Everyone is acquainted with record-calculations of pi and how their (base ten) digits are equally distributed. Closely, but not of course exactly. By contrast, in base two there are seemingly an infinite number of initial digits of pi that contain exactly the same number of zeros and ones: https://oeis.org/A039624 What a difference adding one digit/dimension can make. In base three, the random walk of digits allows the initial 15 and 18 digits of pi to be exactly equidistributed by sheer small-numbers chance [in base four, the first 4 digits (3.021); in base five, the first 75 digits]. I thought I had a shot at finding a third solution in base three but it was not to be: http://gladhoboexpress.blogspot.ca/2016/11/a-deep-walk-in-base-three-pi.html