I'd gotten inspired by @mohitbhoite and noticed that the hackaday circuit sculpture contest was going on. So this is the results of my first attempt!
Putting the ring together turned out better than I expected, despite not having a jig or something similar to ensure the LEDs were all aligned.
The trickiest part was figuring out how to attach the BJTs, given they've got three pins and needed to fit together a little bit.
The code simply turns on and off the transistors. The outermost ring is +5v, and the inner three act as switchable grounds, when the transistor is 'on', the LEDs have a path to ground and so they turn on. There are 12 LEDs total, 4 for each 'rail'. This creates a sort of marqee effect as we've seen in signs around old style theaters and so on.