(If you think that it should be a separeted post, tell me)
Nice post! I remembered this picture that I saw these days in the internet with 10.000 digits.


Trying to recriate it in Mathematica I did this:And get as result this:

Now I tried to use BSplineCurve to get curved lines, but I lost the colors blend between the numbers connections.


Some clue on how can I have color blend on that? Con you improve it?
I also liked this 3D version that look like a cabana.

Neat!
I made a 3D walk by looking at the digits in base 6, here is a quick look at the first 10k steps:


To create the gif I genereted the points in the walk and then interpolated a path between every thousand points. The camera was set to move along the path while looking at a point ahead.

I'm playing with the original graphic: more terms, white background, different color gradients. The generation of the graphic is very memory intensive. If I try with more than 3*10^5 steps, Mathematica invariably crashes on my 32GB system. How might one insert plainly visible, regularly spaced (from initial to final) points?

http://two-n.com/pi shows a different representation of the digits of pi as colored dots.
Note: If you click your mouse in the graphic and drag up or down you can see more digits.

You can see diagonal "edges" or ripples in the images that extend across hundreds of points.
I do realize this is just the eye and brain constructing patterns out of noise.

I wrestled with trying to make the graphic you requested... and failed. I believe some or most of my problem was not being able to exactly reproduce the color palette they used or there may be something else in ArrayPlot makes the result appear different enough to ruin this.

If you go to http://two-n.com/pi  and click on the yellow text to get rid of that and you look from the left 1/4 the width of the box and 1/2 the height of the box from the top then at least my eyes see a light colored diagonal "river" going from the lower left to the upper right and perhaps 3 cm in length. Less than 1 cm above that there is a similar 2cm light colored river. 1/3 of the way from the left and 2/3 of the way from the top I see a steep negative slope river perhaps 4 cm long. At the top of that there is an intersecting 2 cm slightly positive slope river.

Squint your eyes a bit and look for lines or gradual arcs that are at least several cm (on my monitor) where there is a more or less uniform shift in color or brightness as you step across that line or arc.

I did have one idea if it were possible to squeeze out even an extremely tiny advantage by being able to identify and make use of even a few of these rivers.

To be clear to everyone, I have no delusions that I have discovered some deep pattern in the digits of pi which has eluded the brightest mathematical minds for centuries. 

Bill, Not sure, but it could be that these rivers are higher densities of some light color.  Without the tooltips, it is something like this.
For comparison, here it is next to random numbers.  I think I see your rivers in both, but it seems like there are some qualitative differences which might be quantifiable.