to the successive stages of the construction.


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