Footsteps of pi

See also &quot;π plays Pokemon Sapphire&quot;, currently at 372 24-hour segments, and still stuck in the starter town (with a level 76 Sceptile):<p><a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=pegjULYJae4">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=pegjULYJae4</a>

Funny, I did a similar exercise in 2019, <a href="https:&#x2F;&#x2F;earth.hoyd.net&#x2F;visualizing-100k-decimals-of-pi-and-tau-2913&#x2F;" rel="nofollow noreferrer">https:&#x2F;&#x2F;earth.hoyd.net&#x2F;visualizing-100k-decimals-of-pi-and-t...</a> loved how it turned out. Even planned to make it into an art piece on the wall.

I always wondered if some hidden pattern would be exposed when visualising numbers in unconventional ways in numbers with no known pattern such as Pi or prime numbers. A sort of multi-dimensional rendering that suddenly reveals a hidden pattern.

&gt; The colors are arbitrary, and have no deeper meaning<p>I thought that colouring the pattern by the instantaneous velocity of the ball would be an obvious improvement and might uncover further structure.

Would be nice to make not just a decimal direction picture, but one for other bases as well. Binary won&#x27;t work as you just move back and forth along a single line, but ternary should work, and as the minimal base for 2D directions, is less arbitrary than decimal. Then I&#x27;d look at bases 4,5,6,7, and octal as well to see whether the picture depends more on the number or on the base.<p>Another choice is whether to use absolute directions, or relative to the current direction, as in Logo.

Also If you do this for the Square Roots of the integers you can see every integer root is special and has it&#x27;s own kind of shape. And the Squares are also very interesting in that they have no shape in this viewpoint. Just a dot. So you go from infinitesimal chaotic walk patterns to a single dot depending on if the integer is a square or not.<p>maybe there could be a database, online encyclopedia of random-walks

&gt; Here are some more irrational numbers expressed in this way<p>Rather, rational numbers awfully close (in ordinary human terms) to specific, well known irrational numbers. There are, I think, just as many irrational numbers comparably close to any rational number.

I saw one that was similar except that it used triplets of pi digits (3.xyzxyzxyz..) and drew line segments with direction (x,y) and magnitude z.

I wonder if the path stays consistent in other (maybe very high) number bases, or if that general path is random and unique to base-10

Is this not just generating a random walk from a pseudorandom number generator.

a random walk over the 10th roots of unity

&gt; I originally created this image in early 2020 to impress and woo my now-girlfriend, who I adore.<p>Dating gurus hate him for this one weird trick.