hausdorff dimension of PI ???!?!

[hobold]

I recall that there was some measure of "complexity" of any real number, based on continued fractions (http://en.wikipedia.org/wiki/Continued_fraction). The golden ratio was the most complicated number in that sense. I don't know where pi lands on that particular scale.

[kram1032]

You can define a Hausdorff Dimension on anything that has a (mathematically sound) metric by simply replacing the typical euclidean norm in its definition by the metric variant. So by finding some kind of distance measure - e.g. metric - that works on digits of numbers, no matter how arbitrary it is, you can just apply typical box-counting to arrive at an approximate Hausdorff Dimension - the box-counting dimension with respect to the chosen metric - of pi.

Finding the *actual* Hausdorff Dimension would be quite involved, I guess, possibly involving solving wether pi is normal, e.g. wether all digits occur equally often. That's a long standing open question.
Boxcounting could be done relatively quickly on the computer.