You might like some of the papers about random traversals of this graph, e.g. http://front.math.ucdavis.edu/0302.5257 and http://front.math.ucdavis.edu/1601.06391

FAQ writer here. I've got a video with demos https://www.youtube.com/watch?v=38rf9FLhl-8&t=105s from a few years ago. AMA.

BTW I'm fairly sure I wrote the siteswap -> on-screen simulation, in 1988. In Pascal. A few years later three of us wrote http://www.juggling.org/programs/java/MAGNUS/juggler.html in now-ancient Java, which did a ton more stuff.